11th Applied Inverse Problems Conference
September 4 - 8, 2023 | Göttingen, Germany
Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
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Session Overview |
| Date: Monday, 04/Sept/2023 | |
| 8:00am - 9:00am | Reg: Registration Location: ZHG Foyer |
| 9:00am - 9:50am | Opening: President of the University of Göttingen Prof. Metin Tolan Location: ZHG 011 Session Chair: Thorsten Hohage |
| 9:50am - 10:40am | Pl 1: Plenary lecture Location: ZHG 011 Session Chair: Simon Robert Arridge |
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On the sample complexity of inverse problems University of Genoa, Italy
Many inverse problems are modeled by integral or partial differential equations, including, for instance, the inversion of the Radon transform in computed tomography and the Calderón problem in electrical impedance tomography. As such, these inverse problems are intrinsically infinite dimensional and, in theory, require infinitely many measurements for the reconstruction. In this talk, I will discuss recovery guarantees with finite measurements, and with explicit estimates on the sample complexity, namely, on the number of measurements. These results use methods of sampling theory and compressed sensing, and work under the assumption that the unknown either belongs to a finite-dimensional subspace/submanifold or enjoys sparsity properties. I will consider both linear problems, such as the sparse Radon transform, and nonlinear problems, such as the Calderón problem and inverse scattering.
A similar issue arises when applying machine learning methods for solving inverse problems, for instance, to learn the regularizer, which may depend on infinitely many parameters. I will present sample complexity results on the size of the training set, both in the case of generalized Tychonov regularization, and with $\ell^1$-type penalties.
This talk is based on a series of joint works with Á. Arroyo, P. Campodonico, E. De Vito, A. Felisi, T. Helin, M. Lassas, L. Ratti, M. Santacesaria, S. Sciutto and S. I. Trapasso.
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| 10:40am - 11:10am | C1: Coffee Break Location: ZHG Foyer |
| 11:10am - 12:00pm | Pl 2: Plenary lecture Location: ZHG 011 Session Chair: Rainer Kress |
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An unexpected role of transmission eigenvalues in imaging algorithms INRIA, France
Transmission eigenvalues are frequencies related to resonances inside scatterers and by duality to non-scattering for an incident field being an associated eigenvector. Appearing naturally in the study of inverse scattering problems for inhomogeneous media, the associated spectral problem has a deceptively simple formulation but presents a puzzling mathematical structure, in particular it is a non-self-adjoint eigenvalue problem. It triggered a rich literature with a variety of theoretical results on the structure of the spectrum and also on applications for uniqueness results [1].
For inverse shape problems, these special frequencies were first considered as bad values (for some imaging algorithms, e.g., sampling methods) as they are associated with non-injectivity of the measurement operator. It later turned out, as proposed in [2], that transmission eigenvalues can be used in the design of an imaging algorithm capable of revealing density of cracks in highly fractured domains, thus exceeding the capabilities of traditional approaches to address this problem.
This new imaging concept has been further developed to produce average properties of highly heterogeneous scattering media at a fixed frequency (not necessarily a transmission eigenvalue) by encoding a special spectral parameter in the background that acts as transmission eigenvalues [3].
While targeting this unexpected additional value of transmission eigenvalues in imaging algorithms, the talk will also provide an opportunity to highlight some key results and open problems related to this active research area.
[1] F. Cakoni, D. Colton, H. Haddar. Inverse Scattering Theory and Transmission Eigenvalues, CBMS-NSF, 98, 2022.
[2] L. Audibert, L. Chesnel, H. Haddar, K. Napal. Qualitative indicator functions for imaging crack networks using acoustic waves. SIAM Journal on Scientific Computing, 2021.
[3] L. Audibert, H. Haddar, F. Pourre. Reconstruction of average indicators for highly heteregenous scatterers. Preprint, 2023.
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| 12:00pm - 1:30pm | LB1: Lunch Break Location: Mensa |
| 1:30pm - 3:30pm | MS03 1: Compressed Sensing meets Statistical Inverse Learning Location: VG2.103 Session Chair: Tatiana Alessandra Bubba Session Chair: Luca Ratti Session Chair: Matteo Santacesaria |
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Compressed sensing for the sparse Radon transform 1University of Genoa, Italy; 2Politecnico di Torino, Italy
Compressed sensing allows for the recovery of sparse signals from few measurements, whose number is proportional to the sparsity of the unknown signal, up to logarithmic factors. The classical theory typically considers either random linear measurements or subsampled isometries and has found many applications, including accelerated magnetic resonance imaging, which is modeled by the subsampled Fourier transform. In our work, we develop a general theory of infinite-dimensional compressed sensing for abstract inverse problems, possibly ill-posed, involving an arbitrary forward operator. This is achieved by considering a generalized restricted isometry property, and a quasi-diagonalization property of the forward map.
As a notable application, we obtain rigorous recovery estimates for the sparse Radon transform (i.e., with a finite number of angles $\theta_1,\dots,\theta_m$), which models computed tomography. In the case when the unknown signal is $s$-sparse with respect to an orthonormal basis of compactly supported wavelets, we prove exact recovery under the condition $m\gtrsim s,$ up to logarithmic factors.
[1] G. S. Alberti, A. Felisi, M. Santacesaria, I. Trapasso. Compressed sensing for
inverse problems and the sample complexity of the sparse Radon transform, ArXiv
e-prints, arXiv:2302.03577, 2023.
Regularization for learning from unlabeled data using related labeled data Austrian Academy of Sciences, Austria
We consider the problem of learning from unlabeled target datasets using related labeled source datasets, e.g. learning from an image-dataset from a target medical patient using expert-annotated datasets from related source patients. This problem is complicated by (a) missing target labels, e.g. no target expert-annotations of a tumor, and, (b) possible differences in the source and target data generating distributions, e.g. caused by medical patents’ human variations. The major three methods for this problem, are special cases of multiple or cascade regularization methods, i.e., methods involving simultaneously more than one regularization. This talk is based on [1-3] and reviews non-asymptotic (w.r.t. dataset size) error bounds of the major three methods.
[1] W. Zellinger, N. Shepeleva, M.-C. Dinu, H. Eghbal-zadeh, H. D. Nguyen, B. Nessler, S. V. Pereverzyev, B. Moser. The balancing principle for parameter choice in distance-regularized domain adaptation. Advances in Neural Information Processing Systems (NeurIPS). 34: 20798--20811, 2021.
[2] E.R. Gizewski, L. Mayer, B. Moser, D.H. Nguyen, S. Pereverzyev Jr, S.V. Pereverzyev, N. Shepeleva, and W. Zellinger. On a regularization of unsupervised domain adaptation in RKHS. Appl. Comput. Harmon. Anal. 57: 201--227. 2022. https://doi.org/10.1016/j.acha.2021.12.002
[3] M. Holzleitner, S.V. Pereverzyev, W. Zellinger. Domain Generalization by Functional Regression. arXiv preprint arXiv:2302.04724 (2023). https://doi.org/10.48550/arXiv.2302.04724
Random tree Besov priors for detail detection 1Delft University of Technology, Netherlands, The; 2University of Helsinki, Finland
Besov priors are well fitted for imaging since smooth functions with few local irregularities have a sparse expansion in the wavelet basis which is encouraged by the prior. The edge preservation of Besov priors can be enhanced by introducing a new random variable T that takes values in the space of ‘trees’, and which is chosen so that the realizations have jumps only on a small set. The density of the tree, and so the size of the set of jumps, is controlled by a hyperparameter. In this talk I will show how this hyperparameter can be optimized for the data and what the optimal values tell us about behaviour of the signal or image.
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| 1:30pm - 3:30pm | MS12 1: Fast optimization-based methods for inverse problems Location: VG2.102 Session Chair: Tuomo Valkonen |
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Parameter-Robust Preconditioning for Oseen Iteration Applied to Navier–Stokes Control Problems 1Scuola Normale Superiore, Pisa (Italy); 2The University of Edinburgh, Edinburgh (UK)
Optimal control problems with PDEs as constraints arise very often in scientific and industrial problems. Due to the difficulties arising in their numerical solution, researchers have put a great effort into devising robust solvers for this class of problems. An example of a highly challenging problem attracting significant attention is the (distributed) control of incompressible viscous fluid flow problems. In this case, the physics may be described, for very viscous flow, by the (linear) incompressible Stokes equations, or, in case the convection of the fluid plays a non-negligible role in the physics, by the (non-linear) incompressible Navier–Stokes equations. In particular, as the PDEs given in the constraints are non-linear, in order to obtain a solution of Navier–Stokes control problems one has to iteratively solve linearizations of the problems until a prescribed tolerance on the non-linear residual is achieved.
In this talk, we present novel, fast, and parameter-robust preconditioned iterative methods for the solution of the distributed time-dependent Navier–Stokes control problems with Crank-Nicolson discretization in time. The key ingredients of the solver are a saddle-point type approximation for the linear systems, an inner iteration for the $(1, 1)$-block accelerated by a generalization of the preconditioner for convection–diffusion control derived in [2], and an approximation to the Schur complement based on a potent commutator argument applied to an appropriate block matrix. The flexibility of the commutator argument, which is a generalization of the technique derived in [1], allows one to alternatively apply a backward Euler scheme in time, as well as to solve the stationary Navier–Stokes control problem. We show the effectiveness and robustness of our approach through a range of numerical experiments.
This talk is based on the work in [3].
[1] D. Kay, D. Loghin, A.J. Wathen. A Preconditioner for the Steady-State Navier–Stokes Equations, SIAM Journal on Scientific Computing 24: 237–256, 2002.
[2] S. Leveque, J.W. Pearson. Fast Iterative Solver for the Optimal Control of Time-Dependent PDEs with Crank–Nicolson Discretization in Time, Numerical Linear Algebra with Applications 29: e2419, 2022.
[3] S. Leveque, J.W. Pearson. Parameter-Robust Preconditioning for Oseen Iteration Applied to Stationary and Instationary Navier–Stokes Control, SIAM Journal on Scientific Computing 44: B694–B722, 2022.
Sparse Bayesian Inference with Regularized Gaussian Distributions Technical University of Denmark, Denmark
In this talk, we will present a method for Bayesian inference by implicitly defining a posterior distribution as the solution to a regularized linear least-squares problem with randomized data. This method combines Gaussian distributions with the deterministic effects of sparsity-inducing regularization like $l_1$ norms, total variation and/or constraints. The resulting posterior distributions assign positive probability to various low-dimensional subspaces and therefore promote sparsity. Samples from this distribution can be generated by repeatedly solving regularized linear least-squares problems with properly chosen data perturbations, thus, existing tools from optimization theory can be used for sampling. We will discuss some properties of the methodology and discuss an efficient algorithm for sampling from a Bayesian hierarchical model with sparsity structure.
An Accelerated Level-Set Method for Inverse Scattering Problems 1EDF R&D PRISME, 78400, Chatou, France; 2INRIA, Center of Saclay Ile de France and UMA, ENSTA Paris Tech, Palaiseau Cedex, FRANCE; 3School of Mathematical Sciences, Beihang University, Beijing, 100191, CHINA
We propose a rapid and robust iterative algorithm to solve inverse acoustic scattering problems formulated as a PDE constrained shape optimization problem. We use a level-set method to represent the obstacle geometry and propose a new scheme for updating the geometry based on an adaptation of accelerated gradient descent methods. The resulting algorithm aims at reducing the number of iterations and improving the accuracy of reconstructions. To cope with regularization issues, we propose a smoothing to the shape gradient using a single layer potential associated with $ik$ where $k$ is the wave number. Numerical experiments are given for several data types (full aperture, backscattering, phaseless, multiple frequencies) and show that our method outperforms a nonaccelerated approach in terms of convergence speed, accuracy, and sensitivity to initial guesses.
A first-order optimization method with simultaneous adaptive pde constraint solver 1University of Jyväskylä, Finland; 2University of Helsinki, Finland
We consider a pde-constrained optimization problem and based on the nonlinear primal dual proximal splitting method, a nonconvex generalization of the well-known Chambolle-Pock algorithm, we develop a new iterative algorithmic approach to the problem by splitting the inner problem of solving the pde in each step over the outer iterations. In our work we split our pde-problem in a fashion similar to the classical Gauss-Seidel and Jacobi methods, though other iterative schemes may be fruitful too. We show through numerical experiments that significant speed ups can be attained compared to a naive full pde-solve in each step, and we prove convergence under sufficients second-order growth conditions.
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| 1:30pm - 3:30pm | MS14 1: Inverse Modelling with Applications Location: VG1.104 Session Chair: Daniel Lesnic Session Chair: Karel Van Bockstal |
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Scanning biological tissues using thermal-waves University of Leeds, United Kingdom
Many materials in nature possess properties that are unknown and difficult to measure directly. In such a situation, inverse modelling offers a viable alternative where one is trying to infer those unknown properties from appropriate measurements of the main dependent variable(s) governing the physical process under investigation. Our investigation is driven by the fact that knowledge of the properties of biological tissues is essential in monitoring any abnormalities that may be forming and have a major impact on organs malfunctioning. Therefore, these disorders must be detected and treated early to save lives and improve the general health. Within the framework of thermal therapies, e.g. hyperthermia or cryoablation, the knowledge of the tissue temperature and its properties, as well as the blood perfusion rate are of utmost importance. Motivated by such a significant biomedical application, this study investigates the reconstruction of biological properties in the thermal-wave hyperbolic model of bio-heat transfer.
The support of the EPSRC grant EP/W000873/1 on “Transient Tomography for Defect Detection'' is acknowledged.
Identification of the time-dependent part of a heat source in thermoelasticity 1Ghent University, Belgium; 2University of Bucharest, Romania
The isotropic thermoelasticity system of type-III, describing the mechanical and thermal behaviours of a body occupying a bounded domain with a Lipschitz boundary, is considered. The displacement vector and either the normal heat flux or the temperature are prescribed on the boundary.
This talk deals with the theoretical and numerical reconstruction of a time-dependent heat source from the knowledge of an additional weighted integral measurement of the temperature in the framework mentioned above. Firstly, it is proved that the measurement type depends on the available thermal boundary condition, expressed by different conditions on the associated weight function. Secondly, for both thermal boundary conditions, the existence of a unique weak solution for exact data is proved, which is achieved by employing Rothe's method. This approach has the advantage of including a time-discrete numerical scheme for computations. Hence, for each of the two inverse source problems considered in this talk, a numerical algorithm that builds upon a decoupling technique is proposed, and the convergence of these numerical schemes is proved for exact data. Furthermore, the uniqueness of a solution is obtained by using an energy estimate. Finally, using the finite element method, the numerical results obtained for various numerical examples with noisy measurements are presented to validate the convergence and stability of the proposed algorithms. The noisy data are regularised using the nonlinear least-squares method; hence, they can be used as input for the proposed numerical scheme.
The results presented in this talk are published in [1].
[1] K. Van Bockstal, L. Marin. Finite element method for the reconstruction of a time-dependent heat source in isotropic thermoelasticity systems of type-III, Zeitschrift für angewandte Mathematik und Physik 73, 2022.
Uniqueness of determining a space-dependent source for inverse source problems in thermoelasticity Ghent University, Belgium
A thermoelastic system describes the interaction between the changes in the shape of an object $\mathbf{u}(\mathbf{x},t)$ and the fluctuation in the temperature $\theta(\mathbf{x},t)$. We consider an isotropic thermoelastic system of type-III which links the elastic and thermal behaviors of an isotropic material occupying a bounded domain $\Omega \subset \mathbb{R}^d$ with Lipschitz continuous boundary.
In this contribution, we will study and discuss uniqueness results for solutions to several inverse source problems (ISPs). Our main assumption is that either the heat source $h$ or load source $\mathbf{p}$ can be decomposed as a product of a given time-dependent and an unknown space-dependent function. The main goal is to find the spatial component given some measurement of the function(s) $\mathbf{u}(\mathbf{x},t)$ and/or $\theta(\mathbf{x},t).$
More specifically, the first ISP under consideration deals with the determination of the spatial component $\mathbf{f}(\mathbf{x})$ of the load source $\mathbf{p}(\mathbf{x},t) = g(t)\mathbf{f}(\mathbf{x})$ from the final in time measurement $\mathbf{u}(\mathbf{x},T),$ or from the time-average measurement $\int_0^T \mathbf{u}(\mathbf{x},t)\,\mathrm{d}t,$ where $T$ denotes the final time. The second ISP concerns finding $f(\mathbf{x})$ in the heat source $h(\mathbf{x},t) = g(t) f(\mathbf{x}) $ from the time-average measurement $\int_0^T \theta(\mathbf{x},t)\,\mathrm{d}t.$ The uniqueness results are formulated under suitable assumptions on the temporal component $g(t)$ and its derivative. Some examples will be provided showing the necessity of these (sign) conditions on $g.$ The results holds for (homogeneous) Dirichlet boundary conditions on $\mathbf{u}$ and $\theta$ as well as in the case a (homogeneous) Neumann boundary condition for $\theta$ is used. Finally, the in last ISP, we discuss the problem of finding both $\mathbf{f}$ and $f$ simultaneously when a combination of different measurements is available.
