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: Tuesday, 05/Sept/2023 | |
| 9:00am - 10:00am | CP: Calderón Prizes Location: ZHG 011 Session Chair: Barbara Kaltenbacher |
| 10:00am - 10:50am | Pl 3: Plenary lecture Location: ZHG 011 Session Chair: Bastian Harrach |
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On the Fractional Calderon Problem University of Bonn, Germany
Inverse problem for prototypical nonlocal operators such as the fractional Laplacian display strikingly strong uniqueness, stability and single measurement results. These fundamentally rely on global variants of the unique continuation property for these nonlocal operators and dual flexibility properties in the form of Runge approximation results. In this talk, I introduce these properties and discuss some recent results on the relation between the classical and fractional Calderon problems.
This is based on joint work with G. Covi, T. Ghosh, M. Salo and G. Uhlmann.
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| 10:50am - 11:10am | C3: Coffee Break Location: ZHG Foyer |
| 11:10am - 12:00pm | Pl 4: Plenary lecture Location: ZHG 011 Session Chair: Mikko Salo |
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Microlocal applications to the study of marked length spectrum rigidity and lens rigidity in chaotic settings CNRS, France
The boundary rigidity problem asks if the boundary distance function on a simple Riemannian manifold determines the metric. For non simple manifolds with boundary or even for closed Riemannian manifolds, there are corresponding problems named lens rigidity and marked length spectrum rigidity. The general question is essentially reduced to knowing if a conjugacy between two geodesic flows on the unit tangent bundles necessarily comes from an isometry on the underlying Riemannian metrics.
The introduction of microlocal methods to understand the regularity of solutions of transport equations, of invariant distributions for the geodesic flow, has been key in the resolution of such problems that naturally arise when the geodesic flow is chaotic (hyperbolic). We will review recent results in this direction, in collaborations with Bonthonneau, Cekic, Jezequel, Lefeuvre and Paternain.
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| 12:00pm - 1:30pm | LB2: Lunch Break Location: Mensa |
| 1:30pm - 3:30pm | MS02 1: Advances in regularization for some classes of nonlinear inverse problems Location: VG1.102 Session Chair: Bernd Hofmann Session Chair: Robert Plato |
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Deautoconvolution in the two-dimensional case 1Chemnitz University of Technology, Germany; 2University of Würzburg
There is extensive mathematical literature on the inverse problem of deautoconvolution for a function with support in the unit interval $[0,1] \subset \mathcal{R}$, but little is known about the multidimensional situation. We try to fill this gap with analytical and numerical studies on the reconstruction of a real function of two real variables over the unit square from observations of its autoconvolution on $[0,2]^2 \subset \mathcal{R}^2$ (full data case) or on $[0,1]^2$ (limited data case). In an $L^2$-setting, twofoldness and uniqueness assertions are presented for the deautoconvolution problem in 2D. Moreover, its ill-posedness is characterized and illustrated. Extensive numerical
case studies give an overview of the behaviour of stable approximate solutions to the two-dimensional deautoconvolution problem obtained by Tikhonov-type regularization with different penalties and the iteratively regularized Gauss-Newton method.
Efficient minimization of variational functionals via semismooth* Newton methods 1Johann Radon Institue Linz, Austria; 2Johannes Kepler University Linz, Austria
In this talk, we consider the efficient numerical minimization of variational functionals as they appear for example in $L_{p}$ or TV regularization of nonlinear inverse problems. For this, we consider so-called semismooth* Newton methods, which are a class of optimization methods for non-differentiable and set-valued mappings. Based on the concept of (limiting) normal cones, which are a purely geometrical generalization of derivatives, these methods can be shown to converge locally superlinearly under suitable assumptions. Furthermore, we show how they can be applied to efficiently minimize variational functionals with general convex and in some special cases even non-convex penalty terms.
Convergence Nestorov acceleration for linear ill-posed problems Johannes Kepler University Linz, Austria
We show that Nesterov acceleration is an optimal-order iterative regularization method for linear ill-posed problems provided that a parameter is chosen accordingly to the smoothness of the solution.
The central result is a representation of the iteration residual polynomials via Gegenbauer polynomials. This also explains the observed semi-saturation effect of Nesterov iteration.
Analysis of the discrepancy principle for Tikhonov regularization under low order source conditions Universität Siegen, Germany
We study the application of Tikhonov regularization to ill-posed nonlinear operator equations. The objective of this work is to prove low order convergence rates for the discrepancy principle under low order source conditions of logarithmic type. We work within the framework of Hilbert scales and extend existing studies on this subject to the oversmoothing case. The latter means that the exact solution of the treated operator equation does not belong to the domain of definition of the penalty term. As a consequence, the Tikhonov functional fails to have a finite value.
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| 1:30pm - 3:30pm | MS04 1: Statistical and computational aspects of non-linear inverse problems Location: VG2.102 Session Chair: Richard Nickl Session Chair: Sven Wang |
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Surface finite element approximation of Gaussian random fields on Riemannian manifolds Chalmers & University of Gothenburg, Sweden
Whittle-Mat\'{e}rn Gaussian random fields are popular tools in spatial statistics. Interpreting them as solutions to specific stochastic PDEs allow to generalize them from fields on all of $\mathbb{R}^d$ to bounded domains and manifolds. In this talk we focus on Riemannian manifolds and efficient approximations of Gaussian random fields based on surface finite element methods.
Concentration analysis of multivariate elliptic diffusions 1Aarhus University, Denmark; 2University of Mannheim, Germany
We prove concentration inequalities and associated PAC bounds for continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approach via the Poisson equation, which allows us to consider a very broad class of subexponentially ergodic processes. These results add to existing concentration inequalities for additive functionals of diffusion processes which have so far been only available for either bounded functions or for unbounded functions of processes from a significantly smaller class. We demonstrate the usefulness of the results by applying them in the context of high-dimensional drift estimation and Langevin MCMC for moderately heavy-tailed target densities.
A Bernstein-von-Mises theorem for the Calder\'{o}n problem with piecewise constant conductivities University of Bonn, Germany
The talk presents a finite dimensional statistical model for the Calder\'{o}n problem with piecewise constant conductivities. In this setting one can consider a classical i.i.d noise model and the injectivity of the forward map and its linearisation suffice to prove the invertibility of the information operator. This results in a BvM-theorem and optimality guarantees for estimation in Bayesian posterior means.
Bayesian estimation in a multidimensional diffusion model with high frequency data 1Universite Paris-Dauphine; 2Imperial College London
We consider a multidimensional diffusion model describing a particle moving in an insulated inhomogeneous medium under Brownian dynamics. We study Bayesian inference based on discrete high-frequency observations of the particle’s location. Bayesian posteriors (and their posterior means) based on suitable Gaussian priors are shown to estimate the diffusivity function of the medium at the minimax optimal rate over Holder smoothness classes in any dimension. We also show that certain penalized least squares estimators are minimax optimal for estimating the diffusivity.
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| 1:30pm - 3:30pm | MS14 3: Inverse Modelling with Applications Location: VG1.104 Session Chair: Daniel Lesnic Session Chair: Karel Van Bockstal |
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Inverse Problems for Generalized Subdiffusion Equations Tallinn University of Technology, Estonia
The talk focuses on the theoretical investigation of the inverse problem for the Pennes' bioheat wave equation. Uniqueness and existence are the main questions under consideration.
Numerical solution to inverse source problems for a parabolic equation with nonlocal conditions 1Institute of Control Systems of the Ministry of Science and Education, Azerbaijan; 2Baku State University, Azerbaijan
In the report, we consider an inverse problem for a parabolic equation with unknown coefficient depending from only one independent variable: space or time variable.
We consider the following problem of determining unknown coefficient $C_{0} \left(x\right)$ of the linear parabolic equation:
$$
\begin{array}{l} {\frac{\partial v\left(x,t\right)}{\partial t} =a_{0} \frac{\partial ^{2} v\left(x,t\right)}{\partial x^{2} } +a_{1} \frac{\partial v\left(x,t\right)}{\partial x} +a_{2} v\left(x,t\right)+} \\ {+f\left(x,t\right)+F\left(x,t\right),\, \, \, \, \, \, \, \, \, \, \left(x,t\right)\in \Omega =\left\{\left(x,t\right):0<x<l,\, \, 0<t\le T\right\},} \end{array}
$$
under conditions:
$$
k_{1} v\left(x,0\right)+\int _{0}^{T}e^{k\tau } v\left(x,\tau \right)d\tau =\varphi _{0} \left(x\right),\, \, \, \, v\left(x,T\right)=\varphi _{T} \left(x\right),\, \, \, x\in \left[0,\, l\right],
$$
$$
v\left(0,t\right)=\psi _{0} \left(t\right),\, \, \, \, \, v\left(l,t\right)=\psi _{1} \left(t\right),\, \, \, \, t\in \left[0,\, T\right],
$$ and
where $F\left(x,t\right)=B_{0} \left(x,t\right)C_{0} \left(x\right)$ and $k,\, \, k_{1} \ne 0$ are constants.
Two cases are considered. In the first case, the known coefficients $a_{i} ,\, \, i=0,\, ...,2$ are functions of $x$, i.e. $a_{i} =a_{i} \left(x\right)$. The functions $a_{0} (x)>0,$ $a_{1} \left(x\right)$, $a_{2} \left(x\right)$, $\varphi _{0} \left(x\right),\, \varphi _{T} \left(x\right)$, $f\left(x,t\right),\, B_{0} \left(x,t\right)$, $\psi _{0} \left(t\right),\, \psi _{1} \left(t\right)$ are given and satisfy all the conditions of existence and uniqueness of the functions $v\left(x,t\right),\, \, C_{0} \left(x\right)$, which are the solutions to the problem.
We propose a numerical method of solution to the problem, which is based on the use of the method of lines. The initial problem is reduced to the parametric inverse problems with respect to ordinary differential equations. Then, we propose a non-iterative method based on using a special representation of the solutions to the obtained problems [1, 2]. Some of the results of the carried out numerical experiments are given. The obtained results show the efficiency of the proposed approach.
In the second case, similar approaches to numerical solution to the problem of identifying $C_{0} \left(t\right)$ in case $F\left(x,t\right)=B\left(x,t\right)C_{0} \left(t\right)$ are proposed. In this case, the known coefficients $a_{i} ,\, \, i=0,\, ...,2$ are functions of $t$, and instead of the first conditions, we use the following ones:
$$\int _{0}^{l}e^{k\xi } v\left(\xi ,t\right)d\xi =\psi \left(t\right),\, \, \, \, t\in \left[0,\, T\right],$$
$$v\left(x,0\right)=\varphi _{0} \left(x\right),\, \, \, \, x\in \left[0,\, l\right].$$
[1] K.R. Aida-zade, A.B. Rahimov. An approach to numerical solution of some inverse problems for parabolic equations, Inverse Probl. Sci. Eng. 22: 96-111, 2014.
[2] K.R. Aida-zade, A.B. Rahimov. On recovering space or time-dependent source functions for a parabolic equation with nonlocal conditions, Appl. Math. Comp. 419, 2022.
Advances in object characterisation for metal detection inverse problems 1School of Computer Science & Mathematics, Keele University, United Kingdom; 2Department of Mathematics, The University of Manchester, United Kingdom
The location and identification of hidden conducting threat objects using metal detection is an important yet difficult task. Applications include security screening at transport hubs and finding landmines and unexploded ordnance in areas of former conflict. Based on an asymptotic expansion of the perturbed magnetic field, we have derived an economical object description called a magnetic polarizability tensor (MPT), which is a function of the object’s size, shape, material properties and the exciting frequency [1]. The MPT provides the object characterisation in the leading order term of the asymptotic expansion of the perturbed magnetic field as the object size tends to 0 and we have extended this to a complete asymptotic expansion with improved object characterisation provided by generalised MPTs expressed in terms of tensorial and multi-indices [2].
To compute object characterisations, we employ a hp-finite element method, accelerated by an adaptive reduced order model and scaling results [3], to efficiently compute a large dictionary spectral signature characterisations of realistic threat and non-threat objects relevant for security screening [4]. To accurately capture small skin depths and realistic in-homogeneous objects with thin material layers, this involves including thin layers of prismatic boundary layers, which we have shown are in close agreement with measurements [5]. In this talk, we review our latest developments and discuss possible classification strategies [6].
References
[1] P. D. Ledger, W. R. B. Lionheart. The spectral properties of the magnetic polarizability tensor for metallic object characterisation. Mathematical Methods in the Applied Sciences, 43:78–113, 2020.
[2] P.D. Ledger, W.R.B. Lionheart. Properties of generalized magnetic polarizability tensors. Mathematical Methods in the Applied Sciences, To appear 2023. DOI:10.1002/mma.8856
[3] B. A. Wilson, P. D. Ledger. Efficient computation of the magnetic polarizability tensor spectral signature using proper orthogonal decomposition. International Journal for Numerical Methods in Engineering, 122:1940–1963, 2021.
[4] P. D. Ledger, B. A. Wilson, A. A. S. Amad, W. R. B. Lionheart. Identification of metallic objects using spectral magnetic polarizability tensor signatures: Object characterisation and invariants. International Journal for Numerical Methods in Engineering, 122:3941–3984, 2021.
[5] J. Elgy, P.D. Ledger, J.L. Davidson, T. Ozdeger, A.J. Peyton, Computations and measurements of the magnetic polarizability tensor characterisation of highly conducting and magnetic objects, submitted 2023.
[6] B. A. Wilson, P. D. Ledger, and W. R. B. Lionheart. Identification of metallic objects using spectral magnetic polarizability tensor signatures: Object classification. International Journal for Numerical Methods in Engineering 123: 2076-2111, 2022.
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| 1:30pm - 3:30pm | MS18 3: 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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Inverse Problems for Subdiffusion from Observation at an Unknown Terminal Time The Chinese University of Hong Kong
Inverse problems of recovering space-dependent parameters, e.g., initial condition, space-dependent source or potential coefficient, in a subdiffusion model from the terminal observation have been extensively studied in recent years. However, all existing studies have assumed that the terminal time at which one takes the observation is exactly known. In this talk, we present uniqueness and stability results for three canonical inverse problems, e.g., backward problem, inverse source and inverse potential problems, from the terminal observation at an unknown time. The subdiffusive nature of the problem indicates that one can simultaneously determine the terminal time and space-dependent parameter. The analysis is based on explicit solution representations, asymptotic behavior of the Mittag-Leffler function, and mild regularity conditions on the problem data. Further, we present several one- and two-dimensional numerical experiments to illustrate the feasibility of the approach.
UCP and counterexamples to UCP involving generalized ray transforms TIFR Centre for Applicable Mathematics, India
We study generalized Radon-type transforms involving functions and symmetric tensor fields. We show in some instances that unique continuation principle for such transforms holds and we also give explicit counterexamples where such principle does not hold.
An inverse problem related to fractional wave equation National Chengchi University, Taiwan
In this talk, we will focus in an inverse problem for the fractional wave equation with a potential, which can be regarded as a special case of the peridynamics which models the nonlocal elasticity theory. We will discuss the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichlet-to-Neumann map. This talk is prepared based on the work https://doi.org/10.1137/21M1444941
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| 1:30pm - 3:30pm | MS19 1: Theory and algorithms of super-resolution in imaging and inverse problems Location: VG3.103 Session Chair: Habib Ammari Session Chair: Ping Liu |
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Stability and super-resolution of MUSIC and ESPRIT for multi-snapshot spectral estimation CUNY City College, United States of America
We study the spectral estimation problem of estimating the locations of a fixed number of point sources given multiple snapshots of Fourier measurements collected by a uniform array of sensors. We prove novel stability bounds for MUSIC and ESPRIT as a function of the noise standard deviation, number of snapshots, source amplitudes, and support. When the point sources are located in several clumps, we provide explicit upper bounds for MUSIC and ESPRIT in terms of a Super-Resolution Factor (SRF). We also derive a new Cramér-Rao lower bound for the clumps model, and as a result, implies that ESPRIT is comparable to that of the optimal unbiased estimator(s) in terms of the dependence on noise, number of snapshots and SRF. As a byproduct of our analysis, we discover several fundamental differences between the single-snapshot and multi-snapshot problems.
A Mathematical Theory of Computational Resolution Limit and Super-resolution ETH Zurich, Switzerland
Due to the physical nature of wave propagation and diffraction, there is a fundamental diffraction barrier in optical imaging systems which is called the diffraction limit or resolution limit. Rayleigh investigated this problem and formulated the well-known Rayleigh limit. However, the Rayleigh limit is empirical and only considers the resolving ability of the human visual system. On the other hand, resolving sources separated below the Rayleigh limit to achieve so-called “super-resolution” has been demonstrated in many numerical experiments.
In this talk, we will propose a new concept “computational resolution limit” which reveals the fundamental limits in superresolving the number and locations of point sources from a data-processing point of view. We will quantitatively characterize the computational resolution limits by the signal-to-noise ratio, the sparsity of sources, and the cutoff frequency of the imaging system. As a direct consequence, it is demonstrated that $l_0$ optimization achieves the optimal order resolution in solving super-resolution problems. For the case of resolving two point sources, the resolution estimate is improved to an exact formula, which answers the long-standing question of diffraction limit in a general circumstance. We will also propose an optimal algorithm to distinguish images generated by single or multiple point sources. Generalization of our results to the imaging of positive sources and imaging in multi-dimensional spaces will be briefly discussed as well.
Total variation regularized problems: a support stability result 1Inria, France; 2CEREMADE, Université Paris-Dauphine, PSL University; 3Universita di Genova; 4Institut Camille Jordan; 5Ecole Centrale Lyon
The total (gradient) variation has been used in many imaging applications following the seminal work of Rudin, Osher and Fatemi.[1]
In this talk, I will describe a "support stability'' result for total-variation regularized inverse problems: under some assumptions, the solutions at low noise and low regularization have the same number of values as the unknown image, and their level sets converge to those of the unknown image.
It is a joint work with Romain Petit and Yohann De Castro.
[1] L. I. Rudin, S. Osher, E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, Volume 60, Issues 1–4, 1 November 1992, Pages 259-268.
Theoretical and numerical off-the-grid curve reconstruction 1Morpheme (Inria/CNRS), France; 2LJAD (CNRS), France
Recent years have seen the development of super-resolution variational optimisation
optimisation in measure spaces. These so-called off-the-grid approaches offer both theoretical (uniqueness, reconstruction guarantees) and numerical results, with very convincing results in biomedical imaging. However, the gridless variational optimisation is formulated for reconstruction of point sources, which is not always suitable for biomedical imaging applications: more realistic biological structures such as curves should also be reconstructed, to represent blood vessels or filaments for instance.
We propose a new strategy for the reconstruction of curves in an image through an off-the-grid variational framework, inspired by the reconstruction of spikes in the literature. We introduce a new functional CROC on the space of 2-dimensional Radon measures with finite divergence denoted V. Our main contribution lies in the sharp characterisation of the extreme points of the unit ball of the V -norm: there are exactly measures supported on 1-rectifiable oriented simple Lipschitz curves, thus enabling a precise characterisation of our functional minimisers and further opening the avenue for the algorithmic implementation.
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| 1:30pm - 3:30pm | MS21 1: Prior Information in Inverse Problems Location: VG2.103 Session Chair: Andreas Horst Session Chair: Jakob Lemvig |
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Reconstructing spatio-temporal, sparse tomographic data using cylindrical shearlets University of Bath, United Kingdom
In this talk, I will present a motion-aware variational approach, based on a new multiscale directional system of functions called cylindrical shearlets, to reconstruct moving objects from sparse dynamic data. Compared to conventional separable representations, cylindrical shearlets are very efficient in representing spatio-temporal data, since they are better suited to handle the geometry of these data.
We test our approach on both simulated and measured data. Numerical results demonstrate the advantages of our novel approach with respect to conventional multiscale methods.
Fractal priors for imaging using random wavelet trees University of Helsinki, Finland
A novel Bayesian prior distribution family is introduced, based on wavelet transforms.The priors correspond to well-defined infinite-dimensional random variables and can be approximated by finite-dimensional models.
The non-zero wavelet coefficients are chosen in a systematic way so that prior draws have a specific fractal behaviour. This paves the way for new types of signal and image processing methods that can either promote certain fractal properties in the underlying data, or serve as smart "fingerprints" for measured object types. Realisations of the new priors take values in Besov spaces and have singularities only on a small set with a certain Hausdorff dimension. We also introduce an efficient algorithm for calculating the MAP estimator in the denoising problem.
Sampling from a posterior with Besov prior Technical University of Denmark (DTU), Compute, Denmark
Besov priors for Bayesian inverse problems are interesting since they promote various types of regularity on the unknown, especially non-smooth regularity, depending on the choice of basis and parameters of the prior. Besov priors introduces a $p$-norm into the posterior, which makes effective Gaussian samplers inapplicable. Randomize-Then-Optimize (RTO) is an optimization-based sampling algorithm, that computes exact independent samples from a posterior with Gaussian prior and a linear forward operator. We introduce a prior transformation that transforms a Besov prior into a Gaussian prior, which makes Gaussian samplers like RTO applicable. The caveat of the transformation is that the forward operator becomes non-linear even though it originally was linear. To sample from the transformed posterior we use RTO samples as proposals for the Metropolis-Hastings algorithm. We apply this sampling method to a deconvolution problem where the type of Besov prior is varied, to discover the quality of the method and the posterior dependencies on the choice of Besov prior. Our results validate that the computed samples come from the original posterior with Besov prior and shows that the choice of prior basis and parameters has a significant impact on the posterior.
Regularizing Inverse Problems through Translation Invariant Diagonal Frame Decompositions OTH Regensburg, Germany
We consider the challenge of solving the ill-posed reconstruction problem in computed tomography using a translation-invariant diagonal frame decomposition (TI-DFD). First, we review the concept of diagonal frame decompositions (DFD) and their translation-invariant counterparts for general linear operators. Subsequently, we explain how the filter-based regularization methods can be defined using these frame decompositions. Finally, as an example, we introduce the TI-DFD for the Radon transform on $L^2 (\mathbb{R}^2)$ and provide an exemplary construction using the TI wavelet transform. In numerical results, we demonstrate the advantages of our approach over non-translation invariant counterparts.
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| 1:30pm - 3:30pm | MS22 1: Imaging with Non-Linear Measurements: Tomography and Reconstruction from Phaseless or Folded Data Location: VG1.101 Session Chair: Matthias Beckmann Session Chair: Robert Beinert Session Chair: Michael Quellmalz |
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Gradient Methods for Blind Ptychography Helmholtz Center Munich, Germany
Ptychography is an imaging technique, the goal of which is to reconstruct the object of interest from a set of diffraction patterns obtained by illuminating its small regions. When the distribution of light within the region is known, the recovery of the object from ptychographic measurements becomes a special case of the phase retrieval problem. In the other case, also known as blind ptychography, the recovery of both the object and the distribution is necessary.
One of the well-known reconstruction methods for blind ptychography is extended Ptychographic Iterative Engine. Despite its popularity, there was no known analysis of its performance. Based on the fact that it is a stochastic gradient descent method, we derive its convergence guarantees if the step sizes are chosen sufficiently small. The second considered method is a generalization of the Amplitude Flow algorithm for phase retrieval, a gradient descent scheme for the minimization of the amplitude-based squared loss. By applying an alternating minimization procedure, the blind ptychography is reduced to phase retrieval subproblems, which can be solved by performing a few steps of Amplitude Flow. The resulting procedure converges to a critical point at a sublinear rate.
Inversion of the Modulo Radon Transform via Orthogonal Matching Pursuit University of Bremen, Germany
In the recent years, the topic of high dynamic range (HDR) tomography has started to gather attention due to recent advances in the hardware technology. The issue is that registering high-intensity projections that exceed the dynamic range of the detector cause sensor saturation, which, in turn, leads to a loss of information. Inspired by the multi-exposure fusion strategy in computational photography, a common approach is to acquire multiple Radon Transform projections at different exposure levels that are algorithmically fused to facilitate HDR reconstructions.
As opposed to this, a single-shot alternative to the multi-exposure fusion approach has been proposed in our recent line of work which is based on the Modulo Radon Transform, a novel generalization of the conventional Radon transform. In this case, Radon Transform projections are folded via a modulo non-linearity, which allows HDR values to be mapped into the dynamic range of the sensor and, thus, avoids saturation or clipping. The folded measurements are then mapped back to their ambient range using reconstruction algorithms.
In this talk we introduce a novel Fourier domain recovery method, namely the OMP-FBP method, which is based on the Orthogonal Matching Pursuit (OMP) algorithm and Filtered Back Projection (FBP) formula. The proposed OMP-FBP method offers several advantages; it is agnostic to the modulo threshold or the number of folds, can handle much lower sampling rates than previous approaches and is empirically stable to noise and outliers. The effectivity of the OMP-FBP recovery method is illustrated by numerical experiments.
This talk is based on joint work with Ayush Bhandari (Imperial College London).
Phaseless sampling of the short-time Fourier transform University of Vienna, Austria
We discuss recent advances in phaseless sampling of the short-time Fourier transform (STFT). The main focus of the talk lies in the question if phaseless samples of the STFT contain enough information to recover signals belonging to infinite-dimensional function spaces. It turns out, that this problem differs from ordinary sampling in a rather fundamental and subtle way: if the sampling set is a lattice then uniqueness is unachievable, independent of the choice of the window function and the density of the lattice. Based on this discretization barrier, we present possible ways to still achieve unique recoverability from samples: lattice-uniqueness is possible if the signal class gets restricted to certain proper subspaces of $L^2(\mathbb R)$, such as the class of compactly-supported functions or shift-invariant spaces. Finally, we highlight that sampling on so-called square-root lattices achieves uniqueness in $L^2(\mathbb R)$.
Phase Retrieval in Optical Diffraction Tomography TU Berlin, Germany
Optical diffraction tomography (ODT) consists in the recovery of the three-dimensional scattering potential of a microscopic object rotating around its center from a series of illuminations with coherent light. Standard reconstruction algorithms such as the filtered backpropagation require knowledge of the complex-valued wave at the measurement plane. In common physical measurement setups, the collected data only consists in intensities; so only phaseless measurements are available. To overcome the loss of the required phases, we propose a new reconstruction approach for ODT based on three key ingredients. First, the light propagation is modeled using Born's approximation enabling us to use the Fourier diffraction theorem. Second, we stabilize the inversion of the non-uniform discrete Fourier transform via total variation regularization utilizing a primal-dual iteration, which also yields a novel numerical inversion formula for ODT with known phase. The third ingredient is a hybrid input-output scheme. We achieve convincing numerical results showing that the computed 2D and 3D reconstructions are even comparable to the ones obtained with known phase. This indicate that ODT with phaseless data is possible.
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| 1:30pm - 3:30pm | MS28 1: Modelling and optimisation in non-Euclidean settings for inverse problems Location: VG1.108 Session Chair: Luca Calatroni Session Chair: Claudio Estatico Session Chair: Dirk Lorenz |
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A lifted Bregman formulation for the inversion of deep neural networks 1University of Cambridge, United Kingdom; 2Queen Mary University of London, United Kingdom; 3The Alan Turing Institute, United Kingdom
We propose a novel framework for the regularised inversion of deep neural networks. The framework is based on the authors' recent work on training feed-forward neural networks without the differentiation of activation functions. The framework lifts the parameter space into a higher dimensional space by introducing auxiliary variables and penalises these variables with tailored Bregman distances. We propose a family of variational regularisations based on these Bregman distances, present theoretical results and support their practical application with numerical examples. In particular, we present the first convergence result (to the best of our knowledge) for the regularised inversion of a single-layer perceptron that only assumes that the solution of the inverse problem is in the range of the regularisation operator.
A Bregman-Kaczmarz method for nonlinear systems of equations TU Braunschweig, Germany
We propose a new randomized method for solving systems of nonlinear equations, which can find sparse solutions or solutions under certain simple constraints. The scheme only takes gradients of component functions and uses exact or relaxed Bregman projections onto the solution space of a Newton equation. As such, it generalizes the Sparse Kaczmarz method which finds sparse solutions to linear equations, as well as the nonlinear Kaczmarz method, which performs euclidean projections. The relaxed Bregman projection is achieved by using the step size from the nonlinear Kaczmarz method. Local convergence is established for systems with full rank Jacobian under the local tangential cone condition. We show examples in which the proposed method outperforms similar methods with the same memory requirements.
Regularization in non-Euclidean spaces meets numerical linear algebra 1University of Genoa, Italy; 2Istituto di Matematica Applicata e Tecnologie Informatiche, Italy; 3Laboratory of Computer Science, Signals and Systems of Sophia Antipolis, France; 4Eumetsat, Germany
Inverse problems modeled by a functional equation $A(x)=y$ characterized by an ill-posed operator $A:X \longrightarrow Y$ between two non-Euclidean normed spaces $X$ and $Y$ are here considered. The iterative minimization of a functional based on the residual $\|A(x)-y\|_Y$ is a common approach in this setting, where generally no (closed form of the) inverse of $A$ exists. In particular, one-step gradient methods act as implicit regularization algorithms, when combined with an early-stopping criterion to prevent over-fitting of the noise on the data $y$.