The presented work is based on joint work with Dr. Karel Van Bockstal [1].
[1] F. Maes, K. Van Bockstal. Uniqueness for inverse source problems of determining a space-dependent source in thermoeleastic systems, J. Inverse Ill-Posed Probl. 30(6): 845-856, 2022.
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| 1:30pm - 3:30pm | MS15 1: Experimental and Algorithmic Progress in Photoemission Orbital Imaging Location: VG1.102 Session Chair: Russell Luke Session Chair: Stefan Mathias |
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Imaging valence and excited states of fullerenes in momentum space 1University of Mainz, Germany; 2University of Kaiserslautern-Landau
One of the key milestones in advancing the performance of molecular electronic and photonic devices is to gain a comprehensive understanding of the electronic properties and rich excited state dynamics of this class of materials. In this context, momentum-resolved photoemission in combination with photoemission orbital tomography (POT) has been established as a powerful tool to study the band structure of molecular films and to reveal the degree of localization of molecular valence orbitals by their characteristic emission pattern in momentum space.
In this contribution, we exploit these capabilities of POT to study the valence and excited states of fullerenes grown on noble metal surfaces. For the most prototypical fullerene, the buckyball C$_{60}$, we will show that the valence states show signatures of an atomic crystal-like band structure with delocalized $\pi$- and localized $\sigma$-orbitals [1]. This observation differ significantly from our results for thin films of the endohedral fullerene Sc$_3$N@C$_{80}$, where the valence states are strongly localized on the carbon cage of the molecules.
Finally, we provide a first insight into the momentum space signatures of the excited state dynamics of C$_{60}$ thin films obtained by time-resolved two-photon momentum microscopy. For an optical excitation with $3.1\,$eV photons, we are able to identify three characteristic emission patterns even in the small momentum space range accessible by our experiment. These signatures are discussed in the context of the recently proposed charge transfer and Frenkel exciton character of these states [2].
[1] N. Haag, D. Lüftner, F. Haag, J. Seidel, L. Kelly, G. Zamborlini, M. Jugovac, V. Feyer, M. Aeschlimann, P. Puschnig, M. Cinchetti, B. Stadtmüller. Signatures of an atomic crystal in the band structure of a C60 thin film, Phys. Rev. B 101, 2020.
[2] B. Stadtmüller, S. Emmerich, D. Jungkenn, N. Haag, M. Rollinger, S. Eich, M. Maniraj, M. Aeschlimann, M. Cinchetti, S. Mathias. Strong modification of the transport level alignment in organic materials after optical excitation, Nat. Commun. 10, 2019.
Imaging molecular wave functions with photoemission orbital tomography: An introduction Forschungszentrum Jülich, Germany
The photoemission orbital tomography (POT) technique, a variant of angle-resolved photoemission spectroscopy, has been very useful in the characterization of the electronic properties of molecular films. It is a combined experimental and theoretical approach that is based on the interpretation of the photoelectron angular distribution in terms of a one-electron initial state. This includes the unambiguous assignment of emissions to specific molecular orbitals, their reconstruction to real space orbitals in two and three dimensions, the deconvolution of complex spectra into individual orbital contributions beyond the limits of energy resolution, the extraction of detailed geometric information such as molecular orientations, twists and bends, the precise description of the charge balance and transfer at interface, and the detection of momentum-selective hybridization with the substrate, to name only a few examples. In its simplest form, POT relies on the plane-wave approximation for the final state. While this works surprisingly well in many cases, this approximation does have its limitations, most notably for small molecules and with respect to the photon-energy dependence of the photoemission intensity. Regarding the latter, a straightforward extension of the plane wave final state leads to a much-improved description while preserving the simple and intuitive connection between the photoelectron distribution and the initial state.
Time-resolved photoemission orbital tomography of organic interfaces Philipps-Universität Marburg, Germany
Charge transfer across molecular interfaces is reflected in the population of electronic orbitals. For ordered organic layers, time-resolved photoemission orbital tomography (tr-POT) is capable of spectroscopically identifying the involved orbitals and deducing their population from the measured angle-resolved photoemission intensity with high temporal resolution [1]. As examples, I will present recent results obtained for PTCDA and CuPc adsorbed on Cu(100)-2O. We observe two distinct excitation pathways with visible light. While the parallel component of the electric field makes a direct HOMO-LUMO transition, the perpendicular component can transfer a substrate electron into the molecular LUMO. The experimental data are modelled by a density matrix description of the excitation and photoemission process. We find similar LUMO lifetimes for both excitation pathways, whereas the true dephasing times differ by two orders of magnitude.
Future tr-POT experiments will employ a two-pulse coherent control excitation scheme to steer the charge transfer. In some cases, this scheme will allow us to deduce the relative phase of the involved orbitals directly from the experiment. Furthermore, the combination with strong THz excitation and subcycle time resolution will make it possible to monitor charge transfer processes and hybridization during surface bond formation with POT.
[1] R. Wallauer, M. Raths, K. Stallberg, L. Münster, D. Brandstetter, X. Yang, J. Güdde, P. Puschnig, S. Soubatch, C. Kumpf, F. C. Bocquet, F. S. Tautz, U. Höfer. Tracing orbital images on ultrafast time scales, Science 371: 1056-1059, 2021.
https://doi.org/10.1126/science.abf3286
Exciton Photoemission Orbital Tomography: Probing the electron and the hole contributions I. Physical Institute, University of Goettingen, Germany
Time-resolved photoemission orbital tomography is a promising technique for the characterization of light-matter interaction in organic semiconductors. However, its state-of-the-art analysis approach based on density functional theory and the plane-wave model of photoemission cannot account for the correlated many-body nature of excitonic wavefunctions, which nevertheless represent the dominant optoelectronic response of organic semiconductors. Building on the many-body interaction formalisms of the $GW$ approach and the Bethe-Salpeter equation, we present a complete description of the angle-resolved exciton photoemission spectrum, and apply this model to the exemplary exciton relaxation cascade in multilayer C$_{60}$ crystals to investigate an intriguing property of the excitonic wavefunction: In C$_{60}$, and more generally in organic semiconductors, excitons can be of multiorbital nature, with both the electron and hole spread over multiple orbitals. We elucidate how photoemission orbital tomography is uniquely sensitive to this multiorbital nature and exploit it to directly access the hole part of the excitonic wavefunction in addition to its electron counterpart. With this capability, exciton photoemission orbital tomography provides a versatile probe of key exciton properties such as localization, charge transfer, and relaxation dynamics.
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| 1:30pm - 3:30pm | MS18 1: Inverse problems for fractional and nonlocal equations Location: VG1.103 Session Chair: Yi-Hsuan Lin Session Chair: Jesse Railo Session Chair: Mikko Salo |
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On the determination of a coefficient in a space-fractional equation with operators of Abel type 1University of Klagenfurt, Austria; 2Texas A&M University
We consider the inverse problem of recovering an unknown,
spatially-dependent coefficient $q(x)$ from the fractional order equation
$\mathbb{L}_\alpha u = f$ defined
in a region of $\mathbb{R}^2$ from boundary information.
Here $\mathbb{L_\alpha} ={D}^{\alpha_x}_x +{D}^{\alpha_y}_y +q(x)$
where the operators ${D}^{\alpha_x}_x$, ${D}^{\alpha_y}_y$ denote
fractional derivative operators based on the Abel fractional integral.
In the classical case this reduces to $-\triangle u + q(x)u = f$
and this has been a well-studied problem.
We develop both uniqueness and reconstruction results and show how the
ill-conditioning of this inverse problem depends on the geometry of the region
and the fractional powers $\alpha_x$ and $\alpha_y$.
Nonlocality Helps University of Washington and HKUST, United States of America
We give several examples of solutions inverse problems involving lon-range interactions whosecorresponding local problem is not solved.
Fractional p-Calderón problems ETH Zürich, Switzerland
The main purpose of this talk is to discuss two different nonlocal variants of the $p$-Calderón problem.
In the first model the nonlocal operator under consideration is a weighted fractional $p$-Laplacian and, similarly as for the $p$-Laplacian in dimensions $n\geq 3$, it is an open problem, whether it satisfies a unique continuation principle (UCP). However, it will be explained that the variational structure of the problem is still sufficiently nice that one can explicitly reconstruct the weight $\sigma(x,y)$ on the diagonal $D=\{(x,x): x\in W\}$ of the measurement set $W$. This reconstruction formula establishes a global uniqueness result for separable, real analytic coefficients [1].
In the second model, we consider the (anisotropic) fractional $p$-biharmonic operator, which naturally appears in the variational characterization of the optimal fractional Poincaré constant in Bessel potential spaces $H^{s,p}$. In contrast to the one above, this operator satisfies the UCP and so heuristically corresponds to the $p$-Laplacian in dimension $n=2$. Finally, we explain how this can be used to establish a global uniqueness result of the related inverse problem under a monotonicity condition [2].
[1] M. Kar, Y. Lin, and P. Zimmermann. Determining coefficients for a fractional $ p $-Laplace equation from exterior measurements, arXiv:2212.03057, 2022.
[2] M. Kar, J. Railo, and P. Zimmermann. The fractional $ p\, $-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems, arXiv:2208.09528, 2022.
Uniqueness in an inverse problem for the anisotropic fractional conductivity equation University of Bonn, Germany
We study an inverse problem for the fractional anisotropic conductivity equation. Our nonlocal operator is based on the well-developed theory of nonlocal vector calculus, and
differs substantially from other generalizations of the classical anisotropic conductivity operator obtained spectrally. We show that the anisotropic conductivity matrix can be recovered uniquely from fractional Dirichlet-to-Neumann data up to a natural gauge. Our analysis makes use of techniques recently developed for the study of the isotropic fractional elasticity equation, and generalizes them to the case of non-separable, anisotropic conductivities. The motivation for our study stems from its relation to the classical anisotropic Calderòn problem, which is one of the main open problems in the field.
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| 1:30pm - 3:30pm | MS25 1: Hyperparameter estimation in imaging inverse problems: recent advances on optimisation-based, learning and statistical approaches Location: VG0.111 Session Chair: Luca Calatroni Session Chair: Monica Pragliola |
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Automatic Differentiation of Fixed-Point Algorithms for Structured Non-smooth Optimization Saarland University, Germany
A large class of non-smooth practical optimization problems can be written as minimization of a sum of smooth and partly smooth functions. We consider structured non-smooth optimization problems which also depend on a parameter vector and study the problem of differentiating its solution mapping with respect to the parameter which has far reaching applications in sensitivity analysis and parameter learning. We show that under partial smoothness and other mild assumptions, Automatic Differentiation (AD) of the sequence generated by proximal splitting algorithms converges to the derivative of the solution mapping. For a variant of automatic differentiation, which we call Fixed-Point Automatic Differentiation (FPAD), we remedy the memory overhead problem of the Reverse Mode AD and moreover provide faster convergence theoretically. We numerically illustrate the convergence and convergence rates of AD and FPAD on Lasso and Group Lasso problems and demonstrate the working of FPAD on prototypical practical image denoising problem by learning the regularization term.
Learning data-driven priors for image reconstruction: From bilevel optimisation to neural network-based unrolled schemes 1Queen Mary University of London, United Kingdom; 2Physikalisch-Technische Bundesanstalt, Germany; 3Humboldt-Universität zu Berlin; 4Technische Universität Berlin, Germany; 5Finden Ltd, Rutherford Appleton Laboratory, United Kingdom; 6Weierstrass Institute for Applied Analysis and Stochastics, Germany
Combining classical model-based variational methods for image reconstruction with deep learning techniques has attracted a significant amount of attention during the last years. The aim is to combine the interpretability and the reconstruction guarantees of a model-based method with the flexibility and the state-of-the-art reconstruction performance that the deep neural networks are capable of achieving. We introduce a general novel image reconstruction approach that achieves such a combination which we motivate by recent developments in deeply learned algorithm unrolling and data-driven regularisation as well as by bilevel optimisation schemes for regularisation parameter estimation. We consider a network consisting of two parts: The first part uses a highly expressive deep convolutional neural network (CNN) to estimate a spatially varying (and temporally varying for dynamic problems) regularisation parameter for a classical variational problem (e.g. Total Variation). The resulting parameter is fed to the second sub-network which unrolls a finite number of iterations of a method which solves the variational problem (e.g. PDHG). The overall network is then trained end-to-end in a supervised fashion. This results to an entirely interpretable algorithm since the “black-box” nature of the CNN is placed entirely on the regularisation parameter and not to the image itself. We prove consistency of the unrolled scheme by showing that, as the number of unrolled iterations tends to infinity, the unrolled energy functional used for the supervised learning $\Gamma$-converges to the corresponding functional that incorporates the exact solution map of the TV-minimization problem. We also provide a series of numerical examples that show the applicability of our approach: dynamic MRI reconstruction, quantitative MRI reconstruction, low-dose CT and dynamic image denoising.
Learned proximal operators in accelerated unfolded methods with pseudodifferential operators 1University of Bologna; 2University of Bath
In recent years, hybrid reconstruction frameworks has been proposed by unfolding iterative methods and learning a suitable pseudodifferential correction on the part that can provably not be handled by model-based methods. In particular, the inner hyperameters of the method are estimated by using supervised learning techniques.
In this talk, I will present a variant of this approach, where an accelerated iterative algorithm is unfolded and the proximal operator is replaced by a learned operators, as in the PnP framework. The numerical experiments on limited-angle CT achieve promising results.
Masked and Unmasked Principles for Automatic Parameter Selection in Variational Image Restoration for Poisson Noise Corruption 1University of Bologna, Italy; 2University of Naples Federico II, Italy
Due to the statistical nature of electromagnetic waves, Poisson noise is a widespread cause of data degradation in many inverse imaging problems. It arises whenever the acquired data is formed by counting the number of photons irradiated by a source and hitting the image domain. Poisson noise removal is a crucial issue typical in astronomical and medical imaging, where the scenarios are characterized by a low photon count. For the former case, this is related to the acquisition set-up, while in the latter it is desirable to irradiate the patient with lower electromagnetic doses in order to keep it safer. However, the weaker the light intensity, the stronger the Poisson noise degradation in the acquired images and the more difficult the reconstruction problem.
An effective model-based approach for reconstructing images corrupted by Poisson noise is the use of variational methods. Despite the successful results, their performance strongly depends on the selection of the regularization parameter that balances the effect of the regularization term and the data fidelity term. One of the most used approaches for choosing the parameter is the discrepancy principle proposed in [1] that relies on imposing that the data term is equal to its approximate expected values. It works well for mid- and high-photon counting scenarios but leads to poor results for low-count Poisson noise. The talk will address novel parameter
selection strategies that outperform the state-of-the-art discrepancy principles in [1], especially for low-count regime. The approaches are based on decreasing the approximation error in [1] by means of a suitable Montecarlo simulation [2], on applying a so-called Poisson whiteness principle [3] and on cleverly masking the data used for the parameter selection [4], respectively. Extensive experiments are presented which prove the effectiveness of the three novel methods.
[1] M. Bertero, P. Boccacci, G. Talenti, R. Zanella, L. Zanni. A discrepancy principle for Poisson data, Inverse Problems 26(10), 2010. [105004]
[2] F. Bevilacqua, A. Lanza, M. Pragliola, F. Sgallari. Nearly exact discrepancy principle for low-count Poisson image restoration, Journal of Imaging 8(1): 1-35, 2022.
[3] F. Bevilacqua, A. Lanza, M. Pragliola, F. Sgallari. Whiteness-based parameter selection for Poisson data in variational image processing, Applied Mathematical Modelling 117: 197-218, 2023.
[4] F. Bevilacqua, A. Lanza, M. Pragliola, F. Sgallari. Masked unbiased principles for parameter selection in variational image restoration under Poisson noise, Inverse Problems 39(3), 2023. [034002]
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| 1:30pm - 3:30pm | MS29 1: Eigenvalues in inverse scattering Location: VG3.104 Session Chair: Martin Halla Session Chair: Peter Monk |
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Interior transmission eigenvalue trajectories 1Forschungszentrum Jülich GmbH, Germany; 2Karlsruhe Institute of Technology, Germany
Complex-valued eigenvalue trajectories parametrized by a constant index of refraction are investigated for the interior transmission problem. Several properties are derived for the unit disk such as that the only intersection points with the real axis are Dirichlet eigenvalues of the Laplacian. For general sufficiently smooth scatterers in two dimensions the only trajectorial limit points are shown to be Dirichlet eigenvalues of the Laplacian as the refractive index tends to infinity. Additionally, numerical results for several scatterers are presented which give rise to an underlying one-to-one correspondence between these two eigenvalue families which is finally stated as a conjecture.