In this talk, we review iterative methods involving the dual spaces $X^*$ and $Y^*$, showing that they can be fully understood in the context of proximal operator theory, with suitable Bregman distances as proximity measure [1]. Moreover, many relationships of such iterative methods with classical projection algorithms, such us Cimmino and ART (Algebraic Reconstruction Techniques) ones, are discussed, as well as with classical preconditioning theory for structured linear systems arising in numerical linear algebra. Applications to deblurring and inverse scattering problems will be shown.
[1] B. Bonino, C. Estatico, M. Lazzaretti. Dual descent regularization algorithms
in variable exponent Lebesgue spaces for imaging, Numer. Algorithms 92: 149-182, 2023.
[2] M. Lazzaretti, L. Calatroni, C. Estatico. Modular-proximal gradient algorithms in variable exponent Lebesgue spaces, SIAM J. Sci. Compu. 44: 1-23, 2022.
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| 1:30pm - 3:30pm | MS29 3: Eigenvalues in inverse scattering Location: VG3.104 Session Chair: Martin Halla Session Chair: Peter Monk |
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A new family of nearly singular interior transmission eigenvalues for inverse scattering Georg-August Universität Göttingen, Germany
I propose a new family of nearly singular interior transmission eigenvalue problems for inverse scattering. For a known support of the inhomogeneity the eigenvalues allow to identify e.g. the piece-wise constant values of the refractive index. For an unknown support of the inhomogeneity the eigenvalues allow to construct an indicator function, I present an analysis for ideal settings and numerical examples for general cases.
Interior transmission problems related to imaging periodic layers 1INRIA, France; 2ENIT, Tunisia
The extension of sampling methods to the imaging of locally perturbed periodic layers [1] requires the analysis of interior transmission problems in unbounded waveguides. The resulting problem is no longer of Fredholm type and its study necessitates additional tools to those classically used for the case of bounded domains. We shall present a proof of well posdeness in the case of absorbing background using Floquet-Bloch transform and finite dimensional approximation with respect to the Floquet-Bloch variable. The analysis of absorption free problem and related transmission eigenvalues is an open problem that we shall also briefly discuss. We plan to additionally present the so-called differential sampling method where some specific single Floquet-Bloch variable transmission eigenvalue problems shows up. These problems had been adressed in the case of periodically distributed defects in [2].
[1] Y. Boukari, H. Haddar, N. Jenhani. Analysis of sampling methods for imaging a periodic layer and its defects, Inverse Problems, 2023.
[2] F. Cakoni, H. Haddar, T.-P. Nguyen. New interior transmission problem applied to a single Floquet-Bloch mode imaging of local perturbations in periodic media, Inverse Problems, 2018.
Transparent scatterers and transmission eigenvalues of infinite multiplicity 1CNRS, École polytechnique, France; 2Steklov Mathematical Institute, Russia
We give a short review of old and recent results on scatterers with transmission eigenvalues of infinite multiplicity,
including transparent scatterers. Our examples include potentials from the Schwartz class and multipoint potentials of Bethe - Peierls type.
[1] P.G. Grinevich, R.G. Novikov. Russian Mathematical Surveys 77:1021-1028, 2022. https://doi.org/10.4213/rm10080e
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| 1:30pm - 3:30pm | MS38 1: Inverse eigenvalue problems in astrophysics Location: VG2.105 Session Chair: Charlotte Gehan Session Chair: Damien Fournier |
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No planet is an island: what we can learn from how Saturn interacts with its surroundings Canadian Institute for Theoretical Astrophysics (CITA), Canada
Direct observations provide limited information about the deep internal structures and basic properties of gaseous planets, even in our own Solar system. However, more can be learned from how planets interact with their surroundings. I will introduce research focused on interpreting the satellite Cassini's observations of Saturn's gravitational interactions with its rings and satellite moons. Observing these interactions yields information about Saturn's internal oscillation modes and tidally excited waves, the successful inversion of which may provide our best hope for constraining the planet's deep internal structure and rotation state.
Inversion methods in asteroseismology LESIA, France
In this talk, I will review the different kernel-based inversion techniques that have been used in asteroseismology. In particular, I will describe regularised least-squares (RLS) as well as optimally localised averages (OLA) type inversions. These have been applied to rotation and structural profiles as well as to integrated quantities such as the total kinetic rotation energy, the mean density, the acoustic radius, and various evolutionary phase and convective region indicators. I will also briefly show how inverse techniques can lead to more subtle constraints such as inequalities on rotational splittings if one makes certain assumptions on the rotation profile.
Internal structure of giant planets from gravity data IRAP, CNRS, France
The Juno and Cassini spacecrafts have measured the gravity fields of Jupiter and Saturn with exquisite precision. The gravity field can then be projected onto the Legendre polynomials to obtain the gravitational moments, signatures of the density distribution in the planet as a function of radius and angle. In the past few years, a lot of effort has thus been dedicated to create precise methods to calculate gravitational moments from synthetic models and optimise the retrieval of internal structure by comparing with Juno and Cassini data.
In this talk, I will detail how we tackled this inverse problem in the case of Jupiter and Saturn. I will quickly introduce the concentric Maclaurin spheroid method used to calculate gravitational moments, before detailing the recovered internal structures. I will expose the dominant influence of winds on the gravity field and how planetary oscillations can constrain further the recovered density profiles. These results have strong implication for the formation and evolution of the giant planets and solar system in general.
Accurate asteroseismic surface rotation rates for evolved red giants 1Heidelberg Institute for Theoretical Studies, Germany; 2Max Planck Institute for Astrophysics, Germany; 3Stellar Astrophysics Centre, Aarhus University, Denmark; 4Center for Astronomy (ZAH/LSW), Heidelberg University, Germany; 5Department of Astronomy, Yale University, USA
The understanding of the internal stellar rotation and its evolution are important ingredients for the construction of accurate stellar models. We use asteroseismology, the study of global stellar oscillations, to probe the interior rotation of stars, particularly that of red giants. Large systematic errors previously hindered the accurate determination of near-surface rotation rates in evolved red giants e.g. [1]. We have developed a method of effectively eliminating these systematic errors by introducing an extension to a currently used rotational inversion method for red-giant stars [2].
We demonstrate the ability of the new inversion technique to compute accurate envelope rotation rates of stars along the red giant branch (RGB). Furthermore, we show the resulting improvement of our new method compared to other seismic inversion methods. Subsequently, we aim at quantifying systematic uncertainties in asteroseismic rotational inversions occurring due to inaccurate stellar modelling (Ahlborn et al. in prep). More accurate surface rotation rates for evolved red giants will be an important probe to understand the loss of angular momentum in red-giant cores, and an important milestone to improve the theory of rotation in stellar models.
[1] F. Ahlborn, E. P. Bellinger, S. Hekker, S. Basu and G. C. Angelou. Asteroseismic sensitivity to internal rotation along the red-giant branch, Astronomy and Astrophysics, 639:A98, 2020. https://doi.org/10.1051/0004-6361/201936947
[2]
F. Ahlborn, E. P. Bellinger, S. Hekker, S. Basu and D. Mokrytska. Improved asteroseismic inversions for red-giant surface rotation rates, Astronomy and Astrophysics, 668:A98, 2022. https://doi.org/10.1051/0004-6361/202142510
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| 1:30pm - 3:30pm | MS40: Dynamic Imaging Location: VG2.107 Session Chair: Peter Elbau |
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Iterative and data-driven motion compensation in tomography University of Stuttgart, Germany
Most tomographic modalities record the data sequentially, i.e. temporal changes of the object lead
to inconsistent measurements. Consequently, suitable models and algorithms have to be developed
in order to provide artefact free images. In this talk, we provide an overview of different strategies, including a data-driven approach to extract explicit motion maps which can then be incorporated within direct or iterative reconstruction methods for the underlying dynamic inverse problem. Our methods are illustrated by numerical results from real as well as simulated data of different imaging modalities.
Sparse optimization algorithms for dynamic imaging University of Hull, United Kingdom
In this talk we introduce a Frank-Wolfe-type algorithm for sparse optimization in Banach spaces. The functional we want to optimize consist of the sum of a smooth fidelity term and of a convex
one-homogeneous regularizer. We exploit the sparse structure of the variational problem by designing iterates as linear combinations of extremal points of the unit ball of the regularizer. For such iterates we prove global sublinear convergence of the algorithm. Then, under additional structural assumptions, we prove a local linear convergence rate. We apply this algorithm to the problem of particles tracking from heavily undersampled dynamic MRI data. This talk is based on the works cited below.
[1] K.Bredies, M.Carioni, S.Fanzon, D.Walter. Asymptotic linear convergence of Fully-Corrective Generalized Conditional Gradient methods. Mathematical Programming, 2023.
[2] K.Bredies, S.Fanzon. An optimal transport approach for solving dynamic inverse problems in spaces of measures. ESAIM:M2AN, 54(6): 2351-2382, 2020.
[3] K.Bredies, M.Carioni, S.Fanzon, F.Romero. A Generalized Conditional Gradient Method for Dynamic Inverse Problems with Optimal Transport Regularization. Found Comput Math, 2022
[4] K.Bredies, M.Carioni, S.Fanzon. On the extremal points of the ball of the Benamou–Brenier energy. Bull. London Math. Soc., 53: 1436-1452, 2021.
[5] K.Bredies, M.Carioni, S.Fanzon. A superposition principle for the inhomogeneous continuity equation with Hellinger–Kantorovich-regular coefficients. Communications in Partial Differential Equations, 47(10): 2023-2069, 2022.
New approaches for reconstruction in dynamic nano-CT imaging 1Georg-August-University Göttingen, Germany; 2Saarland University Saarbrücken, Germany; 3University of Stuttgart, Germany
Tomographic X-ray imaging on the nano-scale is an important tool to visualize the structure of materials such as alloys or biological tissue. Due to the small scale on which the data acquisition takes place, small perturbances caused by the environment become significant and cause a motion of the object relative to the scanner during the scan. Since this motion is hard to estimate and its incorporation into the reconstruction process strongly increases the numerical effort, we aim at a different approach for a stable reconstruction: We interpret the object motion as a modelling inexactness in comparison to the model in the static case. This inexactness is estimated and included in an iterative regularization scheme called sequential subspace optimization. Data-driven techniques are investigated to estimate the modelling error and to improve the obtained reconstructions.
Artifact reduction for time dependent image reconstruction in magnetic particle imaging Universität Hamburg, Germany
Magnetic particle imaging (MPI) is a preclinical imaging modality exploiting the nonlinear magnetization response of magnetic nanoparticles to applied dynamic magnetic fields. We focus on MPI using a field-free line for spatial encoding because under ideal assumptions such as static objects, ideal magnetic fields and sequential line rotation, the MPI data are obtained by Radon transformed particle distributions. In practice, field imperfections and moving objects occur such that we have to adapt the Radon transform and jointly reconstruct time dependent particle distributions and adapted Radon data by means of total variation regularization.
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| 1:30pm - 3:30pm | MS47 1: Scattering and spectral imaging: inverse problems and algorithms Location: VG3.101 Session Chair: Eric Todd Quinto Session Chair: Gael Rigaud |
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Microlocal properties and injectivity for Ellipsoidal and hyperbolic Radon transforms 1Tufts University, United States of America; 2University of Manchester, England; 3Brigham and Women's Hospital, United States of America
We present novel microlocal results for generalized ellipsoid and hyperboloid Radon transforms in Euclidean Space and we apply our results to Ultrasound Reflection Tomography (URT). We introduce a new Radon transform, $R$, which integrates compactly supported distributions over ellipsoids and hyperboloids with centers on a smooth nypersurface, $S$ in $\mathbb{R}^n$. $R$ is shown to be a Fourier Integral Operator and in our main theorem we prove that $R$ satisfies the Bolker condition if and only if the support of the function is in a connected set that is not intersected by any plane tangent to $S$. In this case, backprojection type reconstruction operators such as the normal operator $R^* R$ do not add artifacts to the reconstruction.
We apply our results to a cylindrical geometry that could be used in URT. We prove injectivity results and investigate the visible singularities in this modality. In addition, we present reconstructions of image phantoms in two dimensions that illustrate our microlocal theory.
Motion detection in diffraction tomography 1TU Berlin, Germany; 2University of Vienna, Austria; 3RICAM, Linz, Austria; 4Christian Doppler Laboratory MaMSi, Vienna, Austria
We study the mathematical imaging problem of optical diffraction tomography (ODT) for the scenario of a rigid particle rotating in a trap created by acoustic or optical forces. Under the influence of the inhomogeneous forces, the particle carries out a time-dependent smooth, but irregular motion. The rotation axis is not fixed, but continuously undergoes some variations, and the rotation angles are not equally spaced, which is in contrast to standard tomographic reconstruction assumptions. Once the time-dependent motion parameters are known, the particle’s scattering potential can be reconstructed based on the Fourier diffraction theorem, considering it is compatible with making the first order Born or Rytov approximation.
The aim of this presentation is twofold: We first need to detect the motion parameters from the tomographic data by detecting common circles in the Fourier-transformed data. This can be seen as analogue to method of common lines from cryogenic electron microscopy (cryo-EM), which is based on the assumption that the assumption that the light travels along straight lines. Then we can reconstruct the scattering potential of the object utilizing non-uniform Fourier methods.
[1] M. Quellmalz, P. Elbau, O. Scherzer, G. Steidl. Motion Detection in Diffraction Tomography by Common Circle Methods 2022.
https://arxiv.org/abs/2209.08086
Deep learning to tackle model inexactness and motion in Compton Scattering Tomography University of Stuttgart, Germany
Modelling the Compton scattering effect leads to many challenges such as non-linearity of the forward model, multiple scattering and high level of noise for moving targets. While the non-linearity is addressed by a necessary linear approximation of the first-order scattering with respect to the sought-for electron density, the multiple-order scattering stands for a substantial and unavoidable part of the spectral data which is difficult to handle due to highly complex forward models. However, the smoothness properties of the operators modelling the different scattering orders suggests that differential operators can be used to reduce the level of multiple scattering. Last but not least, the stochastic nature of the Compton effect may involve a large measurement noise, in particular when the object under study is subject to motion, and therefore time must be taken into account. To tackle these different issues, we discuss in this talk a Bayesian method based on the generalized Golub-Kahan bidiagonalization and explore the possibilities to mimic and improve the stochastic approach with deep neural networks.
Diffusion based regularization for multi-energy CT with limited data Universität Stuttgart, Germany
As shown by the rise of data-driven and learning techniques, the use of specific
features in datasets is essential to build satisfactory solutions to ill-posed
inverse problems suffering limitations, sparsity and large level of noise.
The well-known total-variation regularization has become a standard approach due
to producing good results without any a priori knowledge.
Providing additional information, it is possible to improve the reconstruction
using forward/backward diffusions. An example is the so-called Perona-Malik
functional which is based on a priori on the global contrast.
Such a construction of regularizers is particularly relevant with machine
learning techniques in which a database can provide natural features and
informations such as contrast and edges.
We propose to study this approach and to validate its potential on multi-energy
CT (computerized tomography) subject to limitations such as sparsitiy and
limited angles.
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| 1:30pm - 3:30pm | MS49 1: Applied parameter identification in physics Location: VG3.102 Session Chair: Tram Nguyen Session Chair: Anne Wald |
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Photoacoustic imaging in acoustic attenuating media University Vienna, Austria
Acoustic attenuation describes the loss of energies of propagating waves. This effect is inherently frequency dependent. Typical attenuation models are derived phenomenologically and experimentally without the use of conservation principles.
Because of these general strategy a zoo of models has been developed over decades.
Photoacoustic imaging is a hybrid imaging technique where the object of interest is excited by a laser and the acoustic response of the medium is measured outside of the object. From this the ability of the medium to convert laser excitation into acoustic waves is computationally reconstructed. For photoacoustics, which is a linear inverse problem, we will determine its spectral values, and we shall see that there are two kind of attenuating models, resulting in mildly and severely inverse photoacoustic problems.
Fully Stochastic Reconstruction Methods in Coupled Physics Imaging University College London, United Kingdom
Coupled Physics Imaging methods combine image contrast from one physical process with observations using a secondary process; several modalities in acousto-optical imaging follow this concept wherein optical contrast is observed with acoustic measurements. For the inverse problem both an optical and acoustic model need to be inverted. Classical methods that involve a non-linear optimisation approach can be combined with advances in stochastic subsamplings strategies that are in part inspired by machine learning applications. In such approaches the forward problem is considered deterministic and the stochasticity involves splitting of an objective function into sub functions that approach the fully sampled problem in an expectation sense.
In this work we consider where the forward problem is also solved stochastically, by a Monte Carlo simulation of photon propagation. By adjusting the batch size in the forward and inverse problems together, we can achieve better performance than if subsampling is performed seperately.
Joint work with : S. Powell, C. Macdonald, N. Hänninen, A. Pulkkinen, T. Tarvainen
Some coefficient identification problems from boundary data satisfying range invariance for Newton type methods University of Klagenfurt, Austria
Range invariance is a property that - like the tangential cone condition - enables a proof of convergence of iterative methods for inverse problems. In contrast to the tangential cone condition it can also be verified for some parameter identification problems in partial differential equations PDEs from boundary measurements, as relevant, e.g., in tomographic applications.
The goal of this talk is to highlight some of these examples of coefficient identification from boundary observations in elliptic and parabolic PDEs, among them: combined diffusion and absorption identification (e.g., in steady-state diffuse optical tomography), reconstruction of a boundary coefficient (e.g. in corrosion detection), reconstruction of a coefficient in a quasilinear wave equation (for nonlinearity coefficient imaging).
Traction force microscopy – a testbed for solving the inverse problem of elasticity Heidelberg University, Germany
During the last three decades, the new field of mechanobiology has demonstrated that mechanical forces play a key role for the decision making of biological cells. The standard way to estimate cell forces is traction force microscopy on soft elastic substrates, whose deformations can be tracked with fiducial marker beads. To infer the corresponding cell forces, one can either solve the inverse problem of elasticity, which usually is done in Fourier space, or calculate strain and stress tensors directly from the deformation data. In both cases, some type of regularization is required to deal with experimental noise. Here we discuss recent developments in this field, including microparticle traction force microscopy and machine learning approaches.
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| 1:30pm - 3:30pm | MS52 3: Integral geometry, rigidity and geometric inverse problems Location: VG1.105 Session Chair: Francois Sylvain Monard Session Chair: Plamen Stefanov |
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Marked length spectrum rigidity for Anosov surfaces Sorbonne Université, France
On a closed Riemannian manifold, the marked length spectrum rigidity problem consists in recovering the metric from the knowledge of the lengths of its closed geodesics (marked by the free homotopy classes of the manifold). I will present a solution to this problem for Anosov surfaces namely, surfaces with uniformly hyperbolic geodesic flow (in particular, all negatively-curved surfaces are Anosov). This generalizes to the Anosov setting the celebrated rigidity results by Croke and Otal from the 90s.
Weakly nonlinear geometric optics for the Westervelt equation Leibniz Universität Hannover, Germany
In this talk we will discuss the non-diffusive Westervelt equation, which describes the time evolution of pressure in a medium relative to an equilibrium position. It is a second order quasilinear hyperbolic equation, involving a space dependent parameter which multiplies the nonlinear term. Given a medium with compactly supported but unknown nonlinearity, we would like to recover the latter by probing the medium from different directions with high frequency waves and measuring the exiting wave. To do so, we construct approximate solutions for the forward problem via nonlinear geometric optics and discuss its well posedness. We then explain how the X-ray transform of the nonlinearity can be recovered from the measurements, which allows for it to be reconstructed. Based on joint work with Plamen Stefanov.
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| 1:30pm - 3:30pm | MS58 1: Shape Optimization and Inverse Problems Location: VG2.104 Session Chair: Lekbir Afraites Session Chair: Antoine Laurain Session Chair: Julius Fergy Tiongson Rabago |
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Isogeometric Shape Optimization of Periodic Structures in Three Dimensions 1Universität Basel, Switzerland; 2Università della Svizzera italiana, Switzerland
The optimal design of medical and dental implants, or lightweight structures in aeronautics can be modelled by a periodic structure with an empty, but a-priorily unknown inclusion.
Homogenisation of this periodic scaffold structure, i.e., a material containing periodically arranged, identical copies of a cavity, leads to a macroscopic equation involving an effective material tensor $\mathbf{A}_0(\Omega) \in \mathbb{R}^{d \times d}_{\rm sym}$.
This effective tensor is determined by a microscopic problem, defined on the $d$-dimensional, periodic unit cell $Y := [- \frac{1}{2}, \frac{1}{2} ]^d$, containing the void $\Omega \subset Y$.
The solutions of the respective cell problems
\[
\begin{cases}
\Delta w_i = 0 &\text{in } Y \setminus \overline{\Omega}, \\
\partial_{\boldsymbol{n}} w_i = - \langle \boldsymbol{n}, \, \boldsymbol{e}_i \rangle &\text{on } \partial \Omega,
\end{cases}
\qquad i = 1, \ldots, d,
\]
define the coefficients of the effective tensor by
\[
a_{i, j}(\Omega) = \int_{Y \setminus \overline{\Omega}} \big\langle \boldsymbol{e}_i + \nabla w_i, \ \boldsymbol{e}_j + \nabla w_j \big\rangle \operatorname{d}\!\boldsymbol{y}.
\]
Therefore, effective material tensor on the macroscopic scale is given by the solution of a problem on the microscopic scale.
Considering a sought material tensor $\mathbf{B} \in \mathbb{R}^{d \times d}_{\rm sym}$, which expresses desired material properties, we may ask the following question: Can we find a cavity $\Omega$ such that the effective tensor is as close to $\mathbf{B}$ as possible? In other terms, we want to minimise the tracking type functional
\[
J(\Omega) := \frac{1}{2} \big\| \mathbf{A}_0(\Omega) - \mathbf{B} \big\|_F^2.
\]
In [2], formulae for the shape gradient of the functional $J(\Omega)$ have been derived and numerical examples in two dimensions were presented, whereas in [3], integral equations were used to obtain numerical results in three dimensions.
These examples include simply connected cavities and also more complex cavities of genus greater than zero.
The calculations were performed with the isogeometric C++ library BEMBEL [1].
[1] J. Dölz, H. Harbrecht, S. Kurz, M. Multerer, S. Schöps, F. Wolf. Bembel: The fast isogeometric boundary element C++ library for Laplace, Helmholtz, and electric wave equation, SoftwareX, 11: 100476, 2020.
[2] M. Dambrine, H. Harbrecht. Shape optimization for composite materials and scaffold structures, Multiscale Modeling & Simulation, 18: 1136--1152, 2020.
[3] H. Harbrecht, M. Multerer, R. von Rickenbach. Isogeometric shape optimization of periodic structures in three dimensions, Computer Methods in Applied Mechanics and Engineering, 391: 114552, 2022.
Stokes Traction Method: A Numerical Approach to Volume Constrained Shape Optimization Problems Institute of Mathematics, Czech Academy of Sciences, Czech Republic
Numerically solving shape optimization problems usually takes advantage of the Zolesio-Hadamard form, which writes the shape derivative of the objective function into a boundary integral of the product of the shape grdient and the deformation field. Intuitively, one can choose the deformation field to take the form of the negative of the shape gradient, evaluated on the free boundary, as a gradient descent direction. However, such choice may cause instabilities and oscillations on the free boundary. This issue is a motivation for extending the deformation field to the computational domain in a smooth manner, this method is known as the traction method [1]. In this talk, solenoidal extensions to solve shape optimization problems with volume constraints will be considered. In particular, the deformation field will be extended to the computational domain by solving an incompressible Stokes equations with a Robin data defined as the negative of the shape gradient and the viscosity constant assumed to be sufficiently small. We apply such method to a vorticity maximization problem for the Navier--Stokes equations and compare with it the augmented Lagrangian method used by C. Dapogny et al. [2].
[1] H. Azegami, K. Takeuchi. A smoothing method for shape optimization: traction method using the Robin condition, Int J Comput Methods. 3(1): 21--33, 2006.
[2] C. Dapogny, P. Frey, F. Omnès, Y. Privat. Geometrical shape optimization in fluid mechanics using
FreeFem++, Struct Multidisciplinary Opt 58(6):2761–2788, 2018.
Non-conventional shape optimization methods for solving shape inverse problems 1Kanazawa University, Japan; 2Université Sultan Moulay Slimane, Morocco; 3Université Ibn Zohr, Morocco
We propose non-conventional shape optimization approaches for the resolution of shape inverse problems inspired by non-destructive testing and evaluation. Our main objective is to improve the detection of the concave parts or regions of the unknown inclusion/obstacle/boundary through two different strategies and under shape optimization settings. Firstly, we will introduce the so-called alternating direction method of multipliers or ADMM in shape optimization framework to solve a boundary inverse problem for the Laplacian with Dirichlet condition using a single boundary measurement. Secondly, we will consider a similar problem, but with the Robin condition, and demonstrate how we can effectively detect a void with concavities using several pairs of Cauchy data. We will illustrate the effectiveness of the proposed schemes by testing them to some shape detection problems with pronounced concavities and under noisy data. Examples are given in two and three dimensions.
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| 3:30pm - 4:00pm | C4: Coffee Break Location: ZHG Foyer |
| 4:00pm - 6:00pm | MS02 2: Advances in regularization for some classes of nonlinear inverse problems Location: VG1.102 Session Chair: Bernd Hofmann Session Chair: Robert Plato |
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New results for variational regularization with oversmoothing penalty term in Banach spaces 1Chemnitz University of Technology, Germany; 2University of Siegen, Germany; 3University of Siegen, Germany
In this talk on variational regularization for ill-posed nonlinear problems, we discuss the impact of utilizing an oversmoothing penalty term. This means that the searched-for solution of the considered nonlinear operator equation does not belong to the domain of definition of the penalty functional. In the past years, such variational regularization has been investigated comprehensively in Hilbert scales. Our present results extents those results to Banach scales. This new study includes convergence rates results for a priori and a posteriori choices of the regularization parameter, both for H\"older-type smoothness and low order-type smoothness. An illustrative example intends to indicate the specific characteristics of non-reflexive Banach spaces.
Iterative regularization methods for non-linear ill-posed operator equations in Banach spaces Visvesvaraya National institute of Technology, Nagpur, India
In this talk, we will introduce few simplified iterative regularization methods, in a Banach space setting, to obtain stable approximate solution of nonlinear ill-posed operator equation. We will discuss convergence analysis of these methods under suitable non linearity conditions. Numerical examples will be demonstrated to show applicability of these methods to practical problems.
An Abstract Framework for Stochastic Elliptic Inverse Problems. Rochester Institute of Technology, United States of America
Motivated by the necessity to identify stochastic parameters in a wide range of stochastic partial differential equations, this talk will focus on an abstract inversion framework for stochastic inverse problems. The stochastic inverse problem will be posed as a convex stochastic optimization problem. The essential properties of the solution maps and the solvability of the inverse problem will be discussed. Convergence rates for the stochastic inverse problem without requiring the so-called smallness condition will be presented. We will discuss an application of the abstract framework to estimate stochastic Lam\'e parameters in the system of linear elasticity. We will present numerical results to show the feasibility and efficacy of the developed framework.
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| 4:00pm - 6:00pm | MS04 2: Statistical and computational aspects of non-linear inverse problems Location: VG2.102 Session Chair: Richard Nickl Session Chair: Sven Wang |
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Parameter estimation for boundary conditions in ice sheet models University of Cambridge, United Kingdom
In this work, we are interested in the non-linear inverse problem which consists in retrieving the basal drag factor, an important parameter for scientists who want to understand the dynamics of ice sheets in the Antarctic. This drag factor takes the form of a Robin boundary condition at the bottom of the ice sheet in our PDE problem, and varies spatially along the boundary. Due to the thickness of the ice, the drag cannot be measured directly, and the only data available to us is the velocity of the ice at the surface.
We present a computational routine to estimate posterior densities of parameters for this Robin boundary condition.
MCMC Methods for Low Frequency Diffusion Data Università degli Studi di Torino, Italy
The talk will consider Bayesian nonparametric inference in multi-dimensional diffusion models from low-frequency data. Implementation of Bayesian procedures in such settings is a notoriously delicate task, due to the intractability of the likelihood, often requiring involved augmentation techniques. For the nonlinear inverse problem of inferring the diffusivity function in a stochastic differential equation, we rather propose to exploit the underlying PDE characterization of the transition densities, which allows the numerical evaluation of the likelihood via standard numerical methods for elliptic eigenvalue problems. A simple Metropolis-Hastings-type MCMC algorithm for inference on the diffusivity is then constructed, based on Gaussian process priors. The performance of the the algorithm will be illustrated via the results of numerical experiments. The talk will then discuss theoretical computational guarantees for MCMC methods in the considered inferential problem, based on derived local curvature properties for the log-likelihood, and connected to the `hot spots’ conjecture from spectral geometry.
Joint work with S. Wang (MIT).