A new family of modified interior transmission eigenvalues for a fluid-solid interaction 1University of Delaware; 2University of Oviedo
We study a new family of modified interior transmission eigenvalues for the interaction of a bounded elastic body (the target) embedded in an unbounded compressible inviscid fluid (the acoustic medium). This problem is modelled with the elastodynamic and acoustic equations in the time-harmonic regime, and the interaction of the two media is represented through the dynamic and kinematic boundary conditions; these are are two transmission conditions posed on the wet boundary that represent the equilibrium of forces, and the equality of the normal displacements of the solid and the fluid, respectively.
For such a model problem, we propose a new family of modified interior transmission eigenvalues (mITP eigenvalues), which depends on a tunable parameter $\gamma$ that can help increase the sensitivity of the eigenvalues to changes in the scatterer. We analyze the distribution of the mITP eigenvalues on the complex plane, in particular we show that they are real valued, and that they either fill the whole real line or define a discrete subset with no finite accumulation point. We also justify theoretically that they can be approximated from measurements of the far field pattern corresponding to incident plane waves by solving a collection of modified far field equations.
Furthermore, for a suitable choice of the parameter $\gamma$, our theory is more complete: it includes a proof of the discreteness of the mTIP eigenvalues, an upper bound for them, and a physical interpretation of the largest of them via a Courant min-max principle.
We finally provide numerical results for synthetic data to give an insight of the expected perfomance of the mITP eigenvalues if used as target signatures in applications.
[1] F. Cakoni, D. Colton, S. Meng, P. Monk. Stekloff eigenvalues in inverse scattering, SIAM J. Appl. Math. 76(4): 1737-1763, 2016.
[2] S. Cogar, D. Colton, S. Meng, P. Monk. Modified transmission eigenvalues in inverse scattering theory, Inverse Probl. 33(12): 125002, 2017.
[3] M. Levitin, P. Monk, V. Selgas,.Impedance eigenvalues in linear elasticity, SIAM J. on Appl. Math. 81(6), 2021.
[4] P. Monk, V. Selgas. Modified transmission eigenvalues for inverse scattering in a fluid-solid interaction problem, Research in the Mathematical Sciences 9(3), 2022.
Computation of transmission eigenvalues in singular configurations using a corner perfectly matched layer 1POEMS (CNRS-ENSTA Paris-INRIA), Institut Polytechnique de Paris, Palaiseau, France; 2IDEFIX (EDF-ENSTA Paris-INRIA), Institut Polytechnique de Paris, Palaiseau, France
In scattering, transmission eigenvalues are complex wavenumbers at which there exists an incident field that produces a vanishingly-small scattered far field. These eigenvalues solve the interior transmission eigenvalue problem (ITEP), which is a non-selfadjoint eigenvalue problem formulated on the support of the scatterer. In this work, we consider the discretization of the ITEP in two-dimensional cases where the difference between the parameters of the scatterer and that of the background medium changes sign at some point $O$ on the boundary of the scatterer. This sign change implies the existence of strongly-oscillating singularities localized around $O$, which prevent $H^{1}$-conforming finite element discretizations from approximating transmission eigenvalues, even when the corresponding modes are in $H^{1}$. In this talk we will demonstrate how transmission eigenvalues can be approximated by solving a modified ITEP; the modification consists in applying a suitable perfectly matched layer in a neighborhood of $O$, whose job is intuitively to tame strongly-oscillating singularities without inducing spurious reflections.
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| 1:30pm - 3:30pm | MS34 1: Learned reconstructions for nonlinear inverse problems Location: VG3.103 Session Chair: Simon Robert Arridge Session Chair: Andreas Selmar Hauptmann |
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Continuous generative models for nonlinear inverse problems 1University of Genoa, Italy; 2Technische Universität Berlin, Germany
Generative models are a large class of deep learning architectures, trained to describe a subset of a high dimensional space with a small number of parameters. Popular models include variational autoencoders, generative adversarial networks, normalizing flows and, more recently, score-based diffusion models. In the context of inverse problems, generative models can be used to model prior information on the unknown with a higher level of accuracy than classical regularization methods.
In this talk we will present a new data-driven approach to solve inverse problems based on generative models. Taking inspiration from well-known convolutional architectures, we construct and explicitly characterize a class of injective generative models defined on infinite dimensional functions spaces. The construction is based on wavelet multi resolution analysis: one of the key theoretical novelties is the generalization of the strided convolution between discrete signals to an infinite dimensional setting. After an off-line training of the generative model, the proposed reconstruction method consists in an iterative scheme in the low-dimensional latent space. The main advantages are the faster iterations and the reduced ill-posedness, which is shown with new Lipschitz stability estimates. We also present numerical simulations validating the theoretical findings for linear and nonlinear inverse problems such as electrical impedance tomography.
Data-driven quantitative photoacoustic imaging University of Cambridge, United Kingdom
Photoacoustic imaging faces the challenge of accurately quantifying measurements to accurately reconstruct chromophore concentrations and thus improve patient outcomes in clinical applications. Proposed approaches to solve the quantification problem are often limited in scope or only applicable to simulated data. We use a collection of well-characterised imaging targets (phantoms) as well as simulated data to enable supervised training and validation of quantification methods and train a U-Net on the data set. Our experiments demonstrate that phantoms can serve as reliable calibration objects and that deep learning methods can generalize to estimate the optical properties of previously unseen test images. Application of the trained model to a blood flow phantom and a mouse model highlights the strengths and weaknesses of the proposed approach.
Mapping properties of neural networks and inverse problems 1University of Helsinki, Finland; 2South Dakota State University, USA; 3University of Basel, Switzerland; 4Rice University, USA
We will consider mapping properties of neural networks, in particular, injectivity
of neural networks, universal approximation property of neural networks and
the properties which the ranges of neural networks need to have. Also, we
study approximation of probability measures using neural networks composed
of invertible flows and injective layer and applications of these results in inverse problems.
Data-driven regularization theory of invertible ResNets for solving inverse problems University of Bremen, Germany
Data-driven solution techniques for inverse problems, typically based on specific learning strategies, exhibit remarkable performance in image reconstruction tasks. These learning-based reconstruction strategies often follow a two-step scheme. First, one uses a given dataset to train the reconstruction scheme, which one often parametrizes via a neural network. Second, the reconstruction scheme is applied to a new measurement to obtain a reconstruction. We follow these steps but specifically parametrize the reconstruction scheme with invertible residual networks (iResNets). We demonstrate that the invertibility opens the door to new investigations into the influence of the training and the architecture on the resulting reconstruction scheme. To be more precise, we analyze the effect of different iResNet architectures, loss functions, and prior distributions on the trained network. The investigations reveal a formal link to the regularization theory of linear inverse problems for shallow network architectures and connections to MAP estimation with Gaussian noise models. Moreover, we analytically optimize the parameters of specific classes of architectures in the context of Bayesian inversion, revealing the influence of the prior and noise distribution on the solution.
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| 1:30pm - 3:30pm | MS39: Statistical inverse problems: regularization, learning and guarantees Location: VG2.105 Session Chair: Kim Knudsen Session Chair: Abhishake Abhishake |
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On the Regularized Functional Regression The Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austria
Functional Regression (FR) involves data consisting of a sample of functions taken from some population. Most work in FR is based on a variant of the functional linear model first introduced by Ramsay and Dalzell in 1991. A more general form of polynomial functional regression has been introduced only quite recently by Yao and Müller (2010), with quadratic functional regression as the most prominent case. A crucial issue in constructing FR models is the need to combine information both across and within observed functions, which Ramsay and Silverman (1997) called replication and regularization, respectively. In this talk we are going to present a general approach for the analysis of regularized polynomial functional regression of
arbitrary order and indicate the possibility for using here a technique that has been recently developed in the context of supervised learning. Moreover, we are going to describe of how multiple penalty regularization can be used in the context of FR and demonstrate an advantage of such use. Finally, we briefly discuss the application of FR in stenosis detection.
Joint research with S. Pereverzyev Jr. (Uni. Med. Innsbruck), A. Pilipenko
(IMATH, Kiev) and V.Yu. Semenov (DELTA SPE, Kiev) supported by the consortium
of Horizon-2020 project AMMODIT and the Austrian National Science Foundation (FWF).
Inverse learning in Hilbert scales LUT University Lappeenranta, Finland
We study the linear ill-posed inverse problem with noisy data in the statistical learning setting. Approximate reconstructions from random noisy data are sought with general regularization schemes in Hilbert scale. We discuss the rates of convergence for the regularized solution under the prior assumptions and a certain link condition. We express the error in terms of certain distance functions. For regression functions with smoothness given in terms of source conditions, the error bound can then be explicitly established.
Stability and Generalization for Stochastic Gradient Methods SUNY Albany, United States of America
Stochastic gradient methods (SGMs) have become the workhorse of machine learning (ML) due to their incremental nature with a computationally cheap update. In this talk, I will first discuss the close interaction between statistical generalization and computational optimization for SGMs in the framework of statistical learning theory (SLT). The core concept for this study is algorithmic stability which characterizes how the output of an ML algorithm changes upon a small perturbation of the training data. Our theoretical studies have led to new insights into understanding the generalization of overparameterized neural networks trained by SGD. Then, I will describe how this interaction framework can be used to derive lower bounds for the convergence of existing methods in the task of maximizing the AUC score which further inspires a new direction for designing efficient AUC optimization algorithms.
Causality and Consistency in Bayesian Inference Paradigms University of Copenhagen, Denmark
Bayesian inference paradigms are regarded as powerful tools for solution of inverse problems. However, Bayesian formulations suffer from a number of difficulties that are often overlooked.
The well known, but mostly neglected, difficulty is connected to the use of conditional probability densities. Borel, and later Kolmogorov's (1933/1956), found that the traditional definition of probability densities is incomplete: In different parameterizations it leads to different, conditional probability measures. This inconsistency is generally neglected in the scientific literature, and therefore threatens the objectivity of Bayesian inversion, Bayes Factor computations, and trans-dimensional inversion. We will show that this problem is much more serious than usually assumed.
Additional inconsistencies in Bayesian inference are found in the so-called hierarchical methods where so-called hyper-parameters are used as variables to control the uncertainties. We will see that these methods violate causality, and analyze how this challenges the validity of Bayesian computations.
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| 1:30pm - 3:30pm | MS43 1: Inverse Problems in radiation protection and nuclear safety Location: VG1.108 Session Chair: Lorenz Kuger Session Chair: Samuli Siltanen |
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Passive Gamma Emission Tomography (PGET) of spent nuclear fuel 1Radiation and Nuclear Safety Authority, Finland; 2Helsinki Institute of Physics, University of Helsinki, Finland; 3Department of Mathematical Sciences of the University of Bath, United Kingdom; 4Department of Mathematics and Statistics of the University of Helsinki, Finland
The world’s first deep underground repository for spent nuclear fuel will soon start operations in Eurajoki, Finland. Disposal tunnels have been excavated 430 meters below the ground surface in bedrock, and the spent nuclear fuel will be placed in deposition holes in copper canisters. After the fuel is disposed of, it will be practically unreachable [1]. For safeguarding nuclear material, all fuel items need to be reliably verified prior to disposal in the geological repository. Fuel assembly integrity is investigated to make sure all nuclear material stays in peaceful use.
Passive Gamma Emission Tomography (PGET) is a non-destructive assay method that allows for accurate 2D slice images of the fuel assembly to be reconstructed. Fuel assembly types we have studied are rectangular or hexagonal objects, about 4 meters long and about 15 cm in diameter, consisting of a bunch of 63-126 individual fuel rods in a fixed geometric arrangement. Spent nuclear fuel emits gamma-rays at a very high rate and with specific energies, providing a method to verify the presence of fuel rods in the assemblies. Gamma emission data is collected with the torus-shaped PGET device which has two highly collimated CdZnTe gamma detector banks that rotate a full 360 degree around the fuel assembly, which is placed in the central hole of the device. Gamma-rays are significantly attenuated in the fuel material, and thus the attenuation map of the object is reconstructed simultaneously with the activity map. The mathematical approach to this unique inverse problem is described in another presentation in this minisymposium, while the context of the method and the measurements are presented in more detail in this contribution [2,3].
During 2017-2022, over 100 spent nuclear fuel assemblies have been measured at the Finnish nuclear power plants with the PGET method [3,4]. The imaged fuel has had a range of characteristics and 10 different geometrical types. The measurement campaigns have concentrated on refining the measurement parameters for improving detection of possible empty rod positions. Data acquisition gamma energy windows have been fine-tuned, different sets of angles out of the full 360 angle data have been used in the reconstructions and different methods for quantitatively investigating the image quality have been developed. Even the use of less abundant but higher-energy and higher-penetrating gamma-rays were investigated to improve the detection of missing rods in the central parts of the fuel.
The PGET method has shown to detect individual missing rods with high confidence and has even demonstrated the ability to reproduce intra-rod activity differences. We have also shown that the method is able to distinguish activity differences in the axial direction of the fuel, which we show with a set of axial measurements conducted over a fuel assembly with partial-length fuel rods.
A variety of results from the measurement campaigns will be presented, illustrating the usability of the method for safeguards purposes in the Finnish context.
[1] www.posiva.fi
[2] R. Backholm, T. A. Bubba, C. Bélanger-Champagne, T. Helin, P. Dendooven, S. Siltanen. Simultaneous reconstruction of emission and attenuation in passive gamma emission tomography of spent nuclear fuel, Inv. Probl. Imag. 14: 317-337, 2020.
[3] P. Dendooven, T.A. Bubba. Gamma ray emission imaging in the medical and nuclear safeguards fields, Lecture Notes in Physics 1005: 245-295, 2022.
[4] R. Virta, R. Backholm, T. A. Bubba, T. Helin, M. Moring, S.Siltanen, P. Dendooven, T. Honkamaa. Fuel rod classification from Passive Gamma Emission Tomography (PGET) of spent nuclear fuel assemblies, ESARDA Bulletin 61: 10-21, 2020.
[5] R. Virta, T. A. Bubba, M. Moring, S. Siltanen, T. Honkamaa, P. Dendooven. Improved Passive Gamma Emission Tomography image quality in the central region of spent nuclear fuel, Scientific Reports 12: 12473, 2022.
Bayesian modelling and inference for radiation source localisation 1School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh, United Kingdom; 2Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, United Kingdom; 3Department of Nuclear, Plasma and Radiological Engineering, University of Illinois Urbana Champaign, Champaign, United States; 4School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh, United Kingdom; 5School of Mathematics, University of Edinburgh, Edinburgh, United Kingdom
In this work, we study a Compton Imager, made of an array of scintillation crystals. This imaging system differs from a more classical Compton camera since the sensor array acts simultaneously as a set of scatterers and absorbers. From the recorded data, the objective is to localize the positions of point-like sources responsible for the emission of the measured radiation. The inverse problem is formulated within a Bayesian framework, and a Markov chain Monte Carlo method is investigated to infer the source locations.
Image reconstruction for Passive Gamma Emission Tomography of spent nuclear fuel 1Helsinki Institute of Physics, University of Helsinki, Finland; 2Department of Mathematical Sciences, University of Bath, UK; 3Radiation and Nuclear Safety Authority (STUK), Vantaa, Finland; 4Department of Mathematics and Statistics, University of Helsinki, Finland
A Passive Gamma Emission Tomography (PGET) device is part of the IAEA-approved tools for safeguards inspections of spent nuclear fuel assemblies. In Finland, PGET has been selected to be part of the nuclear safeguards procedures at the geological repository for spent nuclear fuel (SNF), ONKALO [1]. In recent years, we have developed the PGET method for this purpose. This contribution will focus on the data analysis and image reconstruction methods. It will show how the methods chosen are dictated and influenced by the requirements and the physics of the application, as well as the characteristics of the tomographic device that is being used. The characteristics and performance of the reconstruction algorithm will be illustrated with examples from PGET measurements at the SNF storage pools at the Finnish nuclear power plants. The design and operation of the PGET device and the most important results will be discussed in a separate contribution to this minisymposium.
A safeguards inspection aims to verify that all nuclear material is present as declared, to assure that none has been diverted for non-declared use, most specifically the development of nuclear weapons. Because of this requirement, a PGET measurement should not assume any prior information on the object under tomographic investigation. An SNF assembly consists essentially of rods of highly radioactive uranium dioxide, highly attenuating for gamma rays, immersed in non-radioactive water with much lower gamma ray attenuation. Good images thus require some form of attenuation correction. It is in practice very challenging to independently measure an attenuation map of SNF, e.g. by transmission tomography. Also, given the binary nature of the object (fuel rods and water), a good attenuation map needs knowledge of the geometry of the SNF. We have dealt with this conflict between the need of a good attenuation map and the requirement not to use prior information by developing an image reconstruction algorithm that reconstructs a gamma ray emission and attenuation image simultaneously, mathematically treating both in the same way.
The image reconstruction involves 2 steps. The first step is a filtered back-projection (FBP) which uses no prior information at all. Experimentally we observe that the quality of the FBP image is good enough to deduce the SNF assembly geometry. The second step, which produces the final image, is an iterative image reconstruction algorithm which uses the knowledge of the assembly geometry as a regularization term, thus favouring images that resemble the SNF assembly type identified from the FBP image in step 1. The reconstruction problem is formulated as a constrained minimization problem with a least squares data mismatch term (i.e. it implicitly assumes a Gaussian distribution for the noise) and several regularization terms. Next to the geometry regularization term, there are 2 terms related to corrections for the variation of the sensitivity amongst the detectors. Physics knowledge is used to establish upper and lower bounds on the image of attenuation coefficients. The attenuation image is constrained to values between the attenuation coefficient of the relevant gamma ray energy in water (lower bound) and uranium dioxide (upper bound). Most often, imaging is focused on the 662 keV gamma rays emitted by 137Cs, the dominant gamma ray emitter in SNF. The PGET image reconstruction method and its practical implementation will be discussed in some detail. Full details are given in [2-4].
Since 2017, over 100 different SNF assemblies have been measured at the Finnish nuclear power plants. Some representative examples from this vast data set will be used to highlight the performance of the image reconstruction method, especially in identifying missing fuel rods [3,5].
Points for improvement that have been identified over the past few years will be discussed. These are e.g. careful selection of the set of viewing angles, careful selection of the gamma ray energy windows and combining sinograms from different gamma ray energy windows. Improving imaging of the centre of spent fuel assemblies is a major development goal.
[1] www.posiva.fi
[2] P. Dendooven, T. A. Bubba. Gamma ray emission imaging in the medical and nuclear safeguards fields, Lecture Notes in Physics 1005: 245-295, 2022.
[3] R. Virta, R. Backholm, T. A. Bubba, T. Helin, M. Moring, S. Siltanen, P. Dendooven, T. Honkamaa. Fuel rod classification from Passive Gamma Emission Tomography (PGET) of spent nuclear fuel assemblies, ESARDA Bulletin 61: 10-21, 2020.
[4] R. Backholm, T. A. Bubba, C. Bélanger-Champagne, T. Helin, P. Dendooven, S. Siltanen. Simultaneous reconstruction of emission and attenuation in passive gamma emission tomography of spent nuclear fuel, Inv. Probl. Imag. 14: 317-337, 2020.
[5] R. Virta, T. A. Bubba, M. Moring, S. Siltanen, T. Honkamaa, P. Dendooven. Improved Passive Gamma Emission Tomography image quality in the central region of spent nuclear fuel, Scientific Reports 12: 12473, 2022.
Exact inversion of an integral transform arising in passive detection of gamma-ray sources with a Compton camera NC State University, United States of America
This talk addresses the overdetermined problem of inverting the n-dimensional cone (or Compton) transform that integrates a function over conical surfaces in $\mathbb{R}^n$. The study of the cone transform originates from Compton camera imaging, a nuclear imaging method for the passive detection of gamma-ray sources. We present a new identity relating the n-dimensional cone and Radon transforms through spherical convolutions with arbitrary weight functions. This relationship leads to various inversion formulas in n-dimensions under a mild assumption on the geometry of detectors. We present two such formulas along with the results of their numerical implementation using synthetic phantoms.
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| 1:30pm - 3:30pm | MS44 1: Modelling in Earth and planetary sciences by data inversion at various scales Location: VG2.104 Session Chair: Christian Gerhards Session Chair: Volker Michel Session Chair: Frederik J Simons |
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Inverse magnetization problems in geoscience at various scales TU Bergakademie Freiberg, Germany
The inversion of magnetic field data for the underlying magnetization is a frequent problem in geoscience. It occurs at planetary scales, inverting satellite magnetic field information for lithospheric sources, as well as at microscopic scales, inverting for the sources in thin slices of rock samples. All scales have in common that the inverse problem is nonunique and highly instable. Here, we want to provide an overview on this topic and indicate various scenarios for which additional assumptions may ameliorate some of the issues of ill-posedness. This ranges from the assumption of an (infinitely) thin lithosphere (where the Hardy-Hodge decomposition can be used for the characterization of uniqueness) to a priori knowledge about the location or shape of magnetic inclusions within a rock sample (where the Helmholtz decomposition plays a role for the uniqueness aspect).
Slepian concentration problem for polynomials on the Ball TU Bergakademie Freiberg, Germany
The sources of geophysical signals are often spatially localized. Thus, adequate basis functions are required to model such properties. Slepian functions have proven to be a very successful tool.
Here, we consider theoretical properties of the Slepian spatial-spectral concentration problem for the space of multi-variate polynomials on the unit ball in $\mathbb{R}^d$ normalized under Jacobi weights. In particular, we show the phenomena of the step-like shape of the eigenvalue distribution of concentration operators, and characterize the transition by the Jacobi weight $W_{0}$, which serves as an analogue of the $2\Omega T$ rule in the classical Slepian theory. A numerical demonstration is performed for the 3-D ball with Lebesgue weights.
Regularized matching pursuits with a learning add-on for geoscientific inverse problems University of Siegen, Geomathematics Group Siegen, Germany
We consider challenging inverse problems from the geosciences: the downward continuation of
satellite data for the approximation of the gravitational potential as well as the travel time tomography using earthquake data to model the interior of the Earth. Thus, we are able to monitor certain influences on the system Earth, in particular the mass transport of the Earth or its interior anomalies.
For the approximation of these linear(ized) inverse problems, different basis systems can be
utilized. Traditionally, we a-priori either choose a global, e.g. spherical harmonics on the sphere or polynomials on the ball, or a local one, e.g. radial basis functions and wavelets or finite elements.
The Learning Inverse Problem Matching Pursuits (LIPMPs), however, have the unique
characteristic to enable the combination of global and local trial functions for the approximation of inverse problems. The latter is obtained iteratively from an intentionally overcomplete set of
trial functions, the dictionary, such that the Tikhonov functional is reduced. Moreover, the learning add-on allows the dictionary to be infinite such that an a-priori choice of a finite number of trial functions is negligible. Further, it increases the efficiency of the methods.
In this talk, we give details on the LIPMPs and show some current numerical results.
Non-unique Inversions in Earth Sciences - an Underestimated Pitfall? University of Siegen, Germany
Earth exploration is in many cases connected to inverse problems, since often regions of interest cannot be accessed sufficiently. This is the case for the recovery of structures in the Earth's interior. However, it is also present in the investigation of processes at the Earth's surface, e.g. if a sufficient global or regional coverage is required or if remote areas are of interest.
Many of these problems are associated to an instability of the inverse problems, which is why a variety of regularization methods for their stabilization has been developed so far. However, a notable number of the problems is also ill-posed because of a non-unique solution. Phantom anomalies and other artefacts might be possible consequences. In some cases, the mathematical structure of the underlying null spaces is entirely understood (e.g. for a certain class of Fredholm integral equations of the first kind). In other cases, such a theory is still missing. Nevertheless, also for mathematically well described cases, numerical methods often ignore what can be visible and what can be invisible in available data.
The purpose of this talk is to create some more sensitivity regarding the challenges of inverse problems with non-unique solutions.
[1] S. Leweke, V. Michel, R. Telschow. On the non-uniqueness of gravitational and magnetic field data inversion (survey article), in: Handbook of Mathematical Geodesy (W. Freeden, M.Z. Nashed, eds.), Birkhäuser, Basel, 883-919, 2018.
[2] V. Michel. Geomathematics - Modelling and Solving Mathematical Problems in Geodesy and Geophysics. Cambridge University Press, Cambridge, 2022.
[3] V. Michel, A.S. Fokas. A unified approach to various techniques for the non-uniqueness of the inverse gravimetric problem and wavelet-based methods, Inverse Problems 24: 25pp, 2008.
[4] V. Michel, S. Orzlowski. On the null space of a class of Fredholm integral equations of the first kind, Journal of Inverse and Ill-Posed Problems 24: 687-710, 2016.
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| 1:30pm - 3:30pm | MS52 1: Integral geometry, rigidity and geometric inverse problems Location: VG1.105 Session Chair: Francois Sylvain Monard Session Chair: Plamen Stefanov |
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Inverse problem for Yang-Mills-Higgs fields 1University of Helsinki, Finland; 2Fudan University, China; 3University of Cambridge, UK
We show that the Yang-Mills potential and Higgs field are uniquely determined (up to the natural gauge) from source-to-solution type data associated with the classical Yang-Mills-Higgs equations in the Minkowski space. We impose natural non-degeneracy conditions on the representation for the Higgs field and on the Lie algebra of the structure group which are satisfied for the case of the Standard Model. Our approach exploits non-linear interaction of waves to recover a broken non-abelian light ray transform of the Yang-Mills field and a weighted integral transform of the Higgs field.
The Lorentzian scattering rigidity problem and rigidity of stationary metrics Purdue University, United States of America
We study scattering rigidity in Lorentzian geometry: recovery of a Lorentzian metric from the scattering relation $\mathcal{S}$ known on a lateral boundary. We show that, under a non-conjugacy assumption, every defining function $r(x,y)$ of the submanifold of pairs of boundary points which can be connected by a lightlike geodesic plays the role of the boundary distance function in the Riemannian case in the following sense. Its linearization is the light ray transform of tensor fields of order two which are the perturbations of the metric. Next, we study scattering rigidity of stationary metrics in time-space cylinders and show that it can be reduced to boundary rigidity of magnetic systems on the base; a problem studied previously. This implies several scattering rigidity results for stationary metrics.
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| 1:30pm - 3:30pm | MS54 1: The x-ray transform and its generalizations: Theory, methods, and applications Location: VG1.101 Session Chair: Suman Kumar Sahoo |
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Unique continuation for the momentum ray transform 1University of Jyväskylä, Finland; 2National Chengchi University, Taiwan
We will explain the relation between momentum ray transform and the fractional Laplacian. As a consequence of unique continuation property of the fractional Laplacian, one can discuss similar properties for momentum ray transform (of integer and fractional order). In addition, we will also explain the relation between the weighted ray transform and the cone transform, which appeared in different imaging approaches (for example, Compton camera).
This talk is prepared based on the work [1].
[1] J. Ilmavirta, P.-Z. Kow, S. K. Sahoo. Unique continuation for the momentum ray transform, arXiv: 2304.00327, 2023.
The linearized Calderon problem for polyharmonic operators University of Jyvaskyla, Finland
The density of products of solutions to several types of partial differential equations, such as elliptic, parabolic, and hyperbolic equations, plays a significant role in solving various inverse problems, dating back to Calderon's fundamental work. In this talk, we will discuss some density properties of solutions to polyharmonic operators on the space of symmetric tensor fields. These density questions are closely related to the linearized Calderon problem for polyharmonic operators. This is joint work with Mikko Salo.
The star transform and its links to various areas of mathematics 1University of Texas at Arlington, United States of America; 2Dartmouth College, United States of America
The divergent beam transform maps a function $f$ in $\mathbb{R}^n$ to an $n$-dimensional family of its integrals along rays, emanating from a variable vertex location inside the support of $f$. The star transform is defined as a linear combination of divergent beam transforms with known coefficients. The talk presents some recent results about the properties of the star transform, its inversion, and a few interesting connections to different areas of mathematics.
Linearized Calderon problem for biharmonic operator with partial data 1Tata Institute of Fundamental Research (TIFR), India; 2University of Jyvaskyla, Finland
The product of solutions of the Laplace equation vanishing on an arbitrary closed subset of the boundary is dense in the space of integrable functions ([1]). In this talk, we will discuss a similar problem with the biharmonic operator replacing the Laplace operator. More precisely, we show that the integral identity
$$
\int \left [a \Delta u v+ \sum a^1_i \partial_i u v + a^0 u v \right ]\, \mathrm{d}x = 0
$$
for all biharmonic functions vanishing on an proper closed subset of the boundary implies that $(a, a^1, a^0)$ vanish identically. This work is a collaboration with R. S. Jaiswal and S. K. Sahoo.
[1] D. Ferreira, C. E. Kenig, J. Sjostrand, G. Uhlmann. On the linearized local Calderon problem. Math. Res. Lett. 16: 955-970, 2009.
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| 1:30pm - 3:30pm | MS55 1: Nonlinear Inverse Scattering and Related Topics Location: VG3.101 Session Chair: Yang Yang |
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An Inverse Problem for Nonlinear Time-dependent Schrodinger Equations with Partial Data 浙江大学, China, People's Republic of
In this talk, I will present some recent results on solving inverse boundary value problems for nonlinear PDEs, especially for a time-dependent Schrodinger equation with time-dependent potentials with partial boundary Dirichlet-to-Neumann map. After a higher order linearization step, the problem will be reduced to implementing special geometrical optics (GO) solutions to prove the uniqueness and stability of the reconstruction. This is a joint work with my PhD student Xuezhu Lu and Prof. Ru-Yu Lai.
Supercomputing-based inverse modeling of high-resolution atmospheric contaminant source intensity distribution using remote sensing data 1School of Computer Science and Engineering, Sun Yat-sen University, Guangzhou 510006, China; 2Guangdong Province Key Laboratory of Computational Science, Guangzhou 510006, China; 3JülichSupercomputing Centre, Forschungszentrum Jülich, Jülich 52425, Germany; 4Numerical Mathematics, RWTH Aachen University, Aachen 52056, Germany
Atmospheric pollution prevention and control is an important global issue[1]. Observing the emission of harmful trace gases and their atmospheric transport dynamics on a global scale is of great significance for in-depth study of major problems, such as climate change and ecological and environmental change[2, 3]. In recent years, the inverse problem of atmospheric contaminant source intensity distribution has attracted more and more attention from researchers[4]. Due to its mathematical ill-posedness and high computational costs, it is necessary to develop new computational tools[5]. Accurate, rapid, and stable inverse analysis of atmospheric contaminant source intensity distribution, and subsequently using high-resolution numerical simulation methods to predict the local or large-scale, short-term or long-term atmospheric environmental impacts caused by major sudden natural disasters and industrial pollution events, has important scientific significance and practical value.
By using limited information of satellite observation data obtained through NIS (Nonlinear Inverse Scattering) technology[7, 8, 9], we have established a high-throughput parallel computing framework for solving the mathematical and physical inverse problem of high-resolution spatiotemporal atmospheric contaminant sources distribution[10, 11, 12]. A real-time inverse analysis and transport simulation of atmospheric contaminant source intensity distribution with high resolution, stability, and reliability are realized. And thus the high-resolution reanalysis data and prediction information on a global scale are available, which can not be directly obtained by current satellite or optical radar measurement technologies. Considering the influence of complex physical and chemical processes, such as the transport of particles in wind fields and the scattering of particles due to light irradiation, the relationship between unknown source parameters and observed contaminant concentrations is usually nonlinear[2, 5, 7, 13, 14, 15]. Therefore, we comprehensively use numerical simulation, optimization methods, and statistical inference techniques[6, 16, 17]. Taking volcanic eruption and forest fire as examples, based on remote sensing data, we use the jointly developed Lagrangian transport model MPTRAC (Massive Parallel Trajectory Calculation) for forward simulation[11]. Combined with heuristic methods such as segmented strong constraint "product rule" proposed by us, the computational bottleneck of traditional serial regularization methods for solving such inverse problems is overcome[10, 12]. And a million-core supercomputing-based inverse calculation strategy is developed, which greatly reduces time costs while ensuring accuracy and reliability, meeting the needs of future real-time prediction tools.
This study provides practical application scenarios for NIS technology, and plays an important theoretical and practical role in ensuring aviation safety, exploring the mechanism of pollutant degradation, and revealing the causes of global climate change.
[1] N.-N. Zhang et al. Spatiotemporal trends in PM2. 5 levels from 2013 to 2017 and regional demarcations for joint prevention and control of atmospheric pollution in China, Chemosphere 210: 1176-1184, 2018.
[2] S. Huang et al. Inverse problems in atmospheric science and their application, Journal of Physics: Conference Series, IOP Publishing, 2005.
[3] W. Freeden et al. Handbook of geomathematics, 2nd edition. Springer Berlin Heidelberg, 2015.
[4] M. S. Zhdanov. Inverse Theory and Applications in Geophysics, 2nd edition. Elsevier Science, 2015.
[5] J. L. Mueller, S. Siltanen. Linear and nonlinear inverse problems with practical applications, SIAM, 2012.
[6] Y. Bai et al. Computational methods for applied inverse problems, Walter de Gruyter, 2012.
[7] D. Efremenko, A. Kokhanovsky. Foundations of Atmospheric Remote Sensing, Springer, 2021.
[8] W. C. Chew et al. Nonlinear diffraction tomography: The use of inverse scattering for imaging, Int J Imaging Syst Technol. 7(1):16-24, 1996.
[9] T. Hasegawa, T. Iwasaki. Microwave imaging by quasi-inverse scattering, Electron Comm Jpn Pt I. 87(5):52-61, 2004.
[10] Y. Heng et al. Inverse transport modeling of volcanic sulfur dioxide emissions using large-scale simulations, Geoscientific Model Development 9(4): 1627-1645, 2016.