Laplace priors and spatial inhomogeneity in Bayesian inverse problems Massachusetts Institute of Technology, United States of America
Spatially inhomogeneous functions, which may be smooth in some regions and rough in other regions, are modelled naturally in a Bayesian manner using so-called Besov priors which are given by random wavelet expansions with Laplace-distributed coefficients. This talk considers frequentist theoretical guarantees for Bayes methods with Besov priors, in the setting of non-linear inverse problems with Gaussian white noise. Our results are first derived under a general local Lipschitz assumption on the forward map. We then verify the assumption for two non-linear inverse problems arising from elliptic partial differential equations. We also discuss novel convergence rate results for penalized least squares estimators with $\ell_{1}$ wavelet penalty, which have a natural interpretation as maximum a posteriori (MAP) estimators. The true parameter is assumed to belong to some spatially inhomogeneous Besov class. In a setting with direct observations, we complement these upper bounds with a lower bound on the rate of contraction for arbitrary Gaussian priors. Consequently, while Laplace priors can achieve minimax-optimal rates over spatially inhomogeneous classes, Gaussian priors are limited to a (by a polynomial factor) slower contraction rate. This gives information-theoretical justification for the intuition that Laplace priors are more compatible with $\ell_{1}$ regularity structure in the underlying parameter.
Analysis of a localized non-linear ensemble Kalman-Bucy filter with sparse observations Universität Potsdam, Germany
With large scale availability of precise real time data, their incorporation into physically based predictive models, became increasingly important. This procedure of combining the prediction and observation is called data assimilation. One especially popular algorithm of the class of Bayesian sequential data assimilation methods is the ensemble Kalman filter which successfully extends the ideas of the Kalman filter to the non-linear situation. However, in case of spatio-temporal models one regularly relies on some version of localization, to avoid spurious oscillations.
In this work we develop a-priori error estimates for a time continuous variant of the ensemble Kalman filter, known as localized ensemble Kalman--Bucy filter. More specifically we aim for the scenario of sparse observations applied to models from fluid dynamics and space weather.
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| 4:00pm - 6:00pm | MS17: Machine Learning Techniques for Bayesian Inverse Problems Location: VG1.104 Session Chair: Angelina Senchukova |
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Stochastic Normalizing Flows for Inverse Problems via Markov Chains TU Berlin, Germany
Normalizing flows aim to learn the underlying probability distribution of given samples. For this, we train a diffeomorphism which pushes forward a simple latent distribution to the data distribution. However, recent results show that normalizing flows suffer from topolgical constraints and limited expressiveness. Stochastic normalizing flows can overcome these topological constraints and improve the expressiveness of normalizing flow architectures by combining deterministic, learnable flow transformations with stochastic sampling methods. We consider stochastic normalizing flows from a Markov chain point of view. In particular, we replace transition densities by general Markov kernels and establish proofs via Radon-Nikodym derivatives which allows to incorporate distributions without densities in a sound way. Further, we generalize the results for sampling from posterior distributions as required in inverse problems. The performance of the proposed conditional stochastic normalizing flow is demonstrated by numerical examples.
Bayesian computation with Plug & Play priors for inverse problems in imaging 1DTU, Denmark; 2Center for Science of Data, ENS Ulm; 3Universite de Paris, MAP5 UMR 8145,; 4Institut Universitaire de France; 5CMAP, Ecole Polytechnique; 6CNRS; 7Institut Polytechnique de Paris; 8Maxwell Insitute for Mathematical Sciences; 9School of Mathematical and Computer Sciences, Heriot-Watt University
This presentation is devoted to the study of Plug & Play (PnP) methods applied to inverse problems encountered in image restoration. Since the work of Venkatakrishnan et al. in 2013 [1], PnP methods
are often applied for image restoration in a Bayesian context. These methods aim at computing Minimum Mean Square Error (MMSE) or Maximum A Posteriori
(MAP) for inverse problems in imaging by combining an explicit likelihood
and an implicit prior defined by a denoising algorithm. In the literature, PnP methods differ mainly in the iterative scheme used for both optimization and sampling. In the case of optimization algorithms, recent works guarantee the convergence to a fixed point of a certain operator, fixed point which is not necessarily the MAP. In the case of sampling algorithms in the literature, there is no evidence of convergence. Moreover, there are still important open questions concerning the correct definition of the underlying Bayesian models or the computed estimators, as well as their regularity properties, necessary to ensure the stability of the numerical scheme. The aim of this thesis is to develop simple but efficient restoration methods while answering some of these questions. The existence and nature of MAP and MMSE estimators for PnP prior is therefore a first line of study. Three methods with convergence results are then presented, PnP-SGD for MAP estimation and PnP-ULA and PPnP-ULA for sampling. A particular interest is given to denoisers encoded by deep neural networks. The efficiency of these methods is demonstrated on classical image restoration problems such as denoising, deblurring or interpolation. In addition to allowing the estimation of MMSE, sampling makes possible the quantification of uncertainties, which is crucial in domains such as biomedical imaging.
[2] and [3] are the papers related to this talk.
[1] S. Venkatakrishnan, V. Singanallur, C. Bouman, B. Wohlberg. Plug-and-play priors for model based reconstruction, IEEE Global Conference on Signal and Information Processing, 2013. DOI: 10.1109/GlobalSIP.2013.6737048.
[2] R. Laumont, V. De Bortoli, A. Almansa, J. Delon, A. Durmus, M, Pereyra. Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie, SIAM Journal on Imaging Sciences 15(2): 701-737, 2022.
[3] R. Laumont, V. De Bortoli, A. Almansa, J. Delon, A. Durmus, M. Pereyra. On Maximum a Posteriori Estimation with Plug & Play Priors and Stochastic Gradient Descent, Journal of Mathematical Imaging and Vision 65: 140–163, 2023.
Edge-preserving inversion with heavy-tailed Bayesian neural networks priors 1LUT University, Finland; 2University of Potsdam, Germany
We study Bayesian inverse problems where the unknown target function is piecewise constant. Priors based on neural networks with heavy-tailed-distributed weights/biases have been employed due to their discretization-independent property and ability to capture discontinuities. We aim at developing neural network priors whose parameters are drawn from Student's t distributions. The idea is to parameterize the unknown function using a neural network which sets a finite-dimensional inference framework. This requires finding the posterior distribution of the weights/biases of the network representation. The resulting posterior is, however, high-dimensional and multimodal which makes it difficult to characterize using traditional sampling algorithms. Therefore, we explore data assimilation techniques to sample the posterior distribution more effectively. As a numerical example, we consider a simple signal deconvolution to illustrate the properties of the prior.
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| 4:00pm - 6:00pm | MS19 2: Theory and algorithms of super-resolution in imaging and inverse problems Location: VG3.103 Session Chair: Habib Ammari Session Chair: Ping Liu |
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IFF: A Super-resolution Algorithm for Multiple Measurements The Hong Kong University of Science and Technology, Hong Kong S.A.R. (China)
The problem of reconstructing one-dimensional point sources from their Fourier measurements in a bounded interval $[-\Omega, \Omega]$ is known to be challenging in the regime where the spacing of the sources is below the Rayleigh length $\frac{\pi}{\Omega}$. In this talk, we present a super-resolution algorithm, called Iterative Focusing-localization and Filtering (IFF), to resolve closely spaced point sources from their multiple measurements that are obtained by using multiple unknown illumination patterns. The new proposed algorithm requires no prior information about the source numbers and allows for a subsampling strategy that can circumvent the computation of singular-value decomposition for large matrices as in the usual subspace methods.
In the talk, we will also discuss the theoretical results of the methods behind the algorithm. The derived results imply a phase transition phenomenon. Numerical results show that the algorithm can achieve a stable reconstruction for point sources with a minimum separation distance that is close to the theoretical limit as well as the phase transition phenomenon predicted by the theoretical analysis.
Vectorized Hankel Lift: A Convex Approach for Blind Super-Resolution of Point Sources Fudan University, China, People's Republic of
Blind super-resolution is the problem of estimating high-resolution information about a signal from its low-resolution measurements when the point spread functions (PSFs) are unknown. It is a common problem in many scientific and engineering research areas, such as machine learning, signal processing, and computer vision. Blind super-resolution can be cast as a low-rank matrix recovery problem by exploiting the inherent simplicity of the signal and the low-dimensional structure of the PSFs.
In this talk, we will discuss the low-rank matrix recovery problem for blind super-resolution of point sources. The target matrices associated with these problems are not only low rank but also highly structured. Convex approaches are proposed for the corresponding low-rank matrix recovery problems. Theoretical guarantees are established showing that near-optimal sample complexity is sufficient for successful recovery.
Super-resolved Lasso University of Bath, United Kingdom
The behaviour of sparse regularization using the Lasso method is well understood when dealing with discretized linear models. However, the behaviour of Lasso is poor when dealing with models with very large parameter spaces and exact localisation of the sparse support is often not possible due to discretization (gridding) issues. We introduced a new optimization problem known as the super-resolved Lasso, by considering a higher order expansion of the continuous operator, we show that we can precisely recover the support when the 'true' signal lies up to a fraction off the grid. This is joint work with Gabriel Peyre.
Approximate inverse scattering via convex programming MaLGa center, department of Mathematics, University of Genoa
In this work, we propose to apply and adapt known results on convex variational methods for inverse problems to the inverse scattering problem. We rely on approximations to circumvent its nonlinearity, and discuss recovery guarantees and numerical methods.
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| 4:00pm - 6:00pm | MS21 2: Prior Information in Inverse Problems Location: VG2.103 Session Chair: Andreas Horst Session Chair: Jakob Lemvig |
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Regularized, pretrained and subspace-restricted Deep Image Prior for CT reconstruction 1Department of Computer Science, University College London, United Kingdom; 2Department of Engineering, University of Cambridge, United Kingdom; 3Center for Industrial Mathematics, University of Bremen, Germany; 4Department of Mathematics, The Chinese University of Hong Kong, P. R. China; 5Research Unit of Mathematical Sciences, University of Oulu, Finland
Computed tomography (CT) is an important tool in both medicine and industry. By now, a great variety of deep learning (DL) techniques has been developed for inverse imaging tasks including CT reconstruction. In constrast to most DL approaches, the deep image prior (DIP) is an unsupervised framework that does not rely on a large training dataset, but only on the single degraded observation. The central observation with DIP is that the early-stopped optimization an untrained networks can lead to favorable solutions, thus acting as an implicit prior.
We extend the DIP in several ways. First, we add an explicit prior in the form of a total variation regularization term, which can stabilize and improve the reconstruction. Second, we pretrain on a post-processing task with easy-to-generate synthetic data, which induces prior information, learned from the synthetic image class and the operator-specific degradation, into the subsequent unsupervised DIP optimization. This two-stage procedure of supervised pretraining and unsupervised fine-tuning is called the educated DIP (EDIP) and often requires a significantly shorter optimization time in the fine-tuning stage compared to untrained DIP. Finally, we experiment with restricting the parameter space in the fine-tuning stage of EDIP. Using an affine linear subspace, which is expanded around the pretraining parameters with a sparsified basis obtained from many checkpoints saved during the pretraining, both overfitting behaviour can be reduced and second order optimization methods become feasible, enabling more stable and faster reconstruction.
Monitoring of hemorrhagic stroke using Electrical Impedance Tomography University of Eastern Finland, Finland
In this talk, we present recent progress in development of electrical impedance tomography (EIT) based bedside monitoring of hemorrhagic stroke. We present the practical setup and pipeline for this novel application of EIT and the CT prior informed image reconstruction method we have developed for it. Feasibility of the approach is studied with simulated data from anatomically highly accurate simulation models and experimental phantom data from a laboratory setup.
Edge-preserving inversion with $\alpha$-stable priors 1LUT University, Finland; 2Heriot Watt University
The $\alpha$-stable distributions are a family of heavy-tailed and infinitely divisible distributions that are well-suited to edge-preserving inversion in the context of discretization of infinite-dimensional continuous-time statistical inverse problems. In this talk we discuss some of the technical issues arising from the application of such priors.
Optimal learning of high-dimensional classification problems using deep neural networks Katholische Universität Eichstätt-Ingolstadt, Germany
We study the problem of learning classification functions from noiseless training samples, under the assumption that the decision boundary is of a certain regularity. We establish universal lower bounds for this estimation problem, for general classes of continuous decision boundaries. For the class of locally Barron-regular decision boundaries, we find that the optimal estimation rates are essentially independent of the underlying dimension and can be realized by empirical risk minimization methods over a suitable class of deep neural networks. These results are based on novel estimates of the $L^1$ and $L^\infty$ entropies of the class of Barron-regular functions.
This is joint work with Philipp Petersen (University of Vienna).
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| 4:00pm - 6:00pm | MS22 2: Imaging with Non-Linear Measurements: Tomography and Reconstruction from Phaseless or Folded Data Location: VG1.101 Session Chair: Matthias Beckmann Session Chair: Robert Beinert Session Chair: Michael Quellmalz |
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Uniqueness theory for 3D phase retrieval and unwrapping UC Davis, United States of America
We present general measurement schemes with which unique conversion of diffraction patterns into the phase projection for a strong phase object (respectively, the projection for a weak phase object) in each direction can be guaranteed without the knowledge of relative orientations and locations.
This approach has the practical implication of enabling classification and alignment, when relative orientations are unknown, to be carried out in terms of (phase) projections, instead of diffraction patterns.
We also discuss conditions for unique determination of a strong phase object from its phase projection data, including, but not limited to, random tilt schemes densely sampled from a spherical triangle of vertexes in three orthogonal directions and other deterministic tilt schemes.
Interaction Models in Ptychography Helmholtz Munich, Germany
Over the recent years, ptychography became a standard technique for high resolution scanning transmission electron microscopy.
To achieve better and better resolutions, the mathematical model had to be refined several times.
In the simplest approach the measurements can be understood as a discrete, phaseless short-time Fourier transform
$$ I(s) = \left| \mathcal F [\phi \cdot \tau_s w]\right|^2. $$
Here, $\tau_s w$ is an (often unknown) window function, shifted by $s$, $\phi$ the object we would like to recover and $I(s)$
the measured intensity at position $s$. Typically, a few thousands of such measurements are recorded, where $s$ lies on a regular grid. However, for specimens thicker than a few nanometers, this approximation already breaks down.
A more sophisticated interaction model $M(\phi, \tau_s w)$ is needed.
Furthermore, the incoherence of the microscope is a crucial limit and has to be considered as well.
We end up with a model like
$$ I(s) = \sum_j \left| \mathcal F [M( \phi, \tau_s w_j)]\right|^2. $$
In this talk we give an overview over these approaches and discuss their challenges.
We also show reconstructions of experimental data.
Tackling noise in multiple dimensions via hysteresis modulo sampling Imperial College London, United Kingdom
Mapping a multi-dimensional function in a predefined bounded amplitude range can be achieved via a transformation known as the modulo nonlinearity. The recovery of the function from the modulo samples was addressed for the one-dimensional case as part of the Unlimited Sensing Framework (USF) based on uniform samples [1-3], but also based on neuroscience inspired time encoded samples [4]. Alternative analyses implemented de-noising of modulo data in one and multiple dimensions [5]. Extensions of the recovery to multi-dimensional inputs typically amount to a line-by-line analysis of the data on one-dimensional slices. Apart from enabling the reconstruction of a wider class of inputs, this approach does not show an inherent need to apply modulo for high dimensional inputs.
In this talk, we present a modulo sampling operator specifically tailored to multiple dimensional inputs, called multi-dimensional modulo-hysteresis [6]. It is shown that the model can use dimensions two and above to generate redundancy that can be exploited for robust input recovery. This redundancy is particularly made possible by the hysteresis parameter of the operator. A few properties of the new operator are proven, followed by a guaranteed input recovery approach. We demonstrate theoretically and via numerical examples that when the input is corrupted by Gaussian noise the reconstruction error drops asymptotically to 0 for high enough sampling rates, which was not possible for the one-dimensional scenario. We additionally extend the recovery guarantees to classes of non-bandlimited inputs from shift-invariant spaces and include additional simulations with different noise distributions. This work enables extensions to multi-dimensional inputs for neuroscience inspired sampling schemes [4], inherently known for their noisy characteristics.
[1] Bhandari, F. Krahmer, R. Raskar. On unlimited sampling and reconstruction. IEEE Trans. Sig. Proc. 69 (2020) 3827–3839. doi:10.1109/tsp.2020.3041955
[2] Bhandari, F. Krahmer, T. Poskitt. Unlimited sampling from theory to practice: Fourier-Prony recovery and prototype ADC. IEEE Trans. Sig. Proc. (2021) 1131-1141. doi:10.1109/TSP.2021.3113497.
[3] D. Florescu, F. Krahmer, A. Bhandari. The surprising benefits of hysteresis in unlimited sampling: Theory, algorithms and experiments. IEEE Trans. Sig. Proc. 70 (2022) 616–630. doi:10.1109/tsp.2022.3142507
[4] D. Florescu, A. Bhandari. Time encoding via unlimited sampling: theory, algorithms and hardware validation. IEEE Trans. Sig. Proc. 70 (2022) 4912-4924.
[5] H. Tyagi. Error analysis for denoising smooth modulo signals on a graph. Applied and Computational Harmonic Analysis 57 (2022) 151–184.
[6] D. Florescu, A. Bhandari. Multi-Dimensional Unlimited Sampling and Robust Reconstruction. arXiv preprint 2002 arXiv:2209.06426.
Multi-window STFT phase retrieval University of Vienna, Austria
We consider the problem of recovering a function $f\in L^2(\mathbb{R})$ (up to a multiplicative phase factor) from phase-less samples of its short-time Fourier transform $V_g f$, where
$$
V_g f(x,y) =\int_\mathbb{R} f(t) \overline{g(t-x)} e^{-2\pi i y t}\,dt,
$$
with $g\in L^2(\mathbb{R})$ a window function. Recently established dicretization barriers state that in general $f$ is not uniquely determined given $|V_g f(\Lambda)|:=\{|V_g f(\lambda)|, \lambda\in\Lambda\}$ if $\Lambda\subseteq \mathbb{R}^2$ is a lattice (irrespectively of the choice of the window $g$ and the density of the lattice $\Lambda$).
We show that these discretization barriers can be overcome by employing multiple window functions. More precisely, we prove that
$$
\{|V_{g_1} f(\Lambda)|, |V_{g_2} f(\Lambda)|, |V_{g_3} f(\Lambda)|, |V_{g_4} f(\Lambda)|\}
$$
uniquely determines $f\sim e^{i\theta}f$ when $g_1,\ldots,g_4$ are suitably chosen windows provided that $\Lambda$ has sufficient density.
Joint work with Philipp Grohs and Lukas Liehr.
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| 4:00pm - 6:00pm | MS23 1: Recent developments in reconstruction methods for inverse scattering and electrical impedance tomography Location: VG1.103 Session Chair: Roland Griesmaier Session Chair: Nuutti Hyvönen |
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Nonlinear impedance boundary conditions in inverse obstacle scattering Karlsruhe Institute of Technology, Germany
Nonlinear impedance boundary conditions in acoustic scattering are used as a model for perfectly conducting objects coated with a thin layer of a nonlinear medium. We consider a scattering problem for the Helmholtz equation with a nonlinear impedance boundary condition of the form
$$
\dfrac{\partial u}{\partial \nu} + ik\lambda u = g(\cdot,u) \quad \text{on} \ \partial D,
$$
where $\nu$ denotes the unit normal vector, $\lambda \in L^{\infty}(\partial D)$ is a complex-valued impedance function, and $g: \partial D \times \mathbb{C} \to \mathbb{C}$ gives an additional nonlinear term with respect to the total field $u$. The contributed talk is devoted to the well-posedness of the direct problem, the determination of the domain derivative, and the inverse problem, which consists in reconstructing the shape of the scattering object from given far field data. Numerical results are presented relying on an all-at-once regularized Newton-type method based on the linearization of the forward problem and of the domain-to-far-field operator.
Far field operator splitting and completion for inhomogeneous medium scattering Karlsruhe Institute of Technology, Germany
We consider scattering of time-harmonic acoustic waves by an ensemble of compactly supported penetrable scattering objects in 2D.
These scattering objects are illuminated by an incident plane wave.
The resulting total wave is the superposition of incident and scattered wave and solves a scattering problem for the Helmholtz equation.
For guaranteeing uniqueness, the scattered wave must fulfill the Sommerfeld radiation condition at infinity.
In our consideration, measurements of the total wave are replaced by the corresponding far field operator.
This operator contains all information about the scattered wave far away from the scattering objects for all possible illumination directions.
We are interested in two inverse problems.
On the one hand, given a limited observation of this far field operator, we want to determine its missing part, which we refer to as operator completion problem.
'Limited observation' in this context means, that we do not have access to measurements for all illumination directions or that we cannot measure in all observation directions around the scattering objects.
On the other hand, given the far field operator for the ensemble of scattering objects, we want to determine the far field operators of the individual scattering objects.
This is what we refer to as operator splitting problem.
Multiple reflection effects cause, in contrast to the first problem, the nonlinearity of this second problem.
We characterize spaces containing the individual, for the two problems relevant components of the far field operator.
Operators in these spaces turn out to have a low rank and sparsity properties with respect to some known modulated Fourier frame.
Furthermore, this rank and frame can be determined under knowledge of the locations and sizes of the scatterer's components.
In my talk I will suggest two reformulations of the inverse problems, a least squares norm formulation and a $l^1\times l^1$-norm minimization, and appropriate algorithms for solving these formulations numerically.
Moreover, I will present stability results for these reconstructions and support them by numerical experiments.
Uniqueness, error bounds and global convergence for an inverse Robin transmission problem with a finite number of electrodes Goethe University Frankfurt, Germany
Medical imaging and non-destructive testing holds the challenge of determining information of the interior of the body by taking measurements at its boundary. We consider an inverse coefficient problem that is motivated by impedance-based corrosion detection. The aim is to reconstruct an unknown transmission coefficient function in an elliptic PDE from finitely many measurements that correspond to voltage/current measurements on electrodes attached to the domain's outer boundary. We mathematically characterize how many electrodes are required to achieve a desired resolution, we derive stability and error estimates, and we discuss globally convergent numerical reconstruction methods by rewriting the problem as convex non-linear semidefinite optimization problem.
Subspace surrogate methods for Electrical Impedance Tomography Aalto University, Finland
Iterative reconstruction methods for Electrical Impedance Tomography (EIT) often require solving the forward problem multiple times with different inner conductivity parameters using the Finite Element Method (FEM). The solution of the FEM problem requires solving a large sparse linear system of equations during each round of iteration. We present novel subspace methods for solving these linear systems using the same subspace for any conductivity parameter. We report that there seems to be structure in the problem that generally allows the size of these subspaces to stay small, enabling efficient computation.
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| 4:00pm - 6:00pm | MS28 2: Modelling and optimisation in non-Euclidean settings for inverse problems Location: VG1.108 Session Chair: Luca Calatroni Session Chair: Claudio Estatico Session Chair: Dirk Lorenz |
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Gradient descent-based algorithms for inverse problems in variable exponent Lebesgue spaces 1Dip. di Matematica (DIMA), Università di Genova, Italy; 2Dept. of Computer Science, University College London, UK; 3CNRS, UCA, Inria, Laboratoire I3S, Sophia-Antipolis, France
Variable exponent Lebesgue spaces $\ell^{(p_n)}$ have been recently proved to be the appropriate functional framework to enforce pixel-adaptive regularisation in signal and image processing applications (see [1]), combined with gradient descent (GD) or proximal GD strategies. Compared to standard Hilbert or Euclidean settings, however, the application of these algorithms in the Banach setting of $\ell^{(p_n)}$ is not straightforward due to the lack of a closed-form expression and the non-separability property of the underlying norm. We propose the use of the associated separable modular function [2, 3], instead of the norm, to define algorithms based on GD in $\ell^{(p_n)}$ and consider a stochastic GD [3, 4] to reduce the per-iteration cost of iterative schemes, used to solve linear inverse real-world image reconstruction problems.
[1] B. Bonino, C. Estatico, and M. Lazzaretti. Dual descent regularization algorithms in variable exponent Lebesgue spaces for imaging, Numer. Algorithms 92(6), 2023.
[2] M. Lazzaretti, L. Calatroni, and C. Estatico. Modular-proximal gradient algorithms in variable exponent Lebesgue spaces, SIAM J. Sci. Compu. 44(6), 2022.
[3] M. Lazzaretti, Z. Kereta, L. Calatroni, and C. Estatico. Stochastic gradient descent for linear inverse problems in variable exponent Lebesgue spaces, 2023. [https://arxiv.org/abs/2303.09182]
[4] Z. Kereta, and B. Jin. On the convergence of stochastic gradient descent for linear inverse problems in Banach spaces, SIAM J. Imaging Sci. (in press), 2023.
Multiscale hierarchical decomposition methods for images corrupted by multiplicative noise University of Klagenfurt, Austria
Recovering images corrupted by multiplicative noise is a well known challenging task. Motivated by the success of multiscale hierarchical decomposition methods (MHDM) in image processing, we adapt a variety of both classical and new multiplicative noise removing models to the MHDM form. We discuss well-definedness and convergence of the proposed methods. Through comprehensive numerical experiments and
comparisons, we qualitatively and quantitatively evaluate the validity of the considered models. By construction, these multiplicative multiscale hierarchical decomposition methods have the added benefit of recovering many scales of an image, which can provide features of interest beyond image restoration.
Proximal point algorithm in spaces with semidefinite inner product 1TU Braunschweig, Germany; 2University of Graz, Austria
We introduce proximal point algorithms in spaces with semidefinite inner products. We focus our attention in particular on products induced by self-adjoint positive semidefinite linear operators defined on Hilbert spaces. We show convergence for the algorithm under suitable conditions and we investigate applications for splitting methods. As first application, we devise new schemes for finding minimizers of the sum of many convex lower semicontinuous functions and show some applications of these new schemes to congested transport and distributed optimization in the context of Support Vector Machines, investigating their behavior. Finally, we analyze the convergence of the proximal point algorithm letting vary the (semidefinite) metric at each iteration. We discuss applications of this analysis to the primal-dual Douglas-Rachford scheme, investigating adaptive stepsizes for the method.
Asymptotic linear convergence of fully-corrective generalized conditional gradient methods 1University of Graz, Austria; 2University of Twente, The Netherlands; 3Humboldt-Universität zu Berlin, Germany
We discuss a fully-corrective generalized conditional gradient (FC-GCG) method [1] for the minimization of Tikhonov functionals associated with a linear inverse problem, a convex discrepancy and a convex one-homogeneous regularizer over a Banach space. The algorithm alternates between updating a finite set of extremal points of the unit ball of the regularizer [2] and optimizing on the conical hull of these extremal points, where each iteration requires the solution of one linear problem and one finite-dimensional convex minimization problem. We show that the method converges sublinearly to a solution and that imposing additional assumptions on the associated dual variables accelerates the method to a linear rate of convergence. The proofs rely on lifting, via Choquet's theorem, the considered problem to a particular space of Radon measures well as the equivalence of the FC-CGC method to a primal-dual active point (PDAP) method for which linear convergence was recently established. Finally, we present applications scenarios where the stated assumptions for accelerated convergence can be satisfied [3].
[1] Kristian Bredies, Marcello Carioni, Silvio Fanzon and Daniel Walter. Asymptotic linear convergence of fully-corrective generalized conditional gradient methods, 2021. [ArXiv preprint 2110.06756]
[2] Kristian Bredies and Marcello Carioni. Sparsity of solutions for variational inverse problems with finite-dimensional data, Calculus of Variations and Partial Differential Equations 59(14), 2020.
[3] Kristian Bredies, Marcello Carioni, Silvio Fanzon and Francisco Romero. A generalized conditional gradient method for dynamic inverse problems with optimal transport regularization, Foundations of Computational Mathematics, 2022.
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| 4:00pm - 6:00pm | MS38 2: Inverse eigenvalue problems in astrophysics Location: VG2.105 Session Chair: Charlotte Gehan Session Chair: Damien Fournier |
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Mode identification in rapidly-rotating stars: paving the way to inverse methods Universidad de Granada, Spain
Asteroseismology has opened a window on the internal physics of thousands of stars, by relating pulsations of stars to their internal physics. Mode identification, namely the process of associating a measured oscillation frequency to the corresponding mode geometry and properties, is the preliminary step of the seismic analysis. In upper main-sequence stars, that often rotate rapidly, this identification is challenging and largely incomplete, as modes assume complex geometries and frequencies shift under the combined influence of the Coriolis force and centrifugal flattening.
In this contribution, I will describe the various classes of mode geometries that emerge in rapidly rotating stars and their differences with slow rotators. After discussing how their frequencies and periods relate with structural quantities, allowing us to derive constraints on the stellar evolution, I will discuss the approaches developed towards inversion methods.
Progress in Asteroseismology: Where We Stand and Where We'll Go Max Planck Institute for Astrophysics, Germany
Over the past decade, asteroseismic inversion techniques have emerged as crucial tools to help identify the missing physics in our understanding of stellar evolution. In this talk, I will provide a comprehensive overview of the recent progress in asteroseismology and showcase the major advancements in the field, with a focus on novel methods for probing stellar structure and evolution. I will present new inferences into various different types of pulsating stars and improvements in our ability to infer internal stellar dynamics. I will also share our latest research on non-linear inversion methods and the application of inversions to massive stars. Finally, I will discuss the future of asteroseismology, including the expected yield of several forthcoming missions. This talk aims to highlight the remarkable progress in asteroseismology and stimulate discussions on future avenues for continued advancement.