[11] L. Hoffmann et al. Massive-Parallel Trajectory Calculations version 2.2 (MPTRAC-2.2): Lagrangian transport simulations on graphics processing units (GPUs), Geoscientific Model Development 15(7): 2731-2762, 2022.
[12] M. Liu et al. High-Resolution Source Estimation of Volcanic Sulfur Dioxide Emissions Using Large-Scale Transport Simulations, Computational Science – ICCS 2020: 60-73, 2020.
[13] K. Chadan et al. An introduction to inverse scattering and inverse spectral problems, Society for Industrial and Applied Mathematics, 1997.
[14] D. P. Winebrenner, J. Sylvester. Linear and nonlinear inverse scattering, SIAM Journal on Applied Mathematics, 59(2): 669-699, 1998.
[15] V. Isakov. Inverse problems for partial differential equations, Springer, 2006.
[16] E. Haber et al. On optimization techniques for solving nonlinear inverse problems, Inverse problems, 6(5): 1263, 2000.
[17] R. C. Aster et al. Parameter estimation and inverse problems, Elsevier, 2018.
Some progresses in Carleman estimates and their applications in inverse problems for stochastic partial differential equations University of Electronic Science and Technology of China, P. R. China
This talk studies Carleman estimates and their applications for inverse problems of stochastic partial differential equations. By establishing new Carleman estimates for the stochastic parabolic and hyperbolic systems, conditional stability for inverse problems of these systems are proven. Based on the idea of Tikhonov method, regularized solutions are proposed. The analysis of the existence and uniqueness of the regularized solutions, and proofs for error estimate under an a priori assumption are presented. Numerical verification of the regularization, based on the idea of kernel- based learning method, including numerical algorithms and examples are also illustrated.
Inverse scattering problems with incomplete data Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China, People's Republic of
Inverse scattering problems aim to determine unknown scatterers with wave fields measured around the scatterers. However, from the practical point of views, we have only limited information, e.g., limited aperture data phaseless data and sparse data. In this talk, we introduce some data retrieval techniques and the applications in the inverse scattering problems. The theoretical and numerical methods for inverse scattering problems with multi-frequency spase measurements will also be mentioned.
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| 1:30pm - 3:30pm | MS56 1: Inverse Problems of Transport Equations and Related Topics Location: VG2.106 Session Chair: Ru-Yu Lai Session Chair: Hanming Zhou |
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Multiscale Parameter Identification: mesoscopic kernel reconstruction from macroscopic data University of Würzburg, Germany
Motivated from a biological application, we study mesoscopic velocity jump (run-and-tumble) models for particle motion in the phase space. The motion is characterized by a sudden change of direction which is governed by the turning rate. The inverse problem is to determining this mesoscopic turning rate from macroscopic, i.e. directionally averaged data. The lack of directional information in the measurements poses problems in the reconstruction of the mesoscopic quantity. These problems can be leveraged by the use of time dependent interior domain data, as theoretical results on the reconstructability suggest. We then investigate the macroscopic limit behaviour for the inverse problem on the macroscopic regime and present first results on the numerical inversion.
This is joint work with Christian Klingenberg (Würzburg, Germany), Qin Li (Madison, Wisc., USA) and Min Tang (Shanghai, China).
A 1-Wasserstein framework for forward-peaked diffusive transport 1University of Chicago, USA; 2Pontificia Universidad Catolica, Chile
In this talk, I will present a framework to study inverse problems involving forward-peaked diffusive transport. More specifically, we will study Fokker-Planck approximations to highly forward-peaked scattering and the accuracy of its respective approximations via Fermi pencil-beams, in a metric based on the 1-Wasserstein distance. We argue that metrics of this kind are suitable for analyzing the stability of related inverse problems, for instance, in inverse transport, microscopy, and off-axis laser detection. This is joint work with Guillaume Bal.
Recovery of coefficients in nonlinear transport equations University of California Santa Barbara, United States of America
In this talk, we will discuss the determination of coefficients in time-dependent nonlinear transport equations. We consider both cases of time-independent and time-dependent coefficients. The talk is based on joint work with Ru-Yu Lai and Gunther Uhlmann.
Mapping properties and functional relations for the hyperbolic X-ray transform 1University of California Santa Cruz, United States of America; 2Leibniz University Hannover; 3Northwestern University
I will discuss recent results on the range characterization of the X-ray transform on the hyperbolic disk, along with functional relations with distinguished differential operators, and mapping properties in adapted scales of Sobolev spaces.
This is based on joint work with Nikolaos Eptaminitakis (Leibniz University Hannover) and Yuzhou Zou (Northwestern).
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| 1:30pm - 3:30pm | MS57 1: Inverse Problems in Time-Domain Imaging at the Small Scales Location: VG3.102 Session Chair: Eric Bonnetier Session Chair: Xinlin Cao Session Chair: Mourad Sini |
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Inverse wave scattering in the time domain 1DiSAT, Università dell'Insubria, Como, Italy; 2UMR9008 CNRS et Université de Reims Champagne-Ardenne, Reims, France
Let $\Delta_{\Lambda}\le \lambda_{\Lambda}$ be a semi-bounded self-adjoint realization of the Laplace operator with boundary conditions (Dirichlet, Neumann, semi-transparent) assigned on the Lipschitz boundary of a bounded obstacle $\Omega$. Let $u^{\Lambda}_{f}$ and $u^{0}_{f}$ denote the solutions of the wave equations corresponding to $\Delta_{\Lambda}$ and to the free Laplacian $\Delta$ respectively, with a source term $f$ concentrated at time $t=0$ (a pulse). We show that for any fixed $\lambda>\lambda_{\Lambda}\ge 0$ and any fixed $B\subset\subset{\mathbb R}^{n}\backslash\overline\Omega$, the obstacle $\Omega$ can be reconstructed by the scattering data operator
$$
F^{\Lambda}_{\lambda}f(x):=\int_{0}^{\infty}e^{-\sqrt\lambda\,t}\big(u^{\Lambda}_{f}(t,x)-u^{0}_{f}(t,x)\big)\,dt\,,\qquad x\in B\,,\ f\in L^{2}({\mathbb R}^{n})\,,\ \mbox{supp}(f)\subset B\,.
$$
A similar result holds for point scatterers; in this case, the locations of the of scatterers are determined by an analog of $F^{\Lambda}_{\lambda}$ acting in a finite dimensional space.
A new approach to an inverse source problem for the wave equation 1RICAM, Austrian Academy of Sciences, Austria; 2School of Mathematics, Southeast University, P.R. China,
Consider an inverse problem of reconstructing a source term from boundary measurements for the wave equation. We propose a novel approach to recover the unknown source through measuring the wave fields after injecting small particles, enjoying a high contrast, into the medium. For this purpose, we first derive the asymptotic expansion of the wave field, based on the time-domain Lippmann-Schwinger equation. The dominant term in the asymptotic expansion is expressed as an infinite series in terms of the eigenvalues of the Newtonian operator (for the pure Laplacian). Such expansions are useful under a certain scale between the size of the particles and their contrast. Second, we observe that the relevant eigenvalues appearing in the expansion have non-zero averaged eigenfunctions. By introducing a Riesz basis, we reconstruct the wave field, generated before injecting the particles, on the center of the particles. Finally, from these last fields, we reconstruct the source term. A significant advantage of our approach is that we only need the measurements for a single point away from the support of the source.
Simultaneous Reconstruction Of Optical And Acoustical Properties In PA-Imaging Using Plasmonics. 1NCAM, China, People's Republic of; 2RICAM, Austrian Academy of Sciences.
We propose an approach for the simultaneous reconstruction of the electromagnetic
and acoustic material parameters, in the given medium $\Omega$ where to image, using the photoacoustic pressure, measured on a single point of the boundary of $\Omega$, generated by plasmonic nano-particles. We prove that the generated pressure, that we denote by $p^{\star}(x, s, \omega)$, depending on only one fixed point $x \in \partial \Omega$, the time variable $s$, in a large enough interval, and the incidence frequency $\omega$, in a large enough band, is enough to reconstruct both the sound speed, the mass density and the permittivity inside $\Omega$. Indeed, from the behaviour of the measured pressure in terms of time, we can estimate the travel time of the pressure, for arriving points inside $\Omega$, then using the eikonal equation we reconstruct the acoustic speed of propagation, inside
$\Omega$. In addition, we reconstruct the internal values of the acoustic Green’s function. From the singularity analysis of this Green’s function, we extract the integrals along the geodesics, for internal arriving points, of the logarithmic-gradient of the mass density. Solving this integral geometric problem provides us with the values of the mass density function inside $\Omega$. Finally, from the behaviour of $p^{\star}(x, s, \omega)$ with respect to the frequency $\omega$, we detect the generated plasmonic resonances from which we reconstruct the permittivity inside $\Omega$.
Time domain analysis of body resonant-modes for classical waves 1Université de Reims, France; 2Università dell'Insubria, Como-Varese, Italy
We consider the wave propagation in the time domain in the presence of small inhomogeneities having high contrast with respect to a homogeneous background. This can be interpreted as a reduced scalar-model for the interaction of an electromagnetic wave with dielectric nanoparticles with high refractive indices. Such composite systems are known to exhibit a transition towards a resonant regime where an enhancement of the scattered wave can be observed at specific incoming frequencies, commonly referred to as body resonances. The asymptotic analysis of the stationary scattering problem, in the high-index nanoparticles regime, recently provided accurate estimates of the resonant frequencies and useful point-scatterer expansions for the solution in the far-field approximation. A key role in this analysis is played by the Newton operator related to the inhomogeneity support, whose eigenvalues identify the inverse resonant energies. We point out that a characterization of such singular frequencies in a proper sense spectral requires the spectral analysis of the Hamiltonian associated to the time-dependent problem. Here we focus on this problem by introducing the scale-dependent Hamiltonian of
the time-evolution equation. In this framework, we consider the spectral profile with a particular focus on the generalized spectrum close to the branch-cut. We show that, in this region, the resonances are located in small neighbourhoods of the eigenvalues of the inverse Newton operator and provide accurate estimates for their imaginary parts. In particular, this allows a
complete computation of the time-propagator in the asymptotic regime, providing in this way the full asymptotic expansion of the time-domain solution.
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| 3:30pm - 4:00pm | C2: Coffee Break Location: ZHG Foyer |
| 4:00pm - 6:00pm | MS03 2: Compressed Sensing meets Statistical Inverse Learning Location: VG2.103 Session Chair: Tatiana Alessandra Bubba Session Chair: Luca Ratti Session Chair: Matteo Santacesaria |
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SGD for statistical inverse problems LUT University Lappeenranta, Finland
We study a statistical inverse learning problem, where we observe the noisy image of a quantity through an operator at some random design points. We consider the SGD schemes to reconstruct the estimator of the quantity for the ill-posed inverse problem. We develop a theoretical analysis for the minimizer of the regularization scheme using the approach of reproducing kernel Hilbert spaces. We discuss the rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.
Convex regularization in statistical inverse learning problems LUT University, Finland
Statistical inverse learning aims at recovering an unknown function $f$ from randomly scattered and possibly noisy point evaluations of another function $g$, connected to $f$ via an ill-posed mathematical model.
In this talk I blend statistical inverse learning theory with convex regularization strategies and derive convergence rates for the corresponding estimators.
An off-the-grid approach to multi-compartment magnetic fingerprinting University of Bath, United Kingdom
We propose a off-the-grid numerical approach to separate multiple tissue compartments in image voxels and to estimate quantitatively their nuclear magnetic resonance (NMR) properties and mixture fractions, given magnetic resonance fingerprinting (MRF) measurements. One of the challenge is that fine-grid discretisation of the multi-dimensional NMR properties creates large and highly coherent MRF dictionaries that can challenge scalability and precision of the numerical methods for sparse approximation. To overcome this issues, we propose an off-the-grid approach equipped with an extended notion of the sparse group lasso regularisation for sparse approximation using continuous Bloch response models. Through numerical experiments on simulated and in-vivo healthy brain MRF data, we demonstrate the effectiveness of the proposed scheme compared to baseline multi-compartment MRF methods.
This is joint work with Mohammad Golbabaee.
How many Neurons do we need? A refined Analysis. Technische Universität Braunschweig, Germany
We present new results for random feature approximation in kernel methods and discuss the connection to generalization properties of two-layer neural networks in the NTK regime. Here, we aim at improving the results of Nitanda and Suzuki [1] in various directions. More precisely, we aim at overcoming the saturation effect appearing in Nitanda and Suzuki [1] by providing fast rates of convergence for smooth objectives. On our way, we also precisely keep track of the number of hidden neurons required for generalization.
[1] A. Nitanda, T. Suzuki. Functional Gradient Boosting for Learning Residual-like Networks with Statistical Guarantees. Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics. 108: 2981--2991. 2020.
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| 4:00pm - 6:00pm | MS12 2: Fast optimization-based methods for inverse problems Location: VG2.102 Session Chair: Bjørn Christian Skov Jensen |
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Limited memory restarted $\ell^p-\ell^q$ minimization methods using generalized Krylov subspaces 1University of Cagliari, Cagliari, Italy; 2Kent State University, Kent, Ohio
Regularization of certain linear discrete ill-posed problems, as well as of certain regression problems, can be formulated as large-scale, possibly nonconvex, minimization problems, whose objective function is the sum of the $p$-th power of the $\ell^p$-norm of a fidelity term and the $q$-th power of the $\ell^q$-norm of a regularization term, with $0< p,q \leq 2$. We describe new restarted iterative solution methods that require less computer storage and execution time than the methods described by [1]. The reduction in computer storage and execution time is achieved by periodic restarts of the method. Computed examples illustrate that restarting does not reduce the quality of the computed solutions.
[1] G.-X. Huang, A. Lanza, S. Morigi, L. Reichel and F. Sgallari. Majorization–minimization generalized Krylov subspace methods for $\ell_p-\ell_q$ optimization applied to image restoration, BIT Numerical Mathematics 57: 351-378, 2017.
A high order PDE-constrained optimization for the image denoising problem 1University Sultan Moulay Slimane, Morocco; 2University Ibn Zohr, Morroco
In the present work, we investigate the inverse problem of identifying simultaneously the denoised image and the weighting parameter that controls the balance between two diffusion operators for an evolutionary partial differential equation (PDE). The problem is formulated as a non-smooth PDE-constrained optimization model. This PDE is constructed by second- and fourth-order diffusive tensors that combines the benefits from the diffusion model of Perona-Malik in the homogeneous regions, the Weickert model near sharp edges and the fourth-order term in reducing staircasing. The existence and uniqueness of solutions for the proposed PDE-constrained optimization
system are provided in a suitable Sobolev space. Also, an optimization problem for the determination of the weighting parameter is introduced based on the Primal-Dual algorithm. Finally, simulation results show that the obtained parameter usually coincides with the better choice related to the best restoration quality of the image.
A primal dual projection algorithm for efficient constraint preconditioning 1Universität Bayreuth, Germany; 2Zuse Institute Berlin, Germany
We consider a linear iterative solver for large scale linearly constrained quadratic minimization problems that arise, for example, in optimization with partial differential equations (PDEs). By a primal-dual projection (PDP) iteration, which can be interpreted and analysed as a gradient method on a quotient space, the given problem can be solved by computing sulutions for a sequence of constrained surrogate problems, projections onto the feasible subspaces, and Lagrange multiplier updates. As a major application we consider a class of optimization problems with PDEs, where PDP can be applied together with a projected cg method using a block triangular constraint preconditioner. Numerical experiments show reliable and competitive performance for an optimal control problem in elasticity.
An Inexact Trust-Region Algorithm for Nonsmooth Nonconvex Optimization Sandia National Laboratories, United States of America
In this talk, we develop a new trust-region method to minimize the sum of a smooth nonconvex function and a nonsmooth convex function. Our method is unique in that it permits and systematically controls the use of inexact objective function and derivative evaluations. We prove global convergence of our method in Hilbert space and analyze the worst-case complexity to reach a prescribed tolerance. Our method employs the proximal mapping of the nonsmooth objective function and is simple to implement. Moreover, when using a quadratic Taylor model, our algorithm represents a matrix-free proximal Newton-type method that permits indefinite Hessians. We additionally elaborate on potential trust-region sub-problem solvers and discuss local convergence guarantees. We demonstrate the efficacy of our algorithm on various examples from PDE-constrained optimization.