Helioseismic inversions for active latitudes 1Max Planck Institute for Solar System Research, Göttingen, Germany; 2Georg-August-Universität Göttingen, Göttingen, Germany
The eleven-year solar activity cycle is known to affect the acoustic p-modes; higher activity is correlated with increase in mode frequencies and decrease in their lifetimes. This is also seen in the autocorrelation function of the integrated light. Recently, the solar cycle is also observed in travel-time measurements of p-mode wavepackets for multiple skips [1]. In this work, we first construct a forward model to explain the variation in travel-time measurements with solar activity. A simplified model is constructed by considering axisymmetric averages of the magnetic activity associated with perturbations in the near-surface wave-speed. The perturbations are constructed by longitudinally averaging synoptic magnetograms from SDO/HMI and the SOHO/MDI. The maximum correlation between observed and modeled travel-time shifts is as high as $0.92$ for some skips, much less for others. Subsequently, we setup an inverse problem to invert for the latitudinal distribution of solar activity from travel-time observations. This work is a first step towards the goal of retrieving stellar butterfly diagrams from asteroseismic observables.
[1] V. Vasilyev, L. Gizon, 2023, submitted.
Probing solar turbulent viscosity with inertial modes 1Max Planck Institute for Solar System Research, Germany; 2Institut für Astrophysik, Georg-August-Universität Göttingen
Solar inertial modes offer new possibilities to probe the solar interior down to the tachocline, and can be used to constrain such properties as the differential rotation or the spectrum of turbulent energy throughout the convection zone. Linear analysis enables us to compute the discrete eigenfrequencies of these modes [1,2]. However, because the inertial modes overlap in the frequency domain, especially for high azimuthal order $m$, this is not enough: it is necessary to model the power spectral density in the whole inertial frequency range, which can be done by modelling the stochastic source of excitation of the modes by turbulent vorticity.
In this presentation, I will show how this can be achieved in a 2D spherical setting, based on the formalism by [3]. I will then describe how this formalism can be used to relate changes in turbulent properties, with a focus on the turbulent viscosity, to their effects on the whole inertial range power spectral density (forward problem), as well as discuss the corresponding inverse problem.
[1] L. Gizon, D. Fournier, M. Albekioni. Effect of latitudinal differential rotation on solar Rossby waves: Critical layers, eigenfunctions, and momentum fluxes in the equatorial $\beta$ plane, Astron. Astrophys 642: A178, 2020.
[2] Y. Bekki, R.H. Cameron, L. Gizon. Theory of solar oscillations in the inertial frequency range: Linear modes of the convection zone. Astron. Astrophys 662: A16, 2022.
[3] J. Philidet, and L. Gizon. Interaction of solar inertial modes with turbulent convection,
A 2D model for the excitation of linearly stable modes. Astron. Astrophys 673: A124, 2023.
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| 4:00pm - 6:00pm | MS42: Inverse Problems with Anisotropy Location: VG3.104 Session Chair: Kim Knudsen |
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A density property for tensor products of gradients of harmonic functions National Yang Ming Chiao Tung University, Taiwan
In this talk I will present a recent result showing that linear combinations of tensor products of $k$ gradients of harmonic functions, with $k$ at least three, are dense in $C(\overline{\Omega})$, for any bounded domain $\Omega$ in dimension 3 or higher. This kind of density result has applications to inverse problems for elliptic quasilinear equations/systems in divergence form, where the nonlinear part of the "conductivity'' is anisotropic. The talk will be based on two papers written in collaboration with A. Feizmohammadi.
Reconstructing anisotropic conductivities on manifolds University of Jyväskylä, Finland
We study the problem of recovering an electrical anisotropic conductivity from interior power density measurements on a two-dimensional Riemannian manifold. This problem arises in Acousto-Electric Tomography and is motivated by the geometric Calderón problem of recovering the metric from the Dirichlet-to-Neumann map. In contrast to the geometric Calderón problem, we consider a conductive Riemannian manifold and treat the conductivity and metric separately. Assuming that the metric is known, for two-dimensional Riemannian manifolds with genus zero, we highlight in this talk that under certain assumptions on the power density data it is possible to recover the conductivity uniquely and constructively from the data. We illustrate our findings with a numerical experiment and comment on how added noise on the manifold affects the reconstructed conductivity.
Imaging anisotropic conductivities from current densities The Chinese University of Hong Kong
In this talk, we discuss a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. The approach is based on a regularized output least-squares formulation with the standard $L^2$ penalty, which is then discretized by the standard Galerkin finite element method. We discuss the analysis of the discretized problem, especially the convergence of the discrete approximations with respect to the mesh size, using the discrete counterpart of $H$-convergence, when the discretization is uniform or adaptive. We present extensive two-dimensional numerical examples to show the efficiency of the proposed method.
Stability and reconstruction for anisotropic inverse problems. University of Limerick, Ireland
In this talk we investigate the issues of stability and reconstruction in inverse problems in the presence of anisotropy. As is well-known, there is a fundamental obstruction to the unique determination of the anisotropic conductivity of materials. Such obstruction is based on the observation that any deffeomorphism of a domain $\Omega$ that keeps its boundary $\partial\Omega$ fixed, changes the conductivity in $\Omega$ by keeping the boundary measurements unchanged. In this talk we will investigate how to circumvent this obstruction and restore well-posedness in the problem.
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| 4:00pm - 6:00pm | MS45 1: Optimal Transport meets Inverse Problems Location: VG0.111 Session Chair: Marcello Carioni Session Chair: Jan-F. Pietschmann Session Chair: Matthias Schlottbom |
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Efficient adversarial regularization for inverse problems 1University of Bath, United Kingdom; 2University of Twente, Netherlands; 3KTH - Royal Institute of Technology, Sweden; 4University of Cambridge, United Kingdom
We propose a new optimal transport-based approach for learning end-to-end reconstruction operators using unpaired training data for ill-posed inverse problems. The key idea behind the proposed method is to minimize a weighted combination of the expected distortion in the measurement space and the Wasserstein-1 distance between the distributions of the reconstruction and the ground truth. The regularizer is parametrized by a deep neural network and learned simultaneously with an unrolled reconstruction operator in an adversarial training framework. The variational problem is then initialized with the output of the reconstruction network and solved iteratively till convergence. Notably, it takes significantly fewer iterations to converge as compared to variational methods, thanks to the excellent initialization obtained via the unrolled reconstruction operator. The resulting approach combines the computational efficiency of end-to-end unrolled reconstruction with the well-posedness and noise-stability guarantees of the variational setting. We demonstrate with the example of image reconstruction in X-ray computed tomography (CT) that our approach outperforms state-of-the-art unsupervised methods and that it outperforms or is at least on par with state-of-the-art supervised data-driven CT reconstruction approaches.
Data Driven Gradient Flows 1Universität Augsburg, Germany; 2UT Twente, Netherlands
We present a framework enabling variational data assimilation for gradient flows in general metric spaces, based on the minimizing movement (or Jordan-Kinderlehrer-Otto) approximation scheme. After discussing stability properties in the most general case, we specialise to the space of probability measures endowed with the Wasserstein distance. This setting covers many non-linear partial differential equations (PDEs), such as the porous medium equation or general drift-diffusion-aggregation equations, which can be treated by our methods independent of their respective properties (such as finite speed of propagation or blow-up). We then focus on the numerical implementation of our approach using an primal-dual algorithm. The strength of our approach lies in the fact that by simply changing the driving functional, a wide range of PDEs can be treated without the need to adopt the numerical scheme. We conclude by presenting detailed numerical examples.
The quadratic Wasserstein metric for inverse data matching 1The University of Texas at Austin, USA; 2Columbia University, USA; 3Cornell University, USA
This work characterizes, analytically and numerically, two major effects of the quadratic Wasserstein ($W_2$) distance as the measure of data discrepancy in computational solutions of inverse problems. First, we show, in the infinite-dimensional setup, that the $W_2$ distance has a smoothing effect on the inversion process, making it robust against high-frequency noise in the data but leading to a reduced resolution for the reconstructed objects at a given noise level. Second, we demonstrate that, for some finite-dimensional problems, the $W_2$ distance leads to optimization problems that have better convexity than the classical $L^2$ and $\dot{H}^{-1}$ distances, making it a more preferred distance to use when solving such inverse matching problems. This talk is based on the work [1].
[1] B. Engquist, K. Ren, Y. Yang. The quadratic Wasserstein metric for inverse data matching, Inverse Problems 36(5): 055001, 2020.
Quadratic regularization of optimal transport problems TU Braunschweig, Germany
In this talk we consider regularization of optimal transport problems with quadratic terms. We use the Kantorovich for of optimal transport and add a quadratic regularizer, which forces the transport plan to be a square integrable function instead of a general measure. We derive the dual problem and show strong duality and existence of primal and dual solutions to the regularized problem. Then we derive two algorithms to solve the dual problem of the regularized problem: A Gauss-Seidel method and a semismooth quasi-Newton method and investigate both methods numerically. Our experiments show that the methods perform well even for small regularization parameters. Quadratic regularization is of interest since the resulting optimal transport plans are sparse, i.e. they have a small support (which is not the case for the often used entropic regularization where the optimal transport plan always has full measure). Finally we briefly sketch an extension of the results to the more general case of regularization with so-called Young functions which unifies the entropic and the quadratic regularization.
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| 4:00pm - 6:00pm | MS47 2: Scattering and spectral imaging: inverse problems and algorithms Location: VG3.101 Session Chair: Eric Todd Quinto Session Chair: Gael Rigaud |
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V-line tensor tomography 1University of Texas at Arlington, United States of America; 2Dartmouth College, United States of America; 3Indian Institute of Technology Gandhinagar, India
The V-line transform (VLT) maps a function to its integrals along V-shaped trajectories with a vertex inside the support of the function. This transform and its various generalizations appear in mathematical models of several imaging techniques utilizing scattered particles. The talk presents recent results on inversion of generalized VLTs defined on vector fields and symmetric 2-tensor fields in the plane.
Optimal parameter design for spectral CT 1NC State University, United States of America; 2University of Chicago, USA
Spectral CT is an x-ray transmission imaging technique that uses the energy dependence of x-ray photon attenuation to determine elemental composition of an object of interest. Mathematically, forward spectral CT measurements are modeled by a nonlinear integral transform for which no analytical inversion is available. In this talk, I will present some of our recent results on the global uniqueness and the stability of spectral CT reconstructions. These analyses are useful for designing optimal scan parameters, which will be demonstrated using numerical simulations. This is joint work with G. Bal and E. Sidky.
Gamma ray imaging with bidirectional Compton cameras 1FAU Erlangen-Nürnberg; 2Deutsches Elektronen-Synchrotron DESY; 3Universität Hamburg
For in-situ gamma ray spectrometry, Compton cameras are an efficient imaging tool that operate without collimation and therefore attain large sensitivities. Conventionally, Compton cameras are built with separated scattering and absorbing layers. This setup allows detector materials to be tailored to maximize sensitivity and have good energetic or spatial resolution, but often sacrifices the camera's ability to produce spatially resolved images in the whole $4\pi$ field of view resulting in a de facto collimation. We propose the mathematical model of a Compton camera whose detectors are all considered to both scatter and absorb the incoming gamma rays. Since the measurements of the camera do not give any information about the direction of a coincidence of scattering and absorption, we talk of a bidirectional Compton camera. The additional uncertainty is reflected in the operator describing the forward model, which is the weighted sum of two conical Radon transforms. We demonstrate the ability of the system to efficiently image gamma radiation by numerical results on simulated and measured data.
A hybrid algorithm for material decomposition in multi-energy CT Universität Innsbruck, Austria
The aim of multi-energy CT is to reconstruct the distribution of a known set of substances inside a sample by performing CT measurements at different energies. The measurements can be achieved either by using different tube voltages at the source or by means of energy sensitive detectors (e.g. photon counting detectors). In any case the energy dependent absorption of the materials under consideration is used to distinguish the substances in the sample which leads to a nonlinear reconstruction problem. The majority of reconstruction algorithms can be divided into those performing the material decomposition in the sinogram domain and those decomposing the image after inversion of the Radon transform for each energy bin. Both types of algorithms can be implemented very efficiently but also suffer from specific artefacts. More recently one-step algorithms performing decomposition and inversion in one pass have become an active research area. While they eliminate most of the problems of two-step approaches, they are usually computationally costy because they are iterative in nature and the relative similarity of absorption coefficients often leads to poor convergence. We present a method that combines preconditioning in the sinogram domain and an efficient numerical method for the nonlinear problem with a simple and thus fast iteration for the linear part of the problem. Our hybrid method does not suffer from systematic problems like beam hardening or difficulties with not perfectly aligned images for different energy bins. It is iterative but convergence is fast and the computational cost of each iteration is modest.
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| 4:00pm - 6:00pm | MS49 2: Applied parameter identification in physics Location: VG3.102 Session Chair: Tram Nguyen Session Chair: Anne Wald |
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A phase-field approach to shape optimization of acoustic waves in dissipative media Radboud University, The Netherlands
In this talk, we will discuss the problem of finding the optimal shape of a system of acoustic lenses in a dissipative medium. The problem is tackled by introducing a phase-field formulation through diffuse interfaces between the lenses and the surrounding fluid. The resulting formulation is shown to be well-posed and we rigorously derive first-order optimality conditions for this problem. Additionally, a relation between the diffuse interface problem and a perimeter-regularized sharp interface shape optimization problem can be established via the $\Gamma$-limit of the reduced objective.
Parameter identification in helioseismology Max Planck Institute for Solar System Research, Germany
Helioseismology aims at recovering the properties of the solar interior from the observations of line-of-sight Doppler velocities at the surface. Interpreting these observations requires first to solve a forward problem describing the propagation of waves in a highly-stratified medium representing the interior of the Sun. Considering only acoustic waves, the forward problem can be written as
$$\mathcal{L}\psi := -\frac{1}{\rho c^2} (\omega^2 + 2 i \omega \gamma + 2 i \omega \mathbf{u} \cdot \nabla) \psi - \nabla \left( \frac{1}{\rho} \nabla \psi \right) = s,$$
where $\rho$ is the density, $c$ the sound speed, $\mathbf{u}$ the flow and $\psi$ the Lagrangian pressure perturbation. The source term $s$ is stochastic and caused by the excitation of waves by convection. As the signal is incoherent, we cannot study directly the wavefield $\psi$ but only its cross-covariance $C(r_1,r_2,\omega) = \psi(r_1,\omega)^\ast \psi(r_2,\omega)$. Under the hypothesis of energy equipartition, the expectation value of the cross-covariance is proportional to the imaginary part of the Green's function associated to $\mathcal{L}$. The inverse problem is then to reconstruct the parameters $q \in \{\rho, c, \mathbf{u} \}$ from the observations of Im[$G(r_1,r_2,\omega)$] for any two points $r_1$, $r_2$ at the solar surface. To increase the signal-to-noise ratio and reduce the size of the input data, wave travel times are usually extracted from the cross-covariances and serve as input data for the inversions. We will present inversions of large-scale flows (differential rotation and meridional circulation) from travel-time measurements using synthetic and observed data.