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| 4:00pm - 6:00pm | MS14 2: Inverse Modelling with Applications Location: VG1.104 Session Chair: Daniel Lesnic Session Chair: Karel Van Bockstal |
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Boundary identification in cantilever beam equation related to the atomic force microscopy 1University of Malta, Malta; 2Kocaeli University, Turkey; 3University of São Paulo, Brasil
In this work, identification of the shear force in the Atomic Force Microscopy cantilever tip-sample interaction is considered. This interaction is governed by the following dynamic Euler-Bernoulli beam equation.
$$
\left\{ \begin{array}{ll}
\rho_A(x) u_{tt}+\mu(x)u_{t}+ (r(x)u_{xx}+\kappa(x)u_{xxt})_{xx} =0,\, (x,t)\in \Omega_{T},\\ [1pt]
u(x,0)=u_{t}(x,0)=0, ~x \in (0,\ell ), \\ [1pt]
u(0,t)=u_x(0,t)=0,~ \left (r(x)u_{xx}+\kappa(x)u_{xxt}\right)_{x=\ell}=M(t),\\
\qquad \qquad \qquad \left (-(r(x)u_{xx}+\kappa(x)u_{xxt})_x \right )_{x=\ell}=g(t),~t \in [0,T],
\end{array} \right.
$$
where the momentum $M(t)$ is correlated with the transverse shear force $g(t)$ by a certain formula. For the identification of $g(t)$ the deflection on the right hand tip is used as an measured data to minimize the corresponding objective functional by an explicit gradient formula. As a next step, the conjugate gradient algorithm (CGA) is designed for the reconstruction process to have numerical solution of the considered inverse problem. This algorithm is based on the weak solution theory, adjoint problem approach and method of lines combined with Hermite finite elements. Computational results, obtained for noisy output data, are illustrated to show an efficiency and accuracy of the proposed approach, for typical classes of shear force functions with realistic problem parameters.
Recent developments on integral equation approaches for Electrical Impedance Tomography University of Malta, Malta
The talk is focused on recent developments of reconstruction algorithms that can be used to approximate admittivity distributions in Electrical Impedance Tomography. The algorithms are non-iterative and are based on linearized integral equation formulations [1,2] which have been extended to reconstruct the conductivity and/or permittivity distributions of two and three-dimensional domains from boundary measurements of both low and high-frequency alternating input currents and induced potentials [3]. The linearized approaches rely on the solutions to the Laplace equation on a disk and a hemispherical domain subject to appropriate idealized Neumann boundary conditions corresponding to applied spatial varying trigonometric current patterns. Reconstructions from noisy simulated data are obtained from single-time, time-difference and multiple-times data. Moreover, a proposed design of a prototype for a novel integrated circuit based electrical impedance mammographic system embedded in a brassiere will be presented.
[1] C. Hähnlein, K. Schilcher, C. Sebu, H. Spiesberger. Conductivity imaging with interior potential measurements, Inverse Problems in Science and Engineering 19(5): 729-750, 2011.
[2] K-H. Georgi, C. Hähnlein, K. Schilcher, C. Sebu, H. Spiesberger. Conductivity Reconstructions Using Real Data from a New Planar Electrical Impedance Device, Inverse Problems in Science and Engineering, Inverse Problems in Science and Engineering 21(5): 801-822, 2013.
[3] C. Sebu, A. Amaira, J. Curmi. A linearized integral equation reconstruction method of admittivity distributions using Electrical Impedance Tomography, Engineering Analysis with Boundary Elements 150: 103-110, 2023.
Inverse problem of determining time-dependent diffusion coefficient in the time-fractional heat equation 1Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan; 2Ghent University, Belgium
Let $\mathcal{H}$ be a separable Hilbert space
and let $\mathcal{L}$ be operator with a discrete spectrum on $\mathcal{H}.$ For
$$\begin{cases}
&\mathcal{D}_t^\alpha u(t)+a(t)\mathcal{L}u(t)=f(t) \,\, \hbox{in} \,\, \mathcal{H},\,0<t \leq T,\\
&u (0) =h \; \text{in}\; \mathcal{H},
\end{cases}\;\;\;\;(1)
$$
we study
Coefficient inverse problem: Given $f(t), h$ and $E(t), $ find a pair of functions $\{a(t),u(t)\}$ satisfying the problem (1) and the additional condition
$$
F[u(t)]=E(t),\; t\in[0,T],
$$
where $F$ is the linear bounded functional.
As for this kind of inverse problem for parabolic equation, see [1].
Under suitable restrictions on the given data, we prove the existence, uniqueness and continuous dependence of the solution on the data.
[1] Z. Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation, Inverse Problems and Imaging. 11: 875–900, 2017.
Determining an Iwatsuka Hamiltonian through quantum velocity measurement Aix Marseille université, France
The main purpose of this talk is to explain how quantum currents induced by either the classical or the time-fractional Schrödinger equation, associated with an Iwatsuka Hamiltonian, uniquely determine the magnetic potential.
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| 4:00pm - 6:00pm | MS15 2: Experimental and Algorithmic Progress in Photoemission Orbital Imaging Location: VG1.102 Session Chair: Russell Luke Session Chair: Stefan Mathias |
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A minimalist approach to 3D photoemission orbital tomography: how many measurements are enough? University of Göttingen, Germany
Photoemission orbital tomography provides a unique access to the real-space molecular orbitals of well-ordered organic semiconductor layers. Specifically, the application of phase retrieval algorithms to photon-energy- and angle-resolved photoemission patterns enables the reconstruction of full 3D molecular orbitals independent of density functional theory calculations. However, until now this procedure has remained challenging due to the need for densely-sampled, well-calibrated 3D photoemission data. Here, we present an iterative projection algorithm that completely eliminates this challenge: For the benchmark case of the Pentacene frontier orbitals, we demonstrate reconstruction of the full orbital based on a data set containing only seven photoemission momentum maps. Based upon application to simulated data, we discuss the algorithm performance, sampling requirements with respect to the photon energy, optimal measurement strategies and the accuracy of orbital images that can be achieved.
Experimental progress towards time-resolved three-dimensional orbital tomography 1I. Physikalisches Institut, Georg-August-Universität Göttingen; 2Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen
Photoemission Orbital Tomography (POT) is a powerful tool for probing the full electronic structure of oriented molecular thin films which allows a direct comparison of angle-resolved photoemission spectroscopy (ARPES) data with density functional theory calculations. Moreover, the application of numerical phase retrieval algorithms in the POT framework has enabled a complete recovery of the initial molecular orbital independent of theoretical calculations [1, 2]. In combination with ultrafast pump-probe spectroscopy this approach promises to image excited state wavefunctions with Angstrom-level spatial and femtosecond temporal resolution.
However, most POT experiments to date have been restricted to a single probe photon energy, providing only a two-dimensional view of the initial wavefunction. This has limited the access to the full three-dimensional wavefunction to specialized, synchrotron-based facilities, where the implementation of femtosecond time-resolved experiments is challenging. At the same time, a time-resolved access to the third dimension is highly desirable, as it would enable the study of light-induced charge-transfer processes at hybrid molecular interfaces.
We overcome this limitation by implementing an EUV monochromator into the existing high harmonic generation beamline of our femtosecond photoemission setup with the ultimate goal of performing time-resolved three-dimensional orbital tomography. In this talk, I will report on our newly built-up setup and present our first energy-dependent photoemission data of molecules.
[1] P. Puschnig, S. Berkebile, A.J. Fleming, G. Koller, K. Emtsev, T. Seyller, J.D. Riley, C. Ambrosch-Draxl, F.P. Netzer, M.G. Ramsey. Reconstruction of Molecular Orbital Densities from Photoemission Data, Science 326: 702-706, 2009.
[2] G.S.M. Jansen, M. Keunecke, M. Düvel, C. Möller, D. Schmitt, W. Bennecke, F.J.S. Kappert, D. Steil, D.R. Luke, S. Steil, S. Mathia. Efficient orbital imaging based on ultrafast momentum microscopy and sparsity-driven phase retrieval, New J. Phys. 22, 2020.
Element-Selective Structural Information by Hard X-ray Photoelectron Diffraction Johannes Gutenberg-University Mainz, Germany
X-ray photoelectron diffraction (XPD) is a powerful technique that yields detailed structural information of solids and thin films that complements electronic structure measurements. Among the strongholds of XPD we can identify dopant sites, track structural phase transitions, and perform holographic reconstruction. High-resolution imaging of momentum-distributions (momentum microscopy) presents a new approach to core-level photoemission. It yields full-field XPD patterns with unprecedented acquisition speed and richness in details. Beyond the pure intensity-related diffraction information, XPD patterns exhibit pronounced circular dichroism in the angular distribution (CDAD) with asymmetries up to 80%. Experimental results for a number of examples prove that core-level CDAD is a general phenomenon that is independent of atomic number. Calculations using both the Bloch-wave approach and one-step photoemission reveal the origin of the fine structure that represents the signature of Kikuchi diffraction. Comparison to theory allow to disentangle the roles of photoexcitation and diffraction.
[1] O. Fedchenko, A. Winkelmann, K. Medjanik, S. Babenkov, D. Vasilyev, S. Chernov, C. Schlueter, A. Gloskovskii, Yu. Matveyev, W. Drube, B. Schönhense, H.J. Elmers, G. Schönhense. High-resolution hard-x-ray photoelectron diffraction in a momentum microscope – The model case of graphite, New. J. Phys. 21, 2019.
[2] K. Medjanik, O. Fedchenko, O. Yastrubchak, J. Sadowski, M. Sawicki, L. Gluba, D. Vasilyev, S.
Babenkov, S. Chernov, A. Winkelmann, H.J. Elmers, G. Schönhense. Site-specific atomic order and band structure tailoring in the diluted magnetic semiconductor (In, Ga, Mn) As, Phys. Rev. B 103, 2021.
Imaging molecular wave functions with photoemission orbital tomography: Recent developments University of Graz, Austria
This contribution will concentrate on three recent applications of photoemission orbital tomography (POT). First, results of an on-surface synthesized molecular layer will be presented, show-casing how the imaging of molecular orbitals using POT sheds light on surface chemical reactions. Second, it will be demonstrated how POT can be generalized to extended two-dimensional systems. On the example of a strongly-hybridising molecular overlayer on a Cu(110) surface, deep insights into the complicated interplay of bulk states, surface states, and molecular orbitals can be gained from the orbital imaging. Finally, experimental and theoretical results for monolayer graphene will be presented. Here, the photon energy dependence of photoemission intensities indicate limitations of the plane-wave final state approximation which is at the heart of POT. Validated by real-time time-dependent density functional calculations, we develop a simple and intuitive model which accounts for final state scattering, which should allow for the inversion of experimental data to real-space orbital images, thereby going beyond the plane-wave paradigm of POT.
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| 4:00pm - 6:00pm | MS18 2: Inverse problems for fractional and nonlocal equations Location: VG1.103 Session Chair: Yi-Hsuan Lin Session Chair: Jesse Railo Session Chair: Mikko Salo |
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The Calderón problem for directionally antilocal operators Universitat de Barcelona, Spain
The Calderón problem for the fractional Schrödinger equation, introduced by T. Ghosh, M. Salo and G. Uhlmann, satisfies global uniqueness with only one single measurement. This result exploits the antilocality property of the fractional Laplacian, that is, if a function and its fractional Laplacian vanish in a subset, then the function is zero everywhere.
Nonlocal operators which only see the functions in some directions and not on the whole space cannot satisfy an analogous antilocality property. In theses cases, only directional antilocality conditions may be expected.
In this talk, we will consider antilocality in cones, introduced by Y. Ishikawa in the 80s, and its possible implications in the corresponding Calderón problem. In particular, we will see that uniqueness for the associated Calderón problem holds even with a singe measurement, but new geometric conditions are required.
This is a joint work with G. Covi and A. Rüland.
An Inverse Problem for Nonlinear Fractional Magnetic Schrodinger Equation University of Minnesota, United States of America
The study of nonlinear equations arises in many physical phenomena and is an active direction in the field of inverse problems. In this talk, we will discuss inverse problems for the fractional magnetic Schrodinger equation with nonlinear potential and address the crucial techniques that are applied to reconstruct the unknown coefficient from measurements.
Properties of solutions for anisotropic viscoelastic systems 1Rice Univ., U.S.; 2Kanazawa Univ., Japan; 3NCKU, Taiwan; 4Hokkaido Univ., Japan
In this talk I will consider two kinds of anisotropic viscoelastic systems. One is an anisotropic viscoelastic systems which is described as an integro-differential system (ID system) and the other is the so-called extended Maxwell model (EM system) which is schematically described using springs and dashpots. There is a relation between them but they are different systems. I will discuss about their relation and the large time behavior of their solutions. Further, for the EM system, I will discuss about a generation of semigroup and the limiting amplitude principle. The method of proof for the ID system is an energy estimate, and that for the EM system is mainly based on several resolvent estimates. The limiting amplitude principle could be useful for setting up the inverse problem when the measurements may use a sequence of several different time harmonic inputs such as the magnetic resonance elastography.
This is a joint work with Maarten de Hoop, Ching-Lung Lin for the EM system and also including Masato Kimura for the EM system.
Fractional anisotropic Calderon problem on Riemannian manifolds University of California, Irvine, United States of America
We shall discuss some recent progress on the fractional anisotropic Calderon problem on closed Riemannian manifolds of dimensions two and higher. Specifically, we show that the knowledge of the local source-to-solution map for the fractional Laplacian, given on an arbitrary small open nonempty a priori known subset of a smooth closed Riemannian manifold, determines the Riemannian manifold up to an isometry. This can be viewed as a nonlocal analog of the anisotropic Calderon problem in the setting of closed Riemannian manifolds, which is wide open in dimensions three and higher. This is joint work with Ali Feizmohammadi, Tuhin Ghosh, and Gunther Uhlmann.
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| 4:00pm - 6:00pm | MS25 2: Hyperparameter estimation in imaging inverse problems: recent advances on optimisation-based, learning and statistical approaches Location: VG0.110 Session Chair: Luca Calatroni Session Chair: Monica Pragliola |
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Learning a sparsity-promoting regularizer for linear inverse problems 1Università degli Studi di Bologna, Italy; 2Università degli Studi di Genova, Italy; 3Lappeenranta-Lahti University of Technology; 4University of Helsinki
Variational regularization is a well-established technique to tackle instability in inverse problems, and it requires solving a minimization problem in which a mismatch functional is endowed with a suitable regularization term. The choice of such a functional is a crucial task, and it usually relies on theoretical suggestions as well as a priori information on the desired solution.
A promising approach to this task is provided by data-driven strategies, based on statistical learning: supposing that the exact solution and the measurements are distributed according to a joint probability distribution, which is partially known thanks to a training sample, we can take advantage of this statistical model to design regularization operators.
In this talk, I will present a hybrid approach, which first assumes that the desired regularizer belongs to a class of operators (suitably described by a set of parameters) and then learns the optimal one within the class. In the context of linear inverse problems, I will first briefly recap the main results obtained for the family of generalized Tikhonov regularizers: a characterization of the optimal regularizer, and two learning-based techniques to approximate it, with guaranteed error estimates.
Then, I will focus on a class of sparsity-promotion regularizers, which essentially leads to the task of learning a sparsifying transform for the considered data. Also in this case, it is possible to deduce theoretical error bounds between the optimal regularizer and its supervised-learning approximation as the size of the training sample grows.
Noise Estimation via Tractable Diffusion 1Graz University of Technology, Austria; 2Universitätsklinikum Bonn, Germany; 3Université Paris-Dauphine-PSL, France
Diffusion models have recently received significant interest in the imaging sciences.
After achieving impressive results in image generation, focus has shifted towards finding ways to exploit the encoded prior knowledge in classical inverse problems.
In this talk, we highlight another intriguing viewpoint:
Instead of focusing on image reconstruction, we propose to use tractable diffusion models which also allow to estimate the noise in an image.
In particular, we utilize a fields-of-experts-type model with Gaussian mixture experts that admits an analytic expression for a normalized density under diffusion, and can be trained with empirical Bayes.
The normalized model can be used for noise estimation of a given image by maximizing it w.r.t. diffusion time, and simultaneously gives a Bayesian least-squares estimator for the clean image.
We show results on denoising problems and propose possible applications to more involved inverse problems.
Speckle noise removal via learned variational models University of Naples Federico II, Italy
In this talk, we address the image denoising problem in presence of speckle degradation typically arising in ultra-sound images. Variational methods and Convolutional Neural Networks (CNNs) are considered well-established methods for specific noise types, such as Gaussian and Poisson noise. Nonetheless, the advances achieved by these two classes of strategies are limited when tackling the de-speckle problem. In fact, variational methods for speckle removal typically amounts to solve a non-convex functional with the related issues from the convergence viewpoint; on the other hand, the lack of large datasets of noise-free ultra-sound images has not allowed the extension of the state-of-the-art CNN denoiser methods to the case of speckle degradation. Here, we aim at combining the classical variational methods with the predictive properties of CNNs by considering a weighted total variation regularized model; the local weights are obtained as the output of a statistically inspired neural network that is trained on a small and composite dataset of natural and synthetic images. The resulting non-convex variational model, which is minimized by means of the Alternating Direction Method of Multipliers (ADMM) is proven to converge to a stationary point. Numerical tests show the effectiveness of our approach for the denoising of natural and satellite images.
Bayesian sparse optimization for dictionary learning Case Western Reserve University, United States of America
Dictionary learning methods have been used in recent years to solve inverse problems without using the forward model in the traditional optimization algorithms. When the dictionaries are large, and a sparse representation of the data in terms of the atoms is desired, computationally efficient sparse optimization algorithms are needed. Furthermore, reduced dictionaries can represent the data only up to a model reduction error, and Bayesian methods for estimating modeling errors turn out to be useful in this context. In this talk, the ideas of using Bayesian hierarchical models and modeling error methods are discussed.
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| 4:00pm - 6:00pm | MS29 2: Eigenvalues in inverse scattering Location: VG3.104 Session Chair: Martin Halla Session Chair: Peter Monk |
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A duality between scattering poles and transmission eigenvalues in scattering theory Rutgers University, United States of America
Spectral properties of operators associated with scattering phenomena carry essential information about the scattering media. The theory of scattering resonances is a rich and beautiful part of scattering theory and, although the notion of resonances is intrinsically dynamical, an elegant mathematical formulation comes from considering them as the poles of the meromorphic extension of the scattering operator. The scattering poles exist and they are complex with negative imaginary part. They capture physical information by identifying the rate of oscillations with the real part of a pole and the rate of decay with its imaginary part. At a scattering pole, there is a non-zero scattered field in the absence of the incident field. On the flip side of this characterization of the scattering poles one could ask if there are frequencies for which there exists an incident field that doesn’t scatterer by the scattering object. The answer to this question for scattering by inhomogeneous media leads to the introduction of transmission eigenvalues.
In this talk we discuss a conceptually unified approach for characterizing and determining scattering poles and transmission eigenvalues for the scattering problem for inhomogenous media. Our approach explores a duality stemming from interchanging the roles of incident and scattered fields in our analysis. Both sets are related to the kernel of the relative scattering operator mapping incident fields to scattered fields, corresponding to the exterior scattering problem for the transmission eigenvalues, and the interior scattering problem for scattering poles.
Prolate eigensystem and its application in Born inverse scattering Chinese Academy of Sciences, China, People's Republic of
This talk is concerned with the generalized prolate spheroidal wave functions/eigenvalues (in short prolate eigensystem) and their application in two dimensional Born inverse medium scattering problems. The prolate eigenfunctions are eigenfunctions of a Fourier integral operator; they remarkably extend analytically to the whole space, are doubly orthogonal, and are complete in the class of band-limited functions. We first establish a Picard criterion for reconstructing the contrast using the prolate eigensystem, where the reconstruction formula can also be understood in the viewpoint of data processing and analytic extrapolation. Another salient feature associated with the generalized prolate spheroidal wave functions is that the prolate basis for a disk is also a basis for a Sturm-Liouville differential operator. With the help of Sturm-Liouville theory, we estimate the $L^2$ approximation error for a spectral cutoff approximation of $H^s$ functions, $0<s\le1$. This yields a spectral cutoff regularization strategy for noisy data and an explicit stability estimate for contrast in $H^s$ ($0<s\le1$) in the full aperture case. In the limited-aperture and multi-frequency cases, we also obtain spectral cutoff regularization strategies for noisy data and stability estimates for a class of contrast. Numerical examples are currently being investigated and a few preliminary examples are provided to illustrate the application of prolate eigensystem in inverse scattering problems.
Scattering from corners and other singularities LUT University, Finland
I will present a summary of my and my collaborators' work on fixed wavenumber scattering from corners and other geometric shapes of interest from the past 10 years. Our early work showed that in potential scattering, corners produce patterns in the far-field which cannot be cancelled by any other structure nearby or far away. This led to interesting finds such as unique shape determination of polyhedral or pixelated scattering potentials by the far-field made by any single incident wave. It also led to the study of how geometry of the domain affects the distribution of energy of the transmission eigenfunctions. Complete understanding is still away, and different geometrical configurations are being studied. In this talk I present shortly past results and also newer results related to general conical singularities and scattering screens.
The inverse spectral problem for a spherically symmetric refractive index using modified transmission eigenvalues National Technical University of Athens
In recent years, the classic transmission eigenvalue problem has risen in importance in inverse scattering theory. In this work, we discuss the introduction of a modification that corresponds to an artificial metamaterial background [1] and pose the inverse problem for determining a spherically symmetric refractive index from these modified eigenvalues. We show that uniqueness can be established under some assumptions for the magnitude of a fixed wavenumber and the unknown refractive index [2].
[1] D. Gintides, N. Pallikarakis, K. Stratouras. On the modified transmission eigenvalue problem with an artificial metamaterial background, Res. Math. Sci. 8, 2021.
[2] D. Gintides, N. Pallikarakis, K. Stratouras. Uniqueness of a spherically symmetric
refractive index from modified transmission eigenvalues, Inverse Problems 38, 2022.
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| 4:00pm - 6:00pm | MS34 2: Learned reconstructions for nonlinear inverse problems Location: VG3.103 Session Chair: Simon Robert Arridge Session Chair: Andreas Selmar Hauptmann |
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Learned iterative model-based approaches in quantitative photoacoustic tomography 1University of Oulu, Finland; 2Centrum Wiskunde & Informatica
Quantitative photoacoustic tomography (QPAT) is an upsurging imaging modality which can provide high-resolution tissue images based on optical absorption. Classical reconstruction methods rely on sufficient prior information to overcome noisy and imperfect data. As these methods utilise computationally expensive forward models, the computation becomes slow, delimiting the possibilities of QPAT in time-critical applications. As an alternative approach, deep learning-based reconstruction methods have been proposed to allow fast computation of accurate reconstructions. In our work, we adopt the model-based learned iterative approach to solve the nonlinear optical problem of QPAT. In the learned iterative model-based approach, the forward operator and its derivative are iteratively evaluated to compute an update step direction, which is then fed to the network. The learning task is formulated as greedy, requiring iterate-wise optimality, as well as in an end-to-end manner, where all updating networks are trained jointly. We formulated these training schemes and evaluated the performances when the step direction was computed with gradient descent and with the Gauss-Newton method.
Autocorrelation analysis for cryo-EM with sparsifying priors Princeton University, United States of America
Cryo-electron microscopy is a non-linear inverse problem that aims to reconstruct 3-D molecular structures from randomly oriented tomographic projection images, taken at extremely low signal-to-noise-ratio.
This talk presents new results for using the method of moments to reconstruct sparse molecular structures. We prove that molecular structures modeled as sparse sums of Gaussians can be uniquely recovered from the autocorrelations of the images, which significantly lowers the sample complexity of the problem compared to previous results. Moreover, we provide practical reconstruction algorithms inspired by crystallographic phase retrieval.
The full reconstruction pipeline includes estimating autocorrelations from projection images, using rotation-invariant principal component analysis made possible by recent improvements to approximation algorithms into the Fourier-Bessel basis.
Model corrections in linear and nonlinear inverse problems 1University of Oulu, Finland; 2University College London, UK
Solving inverse problems in a variational formulation requires repeated evaluation of the forward operator and its derivative. This can lead to a severe computational burden, especially so for nonlinear inverse problems, where the derivative has to be recomputed at every iteration. This motivates the use of faster approximate models to make computations feasible, but due to an arising approximation error the need to introduce a designated correction arises.
In this talk we first discuss the concept of learned model corrections applied to linear inverse problems, when computationally fast but approximate forward models are used. We then proceed to examine the possibility to approximate nonlinear models with a linear one and then solve the linear problem instead, avoiding differentiation of the nonlinear model. To correct for the arising approximation errors, we sequentially estimate the error between linear and nonlinear model and update a correction term in the variational formulation. In both cases we discuss convergence properties to solutions of the variational problem given the accurate models.
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| 4:00pm - 6:00pm | MS35: Edge-preserving uncertainty quantification for imaging Location: VG2.105 Session Chair: Amal Mohammed A Alghamdi Session Chair: Jakob Sauer Jørgensen |
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Efficient Bayesian computation for low-photon imaging problems 1Heriot-Watt University, United Kingdom; 2University of Edinburgh, United Kingdom
This talk presents a new and highly efficient MCMC methodology to perform Bayesian inference in low-photon imaging problems, with particular attention to situations involving observation noise processes that deviate significantly from Gaussian noise, such as binomial, geometric and low-intensity Poisson noise. These problems are challenging for many reasons. From an inferential viewpoint, low photon numbers lead to severe identifiability issues, poor stability and high uncertainty about the solution. Moreover, low-photon models often exhibit poor regularity properties that make efficient Bayesian computation difficult; e.g., hard non-negativity constraints, non-smooth priors, and log-likelihood terms with exploding gradients. More precisely, the lack of suitable regularity properties hinders the use of state-of-the-art Monte Carlo methods based on numerical approximations of the Langevin stochastic differential equation (SDE) or other similar dynamics, as both the continuous-time process and its numerical approximations behave poorly. We address this difficulty by proposing an MCMC methodology based on a reflected and regularised Langevin SDE, which is shown to be well-posed and exponentially ergodic under mild and easily verifiable conditions. This then allows us to derive four reflected proximal Langevin MCMC algorithms to perform Bayesian computation in low-photon imaging problems. The proposed approach is illustrated with a range of experiments related to image deblurring, denoising, and inpainting under binomial, geometric and Poisson noise.
Advancements of $\alpha$-stable priors for Bayesian inverse problems 1Heriot Watt University, United Kingdom; 2LUT, Finland
In this talk, we will summarize the recent advacements made for non-Gaussian
process priors for statistical inversion. This will be primarily focused on $\alpha$-stable
distributions which provide a natural generalization of a family of distributions, such as the normal and Cauchy. We discuss recently proposed priors which include various Cauchy priors, hierarchical and neural-network based $\alpha$-stable priors. The focus will be computational where we demonstrate their gains on a range of examples for fully Bayesian and MAP-based estimation. We also provide some theoretical insights which include error bounds.
Edge preserving Random Tree Besov Priors 1Delft University of Technology, Netherlands; 2University of Helsinki, Finland
Gaussian process priors are often used in practice due to their fast computational properties. The smoothness of the resulting estimates, however, is not well suited for modelling functions with sharp changes. We propose a new prior that has same kind of good edge-preserving properties than total variation or Mumford-Shah but correspond to a well-defined infinite dimensional random variable. This is done by introducing a new random variable $T$ that takes values in the space of ‘trees’, and which is chosen so that the realisations have jumps only on a small set.
CUQIpy - Computational Uncertainty Quantification for Inverse problems in Python Technical University of Denmark (DTU), Denmark
In this talk we present CUQIpy (pronounced ”cookie pie”) - a new computational modelling environment in Python that uses uncertainty quantification (UQ) to access and quantify the uncertainties in solutions to inverse problems. The overall goal of the software package is to allow both expert and non-expert (without deep knowledge of statistics and UQ) users to perform UQ related analysis of their inverse problem while focusing on the modelling aspects. To achieve this goal the package utilizes state-of-the-art tools and methods in statistics and scientific computing specifically tuned to the ill-posed and often large-scale nature of inverse problems to make UQ feasible. We showcase the software on problems relevant to imaging science such as computed tomography and partial differential equation-based inverse problems. CUQIpy is developed as part of the CUQI project at the Technical University of Denmark and is available at https://github.com/CUQI-DTU/CUQIpy.
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| 4:00pm - 6:00pm | MS43 2: Inverse Problems in radiation protection and nuclear safety Location: VG1.108 Session Chair: Lorenz Kuger Session Chair: Samuli Siltanen |
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Gamma spectrum analysis in nuclear decommissioning 1Friedrich-Alexander-Universität Erlangen-Nürnberg; 2Deutsches Elektronen-Synchrotron DESY; 3Universität Hamburg
In the radiological characterisation of nuclear power stations, gamma spectroscopy builds the basis for many further investigation methods. The measurements of scintillation detectors in, e.g. a Compton camera, can be used to identify a priori the present radioactive nuclides. We formulate this gamma spectrum analysis problem as a Bayesian inverse problem with Poisson-distributed data. Techniques from convex analysis are used to compute the resulting maximum likelihood estimator given by a list of present nuclides and their corresponding intensities. The approach is tested on coincidence data measured with a Compton camera in potential use cases.
Practical gamma ray imaging with monolithic scintillation detector Compton cameras 1FAU Erlangen-Nürnberg, Germany; 2Deutsches Elektronen-Synchrotron DESY, Germany; 3Universität Hamburg, Germany
Compton cameras are stationary, uncollimated gamma ray imaging devices that use the Compton effect to reconstruct a spatially resolved activity distribution. Due to the missing collimation, the cameras exhibit high sensitivities, particularly so for setups with large detectors in close distance of each other. This allows reconstruction in relatively low-count regimes and hence a flexible application in areas of low activity. For cameras with spatially non-resolved detectors, the size of the detectors however results in large angular uncertainty and distorted measurements due to multiple scattering contributions in the data. In this talk, we address the design and corresponding modeling approaches of Compton cameras with such monolithic, spatially non-resolved scintillation detectors. Numerical results on measured data support the theoretical considerations.
Machine Learning Techniques applied to Compton Cameras Hellma Materials GmbH, Germany
Compton cameras have a long tradition in $\gamma$-ray astronomy and increasingly find new applications in radiation protection and nuclear safety. We have built three prototypes of Compton cameras to assist the decommissioning process of safely removing a nuclear facility from service and reducing residual radioactivity to permissible levels. In this talk, the inverse problem of Compton cameras is addressed with two techniques, one based on a Bayesian framework, and another graph-based approach that makes full use of the discrete nature of ionizing radiation interactions with matter. We have implemented a new physics concept in our Compton cameras whereby we no longer label radiation detectors according to their function as either scattering or absorbing detectors but rather characterize them by their materials, most importantly their effective atomic number $Z_\text{eff}$. As essentially no detector exists that would exclusively absorb radiation, we propose to record coincidence events between all pairs of detectors in which at least one detector material has $Z_\text{eff} > 30$. This new trigger condition includes coincidences of detector pairs where both materials have $Z_\text{eff} > 30$ which would traditionally be labeled absorbing detectors and would normally not be recorded by a Compton camera. These changes in the electronics setup of our Compton cameras yield an enlarged data sample available for subsequent inversion treatment.
We will present experimental results obtained with the relevance vector machine and a graph heuristic used for assigning coincidence events to emission points. The measurements were carried out at the radiation laboratory of the University of Applied Sciences Zittau/Görlitz. We gratefully acknowledge financial support by the Federal Ministry of Education and Research (BMBF) through the FORKA program under Grant No. 15S9431A-D.
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| 4:00pm - 6:00pm | MS44 2: Modelling in Earth and planetary sciences by data inversion at various scales Location: VG2.104 Session Chair: Christian Gerhards Session Chair: Volker Michel Session Chair: Frederik J Simons |
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Efficient Parameter Estimation of Sampled Random Fields 1Princeton University, United States of America; 2Queen Mary, University of London, UK; 3Imperial College, London, UK; 4Ecole Polytechnique Federale de Lausanne, Switzerland
Describing and classifying the statistical structure of topography and
bathymetry is of much interest across the geophysical
sciences. Oceanographers are interested in the roughness of seafloor
bathymetry as a parameter that can be linked to internal-wave
generation and mixing of ocean currents. Tectonicists are searching
for ways to link the shape and fracturing of the ocean floor to build
detailed models of the evolution of the ocean basins in a
plate-tectonic context. Geomorphologists are building time-dependent
models of the surface that benefit from sparsely parameterized
representations whose evolution can be described by differential
equations. Geophysicists seek access to parameterized forms for the
spectral shape of topographic or bathymetric loading at various
(sub)surface interfaces in order to use the joint structure of
topography and gravity for inversions for the effective elastic
thickness of the lithosphere. A unified geostatistical framework
involves the Matérn process, a theoretically well justified
parameterized form for the spectral-domain covariance of Gaussian
processes. We provide a computationally and statistically efficient
method for estimating the parameters of a stochastic covariance model
observed on a regular spatial grid in any number of dimensions. Our
proposed method makes important corrections to the well-known Whittle
likelihood to account for large sources of bias caused by boundary
effects and aliasing. We generalise the approach to flexibly allow for
significant volumes of missing data including those with
lower-dimensional substructure, and for irregular sampling
boundaries. We provide detailed implementation guidelines which
maintains the computational scalability of Fourier and Whittle-based
methods for large data sets.
Co-estimation of core and lithospheric signals in satellite magnetic data Technical University of Denmark, Denmark
Satellite observations of the geomagnetic field contain signals originating both from electrical currents in the core and from magnetized rocks in the lithosphere. At short wavelengths the lithospheric signal dominates, obscuring the signal from the core. Here we present details of a method to co-estimate separate models for the core and lithospheric fields, which are allowed to overlap in spherical harmonic degree, that makes use of prior information regarding the sources. Using a maximum entropy method, we estimate a time-dependent model of the core field together with a static model of the lithospheric field that satisfy the constraints provided by satellite observations as well as statistical prior information, but are otherwise maximally non-committal with regard to the distribution of radial magnetic field at the source surfaces. Tests based on synthetic data are encouraging, demonstrating it is possible to retrieve parts of the core field beyond degree 13 and the lithospheric field below degree 13. Results will be presented from our new model of the time-dependent core surface field up to spherical harmonic degree 30 and implications regarding our understanding of the core dynamo discussed.