Parameter identification in magnetization models for large ensembles of magnetic nanoparticles University of Bremen, Germany
Magnetic particle imaging (MPI) is a tracer based imaging modality which exploits the magnetization behavior of magnetic nanoparticles (MNPs) to obtain spatially distributed concentration images from voltage measurements. Proper modeling, which is still an unsolved problem in MPI, relies on magnetization dynamics of individual nanoparticles which typically include Neel and Brownian magnetic moment rotation dynamics. In the context of MPI large ensembles of MNPs and their magnetization behavior need to be considered. Taking into account Neel/Brownian rotation, the ensembles magnetization behavior can be described by a Fokker-Planck equation, i.e., a linear parabolic PDE which models the temporal evolution of the probability that the magnetic moment of a nanoparticle has a certain orientation. The resulting behavior is strongly influenced by time-dependent parameters in the PDE. In this talk we discuss the physical modeling as well as time-dependent parameter identification problems related to the magnetization dynamics based on a Fokker-Planck equation.
Lipschitz stable determination of polyhedral conductivity inclusions from local boundary measurements Università degli Studi di Milano, Italy
In this talk, we consider the problem of determining a polyhedral conductivity inclusion embedded in a homogeneous isotropic medium from boundary measurements.
Specifically, we prove global Lipschitz stability for the polyhedral inclusion from the local Dirichlet-to-Neumann map.
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| 4:00pm - 6:00pm | MS54 3: The x-ray transform and its generalizations: Theory, methods, and applications Location: VG1.105 Session Chair: Suman Kumar Sahoo |
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The Calderón problem for space-time fractional parabolic operators with variable coefficients 1TIFR CAM, Bangalore, India; 2RICAM, Austria
We study an inverse problem for variable coefficient fractional parabolic operators of
the form $(\partial_t − \textrm{div}(A(x)\nabla_x))^s + q(x, t)$ for $s \in (0, 1)$ and show the unique recovery of q from exterior measured data. Similar to the fractional elliptic case, we use Runge type approximation argument which is obtained via a global weak unique continuation property. The proof of such a unique continuation result involves a new Carleman estimate for the associated variable coefficient extension
operator. In the latter part of the work, we prove analogous unique determination results for fractional parabolic operators with drift.
[1] A. Banerjee, S. Senapati. The Calderón problem for space-time fractional parabolic operators with variable coefficients, arXiv: 2205.12509, 2022.
Rich tomography reconstruction problems in applications. University of Manchester, United Kingdom
Rich tomography refers generally to problems in which the image has more than just a scalar per voxel, and often the measurement is more than one scalar per source and detector pair. In this talk I will give a number of examples of real problems where the data collection systems exist and I will review their mathematical formulation, what is known and what is yet to be determined about the reconstruction problem (as well as sufficiency of data, range characterization and stability,
Small Angle X-ray Scattering (SAXS) tomography is an example where a diffraction pattern is measured for each ray, and the inverse problem is to determine the `reciprocal space map' a function of three variables at each point.
Several techniques involve the imaging of strain in a crystalline or polycrystalline material. I will show the formulation of the problem where the measurement uses neutrons and electrons. In polycrystalline materials the texture, or distribution of crystal orientations over a given scale, is often an `nuisance variable' but can be of interest in its own right. I will suggest some possible mathematical challenges.
Finally the imaging of magnetic fields is a vector tomography problem and I will contrast the method using polarimetric tomography with neutrons and also a method using electron tomography.
Localized artifacts in medical imaging ETH Zurich, Switzerland
Medical imaging reconstruction is typically regularized with methods that lead to stability in an $L^2$ sense. However, we argue that the $L^2$ norm is not always a good metric with which to assess the quality of image reconstruction. For instance, two objects might be close in $L^2$, while one of them carries a localized, clearly visible artifact, not present in the other. While this issue has been raised for deep learning based algorithms, we show as an example, that the classical regularization method of compressed sensing for MRI is also not protected from such possible instabilities. This is joint work with Giovanni S. Alberti (University of Genoa) and Tandri Gauksson (ETH Zurich).
Explicit inversion of momentum ray transform IISER Bhopal, India
The inversion of the ray transform serves as an important mathematical tool for investigating object properties from external measurements with extensive applications spanning medical imaging to geophysics. However, the inversion of the ray transform on symmetric tensor fields is constrained by the presence of an infinite dimensional null space. One natural question is whether we can utilize supplementary data in the form of higher order moments of the ray transform for the explicit recovery of the entire tensor field. In this talk, we will focus our attention on normal operators associated to momentum ray transforms (the composition of the transform with its formal $L^2$ adjoint), and introduce an approach for the explicit reconstruction of entire symmetric $m$ tensor field from this data.
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| 4:00pm - 6:00pm | MS57 3: Inverse Problems in Time-Domain Imaging at the Small Scales Location: VG2.107 Session Chair: Eric Bonnetier Session Chair: Xinlin Cao Session Chair: Mourad Sini |
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The electromagnetic waves generated by a cluster of nanoparticles with high refractive indices and corresponding effective medium theory. RICAM, Austrian Academy of Science, Austria
We estimate the electromagnetic fields generated by a cluster of dielectric nanoparticles which are small scaled but enjoy high contrast of their relative permittivity, embedded into a background made of a vacuum. Under certain ratio between their size and contrast, these nanoparticles generate resonances, called dielectric resonances. We first characterize the dominant field generated by a cluster of such dielectric-resonating nanoparticles. In this point-interaction approximation, the nanoparticles can be distributed to occupy volume-like domains or low dimensional hypersurfaces where periodicity is not required. Then we investigate the corresponding effective electromagnetic medium with periodic distribution under some mild assumptions. We show that even though the dielectric nanoparticles are merely generated by the contrasts of their permittivity (and not their permeability), the effective medium is a perturbation of the permeability and not the permittivity. Both of the cases for the effective permeability being positive and negative are studied.
Galerkin Foldy-Lax asymptotic models for time-domain scattering by small particles INRIA, France
Foldy-Lax models are asymptotic models of wave scattering
by multiple obstacles, in the regime when their characteristic size tends to
zero. While frequency-domain Foldy-Lax models are now fairly well-studied
(see e.g. [1,2,3]), their time-domain
counterparts were considered only very recently, see [4,5].
This talk is dedicated to the derivation, stability and convergence analysis
of a time-domain Foldy-Lax model for sound-soft scattering by small obstacles. We start with the analysis of the time-domain counterpart of the respective frequency-domain model for circular scatterers from [6] and show that it is unstable for
some geometric configurations. To stabilize it, we propose its reinterpretation
as a perturbed Galerkin discretization of a single layer boundary integral equation.
Its unperturbed Galerkin discretization is then automatically stable due
to a coercivity-like property of the underlying operator and thus serves as a
basis to derive the stabilized model.
Let us remark that this reinterpretation provides us with an alternative way to derive asymptotic models as Galerkin discretizations of boundary integral formulations with well-chosen basis functions.
We will present the convergence analysis of the new model,
discuss its numerical implementation, and illustrate our findings with numerical experiments.
[1] P. Martin. Multiple scattering, vol. 107 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2006.
[2] D.P. Challa, M. Sini. On the justification of the Foldy-Lax approximation for the acoustic scattering by small rigid bodies of arbitrary shapes, Multiscale Model. Simul., 12 ,
pp. 55–108, 2014.
[3] D.P. Challa, M. Sini. The Foldy-Lax approximation of the scattered waves by many small bodies for the Lamé system, Math. Nachr., 288, pp. 1834–1872, 2015.
[4] H. Barucq, J. Diaz, V. Mattesi, S. Tordeux. Asymptotic behavior of acoustic waves scattered by very small obstacles, ESAIM Math. Model. Numer. Anal., 55, pp. S705–S731, 2021.
[5] M. Sini, H. Wang, Q. Yao. Analysis of the acoustic waves reflected by a cluster of small holes in the time-domain and the equivalent mass density, Multiscale Model. Simul., 19, pp. 1083–1114, 2021.
[6] M. Cassier, C. Hazard. Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: mathematical justification of the Foldy-Lax model, Wave Motion, 50, pp. 18–28, 2013.
Electromagnetic waves generated by a moving dielectric under the special relativity assumptions Indian Institute of Science Education and Research Bhopal, India
In this talk, we will consider the electromagntic waves generated by an in-
clusion moving at a uniformly constant speed. We will discuss the direct and inverse scattering problem for the corresponding transmission problem. We
first show that the scattering electromagnetic fields satisfy a related Lippmann-Schwinger system of equations and the solutions of this system of integral equations can be written in terms of the Neumann series under certain assumption on the ratio between the speed of the moving object and the speed of light in
the vaccum. Finally, as an application of this result, we will prove that the
far-field map uniquely determines the unknown moving object.
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| 4:00pm - 6:00pm | MS58 2: Shape Optimization and Inverse Problems Location: VG2.104 Session Chair: Lekbir Afraites Session Chair: Antoine Laurain Session Chair: Julius Fergy Tiongson Rabago |
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Minimization of blood damage induced by non-newtonian fluid flows in moving domains Institute of Mathematics of the Czech Academy of Sciences, Czech Republic
The use of blood pumps may be necessary for people with heart problems, but there are
potential risks of complications associated with this type of device, in particular hemolysis
(destruction of red blood cells). Many engineering works are interested in the parametric optimization of these pumps to minimize hemolysis. In order to generalize
this approach in the present work, we study the shape continuity of a coupled system of PDE
modeling blood flows and hemolysis evolution in moving domains, governed respectively by
non-Newtonian Navier-Stokes and by transport equations.
First, the shape continuity of the blood fluid velocity $u$ is shown. This development, which
extends the one led in [1], is based on the recent progress made in [2]. Indeed, the non-
Newtonian stress for blood flows can be described by the following rheological law:
$$
S(Du) := (1 + |Du|)^{q−2} Du ,
$$
where $S(Du)$ is the stress tensor, the symmetric gradient is given by $Du := \frac{1}{2} (\nabla u + \nabla u^{\top})$, and where $q < 2$. Such fluids are called shear thinning fluids. Yet in [2], an existence result is provided for the case $q > 6/5$ in moving domains, by means of the study of Generalized Bochner spaces and the Lipschitz truncation method. Thus, these techniques are extended to the present framework of a sequence of converging moving domains.
After calculating the blood flow solutions, the velocity and stress field of the fluid are used as the coefficients for the transport equation governing the evolution of the hemolysis rate $h$:
$$
\partial_t h + u \cdot \nabla h = | S(Du) |^{\gamma} (1 − h),
$$
where the right hand side plays the role of a source term with saturation, for some $\gamma > 1$. From this, the shape continuity of the hemolysis rate is also proved.
Finally, these results allow to show the existence of minimum for a class of shape optimization
problems based on the minimization of the hemolysis rate, in the framework of moving domains. The lack of uniqueness for shear thinning fluids solutions prevents the study of shape sensitivity from being pursued, so that an extension of this work for the purpose of computing a shape gradient must somehow consider a regularization of the present model.
[1] J. Sokolowski, J. Stebel. Shape optimization for non-Newtonian fluids in time-dependent domains, Evol. Equ. Control Theory, 3(2):331–348, 2014.
[2] P. Nagele, M. Ruzicka. Generalized Newtonian fluids in moving domains, J. Differential Equations, 264(2):835–866, 2018.
On the new coupled complex boundary method for shape inverse problem with the Robin homogeneous condition 1Sultan Moulay Slimane University, Béni Mellal, Morocco; 2Kanazawa University, Kanazawa, Japan
We consider the problem of identifying an unknown portion $\Gamma$ of the boundary with a Robin condition of a d-dimensional $(d = 2;3)$ body $\Omega$ by a pair of Cauchy data $(f; g)$ on the accessible part $\Sigma$ of the boundary of a harmonic function $u$. For a fixed constant impedance $\alpha$, it is known [1] that a single measurement of $(f; g)$ on $\Sigma$ can give rise to infinitely many different domains . Nevertheless, a well-known approach to numerically solve the problem and obtain fair detection of the unknown boundary is to apply shape optimization methods. First, the inverse problem is recast into three different shape optimization formulations and the shape derivative of the cost function associated with each formulations are obtained [3]. Second, in this investigation, a new application of the so-called coupled complex boundary method – first put forward by Cheng et al. [2] to deal with inverse source problems – is presented to resolve the problem. The over-specified problem is transformed to a complex boundary value problem with a complex Robin boundary condition coupling the Cauchy pair on the accessible exterior boundary. Then, the cost function constructed by the imaginary part of the solution in the whole domain is minimized in order to identify the interior unknown boundary. The shape derivative of the complex state as well as the shape gradient of the cost functional with respect to the domain are computed. In addition, the shape Hessian at the critical point is characterized to study the ill-posedness of the problem. Specifically, the Riesz operator corresponding to the quadratic shape Hessian is shown to be compact. Also, with the shape gradient information, we devise an iterative algorithm based on a Sobolev gradient to solve the minimization problem. The numerical realization of the scheme is carried out via finite element method and is tested to various concrete example of the problem, both in two and three spatial dimensions.
[1] F. Cakoni, R. Kress. Integral equations for inverse problems in corrosion detection from partial cauchy data, Inverse Prob. Imaging, 1:229–245, 2007.
[2] X. L. Cheng, R. F. Gong, W. Han, X. Zheng. A novel coupled complex boundary method for solving inverse source problems, Inverse Problems, 30, 055002, 2014.
[3] L. Afraites, J. F. T. Rabago. Shape optimization methods for detecting an unknown boundary with the Robin condition by a single measurement, Discrete Contin. Dyn. Syst. - S, 2022. [10.3934/dcdss.2022196]
Shape optimization approach for sharp-interface reconstructions in time-domain full waveform inversion. University of Duisburg-Essen, Germany
Velocity models presenting sharp interfaces are frequently encountered in seismic imaging, for instance for imaging the subsurface of the Earth in the presence of salt bodies. In order to mitigate the oversmoothing of classical regularization strategies such as the Tikhonov regularization, we propose a shape optimization approach for sharp-interface reconstructions in time-domain full waveform inversion.
Using regularity results for the wave equation with discontinuous coefficients, we show the shape differentiability of the cost functional measuring the misfit between observed and predicted data, for shapes with low regularity. We propose a numerical approach based on the obtained distributed shape derivative and present numerical tests supporting our methodology.
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