Transdimensional joint inversion of gravity and surface wave phase velocities Kiel University, Germany
A fundamental choice for any geophysical inversion is the parametrization of the subsurface. Voxels and coefficients of basis functions (i.e., spherical harmonics) often are a natural choice, especially since they can simplify forward calculations. An alternative approach is to use a finite collection of discrete anomalies, which leads to transdimensional (TD) techniques, when considered through a Bayesian lens. The most popular form of TD inversion uses a variable number of Voronoi cells as spatial representation.
The TD approach addresses the issues of non-uniqueness and lack of resolution in a special way: Instead of smoothing or damping the solutions, the spatial structure of the model is controlled by weighing the number of elements against the achieved data fit. This gives it an intrinsic adaptive behaviour, useful for heterogeneous data coverage. Furthermore, geophysical sensitivity often changes with depth, which TD approaches can also adapt to. In a joint inversion context for several properties (like seismic velocities and densities), spatial coupling between different sought parameters is automatically guaranteed.
In this contribution I will present some examples for using TD inversions on global gravity and surface wave data to simultaneously determine the velocity and density structure within the Earth’s mantle.
The inverse problem of micromagnetic tomography in rock- and paleomagnetism Norwegian University of Science and Technology, Norway
The intrinsic non-uniqueness of potential-field inversion of surface scanning data can be circumvented by solving for the potential field of known individual source regions. A uniqueness theorem characterizes the mathematical background of the corresponding inversion problem, and determines when a potential-field measurement on a surface uniquely defines the magnetic potentials of the individual source regions. For scanning magnetometers in rock magnetism, this result implies that dipole magnetization vectors of many individual magnetic particles can be reconstructed from surface scans of the magnetic field. It is shown that finite sensor size still retains this conceptual uniqueness. The technique of micromagnetic tomography (MMT) combines X-ray micro computed tomography and scanning magnetometry to invert for the magnetic potential of individual magnetic grains within natural and synthetic samples. This provides a new pathway to study the remanent magnetization that carries information about the ancient
geomagnetic field and is the basis of all paleomagnetic studies. MMT infers the magnetic potential of individual grains by numerical inversion of surface magnetic measurements using spherical harmonic expansions. Because the full magnetic potential of the individual particles in principle is uniquely determined by MMT, not only the dipole but also more complex, higher order, multipole moments can be recovered. Even though a full reconstruction of complex magnetization structures inside the source minerals is mathematically impossible, these additional constraints by far-field multipole terms can substantially reduce the number of possible micromagnetic energy minima.
For complex particles with many micromagnetic energy minima it is possible to include the far-field constraints into the micromagnetic minimization algorithm.
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| 4:00pm - 6:00pm | MS52 2: Integral geometry, rigidity and geometric inverse problems Location: VG1.105 Session Chair: Francois Sylvain Monard Session Chair: Plamen Stefanov |
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Resonant forms at zero for dissipative Anosov flows 1University of Zurich, Switzerland; 2University of Cambridge, UK
The Ruelle Zeta Function of a chaotic (Anosov) flow is a meromorphic function in the complex plane defined as an infinite product over closed orbits. Its behaviour at zero is expected to carry interesting topological and dynamical information, and is encoded in certain resonant spaces of differential forms for the action of the Lie derivative on suitable spaces with anisotropic regularity. In this talk, I will introduce a new notion of helicity (average self-linking), and explain how this can be used to compute the resonant spaces for any Anosov flow in 3D, with particular emphasis in the dissipative (non volume-preserving) case. A prototype example is given by the geodesic flow of an affine connection with torsion and we shall see that for such a flow the order of vanishing drops by 1 in relation to the usual geodesic flow due to the Sinai-Ruelle-Bowen measure having non-zero winding cycle.
Ray transform problems arising from seismology University of Jyväskylä, Finland
Many different ray transform problems arise from seismology. My examples are periodic ray transform problems in the presence of interfaces, linearized travel time tomography in strong anisotropy, and a partial data problem originating from shear wave splitting. I will discuss the underlying inverse problems and the arising integral geometry problems. This talk is based on joint work with de Hoop, Katsnelson, and Mönkkönen.
X-ray mapping properties and degenerately elliptic operators 1Northwestern University; 2University of California, Santa Cruz; 3Indian Institute of Technology Gandhinagar
We discuss recent results regarding $C^\infty$-isomorphism properties of weighted normal operators of the X-ray transform on manifolds with boundary, in joint work [1] with Francois Monard and Rohit Kumar Mishra. The crux of the result depends on understanding the Singular Value Decomposition of weighted X-ray transforms/backprojection operators, which itself can be obtained via intertwining with certain degenerately elliptic differential operators. We also discuss recent work [2] with Francois Monard on developing tools to study such degenerately elliptic operators even further. Such tools include a scale of Sobolev spaces which take into account behavior up to the boundary, as well as generalizations of Dirichlet and Neumann traces called boundary triplets associated to degenerately elliptic operators which pick out the first and second most singular terms of a function near the boundary.
[1] R. Mishra, F. Monard, Y. Zou. The $C^\infty$-isomorphism property for a class of singularly-weighted X-ray transforms. Inverse Problems 39: 024001, 2023. https://doi.org/10.1088/1361-6420/aca8cb
[2] F. Monard, Y. Zou. Boundary triples for a family of degenerate elliptic operators of Keldysh type, arXiv: 2302.08133, 2023.
The range of the non-Abelian X-ray transform University of Bonn, Germany
We discuss a nonlinear analogue of the Pestov-Uhlmann range characterisation for geodesic X-ray transforms on simple surfaces. The transform under consideration takes as input matrix-valued and possibly direction dependent functions (which may encode magnetic fields or connections on a vector bundle) and outputs their 'scattering data' at the boundary. The range of this transform can be completely described in terms of boundary objects, and this description is reminiscent of the Ward correspondence for anti-self-dual Yang-Mills fields, but without solitonic degrees of freedom. The talk is based on joint work with Gabriel Paternain.
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| 4:00pm - 6:00pm | MS54 2: The x-ray transform and its generalizations: Theory, methods, and applications Location: VG1.101 Session Chair: Suman Kumar Sahoo |
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Inversion of the momenta X-ray transform of symmetric tensor fields in the plane Johann Radon Institute (RICAM), Austria
The X-ray transform of symmetric tensor fields recovers the tensor field only up to a potential field. In 1994, V. Sharafutdinov showed that augmenting the X-ray data with several momentum-ray transforms establishes uniqueness, with a most recent work showing stability of the inversion. In this talk, I will present a different approach to stably reconstruct symmetric tensor fields compactly supported in the plane.
The method is based on the extension of Bukhgeim's theory to a system of $A$-analytic maps.
This is joint work with H. Fujiwara, D. Omogbhe and A. Tamasan.
Simultaneous recovery of attenuation and source density in SPECT and multibang regularisation University of Manchester, United Kingdom
I will discuss results about simultaneous recovery of the attenuation and source density in the SPECT inverse problem, which is given mathematically by the attenuated ray transform. Assuming the attenuation is piecewise constant and the source density piecewise smooth we show that, provided certain conditions are satisfied, it is possible to uniquely determine both. I will also discuss a numerical algorithm that allows for determination of both parameters in the case when the range of the piecewise constant attenuation is known and look at some synthetic numerical examples. This is based on joint work with Philip Richardson.
Inversion of a restricted transverse ray transform on symmetric $m$-tensor fields in $\mathbb{R}^3$ Indian Institute of Technology Gandhinagar, India
In this work, we study a restricted transverse ray transform on symmetric $m$-tensor fields in $\mathbb{R}^3$ and provide an explicit inversion algorithm to recover the unknown $m$-tensor field. We restrict the transverse ray transform to all lines passing through a fixed curve $\gamma$ satisfying the Kirillov-Tuy condition. This restricted data is used to find the weighted Radon transform of components of the unknown tensor field, which we use to recover components of the tensor field explicitly.
Inverse problems, unique continuation and the fractional Laplacian University of Cambridge, United Kingdom
The Calderón problem is a famous nonlinear model inverse problem: Do voltage and current measurements on the boundary of an object determine its electric conductivity uniquely? X-ray computed tomography is a famous linear model inverse problem studied via Radon transforms. We discuss how the fractional Laplacians pop up in the analysis of Radon transforms. We then discuss recent results on the unique continuation of the fractional Laplacians and the related Caffarelli-Silvestre extension problem for $L^p$ functions. We explain some of the implications to the analysis of Radon transforms with partial data and its further generalizations. Finally, we discuss the role of unique continuation in recent mathematical studies of the Calderón problem to nonlocal equations.
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| 4:00pm - 6:00pm | MS55 2: Nonlinear Inverse Scattering and Related Topics Location: VG3.101 Session Chair: Yang Yang |
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Imaging with two-photon absorption optics Auburn University, United States of America
In this talk, we briefly talk about the inverse boundary/medium problems with the semilinear transport model which naturally rises from two-photon absorption optics. The model can be formally derived from a paraxial setting of a nonlinear absorption wave model. For the related inverse problems, we consider two cases. For the inverse boundary problem, we adopted the usual linearization technique and prove the uniqueness of reconstruction. For the inverse medium problems, we consider the problem from photoacoustic imaging specifically in static and time-dependent settings and prove the uniqueness of the reconstruction for the absorption coefficients.
Hopf lemma for fractional diffusion equations and application to inverse problem Ningbo University, China, People's Republic of
In this talk, we will discuss an inverse problem of determining the spatially dependent source term and the Robin boundary coefficient in a time-fractional diffusion equation, with the aid of extra measurement data at a subdomain near the accessible boundary. Firstly, the spatially varying source is uniquely determined in view of the unique continuation principle and Duhamel principle for the fractional diffusion equation. The Hopf lemma for a homogeneous time-fractional diffusion equation is proved and then used to prove the uniqueness of recovering the Robin boundary coefficient. Numerically, based on the theoretical uniqueness result, we apply the classical Tikhonov regularization method to transform the inverse problem into a minimization problem, which is solved by an iterative thresholding algorithm. Finally, several numerical examples are presented to show the accuracy and effectiveness of the proposed algorithm.
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| 4:00pm - 6:00pm | MS56 2: Inverse Problems of Transport Equations and Related Topics Location: VG2.106 Session Chair: Ru-Yu Lai Session Chair: Hanming Zhou |
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Reconstruction of the doping profile in Vlasov-Poisson 1Simon Fraser University, Canada; 2University of Minnesota, USA; 3University of Wisconsin Madison, USA
In this talk we show how the singular decomposition method can be applied to recover the doping profile in the Vlasov-Poisson equation.
Quantitative reconstructions in inverse transport problems Columbia University, United States of America
In many practical applications of inverse problems, the data measured contain unknown normalization constants due to the unknown strength of the illumination sources. In such cases, it is usually impossible to quantitatively reconstruct the coefficients of interest. We show, mainly computationally, that one can actually have quantitative reconstructions for some inverse transport problems where redundancy in data helps us to eliminate the unknown normalization constant encoded in the illumination sources.
Imaging with two-photon absorption optics Auburn University, United States of America
In this talk, we briefly talk about the inverse boundary/medium problems with the semilinear transport model which has a natural application in two-photon absorption optics. For the inverse boundary problem, we adopted the usual linearization technique and prove the uniqueness of reconstruction. For the inverse medium problems, we consider the problem from photoacoustic imaging specifically in static and time-dependent settings and prove the uniqueness of the reconstruction for the absorption coefficients.
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| 4:00pm - 6:00pm | MS57 2: Inverse Problems in Time-Domain Imaging at the Small Scales Location: VG3.102 Session Chair: Eric Bonnetier Session Chair: Xinlin Cao Session Chair: Mourad Sini |
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A mathematical theory of resolution limits for dynamic super-resolution in particle tracking problems ETH Zurich, Switzerland
Particle tracking in a live cell environment is concerned with reconstructing the trajectories, locations, or velocities of the targeting particles, which holds the promise of revealing important new biological insights. The standard approach of particle tracking consists of two steps: first reconstructing statically the source locations in each time step, and second applying tracking techniques to obtain the trajectories and velocities. In contrast to the standard approach, the dynamic reconstruction seeks to simultaneously recover the source locations and velocities from all frames, which enjoys certain advantages. In this talk, we will present a rigorous mathematical analysis for the resolution limit of reconstructing source number, locations, and velocities by general dynamical reconstruction in particle tracking problems, by which we demonstrate the possibility of achieving super-resolution for dynamic reconstruction. We show that when the location-velocity pairs of the particles are separated beyond certain distances (the resolution limits), the number of particles and the location-velocity pair can be stably recovered. The resolution limits are related to the cut-off frequency of the imaging system, signal-to-noise ratio, and the sparsity of the source. By these estimates, we also derive a stability result for a sparsity-promoting dynamic reconstruction. In addition, we further show that the reconstruction of velocities has a better resolution limit which improves constantly as the particles move. The result is derived from a crucial observation that the inherent cut-off frequency for the velocity recovery can be viewed as the cut-off frequency of the imaging system multiplied by the total observation time, which may lead to a better resolution limit than the one for each diffraction-limited frame. In addition, we propose super-resolution algorithms for recovering the number and values of the velocities in the tracking problem and demonstrate theoretically or numerically their super-resolution capability.
Heat Generation Using Lorentzian Nanoparticles. The Full Maxwell System 1Radon Institute (RICAM), Austrian Academy of Sciences, Austria; 2Johannes Keplar Universität Linz, Austria
We analyze and quantify the amount of heat generated by a nanoparticle, injected in a background
medium, while excited by incident electromagnetic waves. These nanoparticles are dispersive with
electric permittivity following the Lorentz model. The purpose is to determine the quantity of heat
generated extremely close to the nanoparticle (at a distance proportional to the radius of the
nanoparticle). We show that by exciting the medium with incident frequencies close to
the Plasmonic or Dielectric resonant frequencies, we can generate any desired amount of heat close
to the injected nanoparticle while the amount of heat decreases away from it. These results offer a
wide range of potential applications in the areas of photo-thermal therapy, drug delivery, and
material science, to cite a few.
To do so, we employ time-domain integral equations and asymptotic analysis techniques to study
the corresponding mathematical model for heat generation. This model is given by the heat
equation where the body source term comes from the modulus of the electric field generated by
the used incident electromagnetic field. Therefore, we first analyze the dominant term of this
electric field by studying the full Maxwell scattering problem in the presence of Plasmonic or All-dielectric
nanoparticles. As a second step, we analyze the propagation of this dominant electric
field in the estimation of the heat potential. For both the electromagnetic and parabolic models,
the presence of the nanoparticles is translated into the appearance of large scales in the contrasts
for the heat-conductivity (for the parabolic model) and the permittivity (for the full Maxwell
system) between the nanoparticle and its surrounding.
Lipschitz stability for some inverse problems for a hyperbolic PDE with space and time dependent coefficients 1TIFR CAM, Bangalore, India; 2RICAM, Austria; 3University of Delaware, USA
We study stability aspects for the determination of space and time-dependent lower order perturbations of the wave operator in three space dimensions with point sources. The problems under consideration here are formally determined and we establish Lipschitz stability results for these problems. The main tool in our analysis is a modified version of Bukgheĭm-Klibanov method based on Carleman estimates.
[1] V. P. Krishnan, S. Senapati, Rakesh. Stability for a formally determined inverse problem for a hyperbolic PDE with space and time dependent coefficients, SIAM J. Math. Anal. 53, no. 6, 6822–6846, 2021.
[2] V. P. Krishnan, S. Senapati, Rakesh. Point sources and stability for an inverse problem for a hyperbolic PDE with space and time dependent coefficients, J. Differential Equations 342, 622–665, 2023.
Scattering of electromagnetic waves by small obstacles EPI Makutu, Pau University, Inria, LMAP UMR CNRS 5142
We develop fast, accurate and efficient numerical methods for solving the time harmonic scattering problem of electromagnetic waves in 3D by a multitude of obstacles for low and medium frequencies. Taking into account a large number of heterogeneities can be costly in terms of computation time and memory usage, particularly in the construction process of the matrix. We consider a multi-scale diffraction problem in low-frequency regimes in which the characteristic length of the obstacles is small compared to the incident wavelength. We use the matched asymptotic expansion method which allows for the model reduction. Two types of approximations are distinguished : near-field or quasi-static approximations that descibe the phenomenon at the microscopic scale and far-field approximations that describe the phenomenon at a long distance. In the latter one, small obstacles are no longer considered as geometric constraintsand can be modelled by equivalent point-sources which are interpreted in terms of electromagnetic multipoles.
[1] J. Labat, V. Péron, S. Tordeux. Equivalent multipolar point-source modeling of small spheres for fast and accurate electromagnetic wave scattering computations, Wave Motion 92: 102409, 2020.
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