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 | |
| Location: VG1.104 |
| Date: Monday, 04/Sept/2023 | |
| 1:30pm - 3:30pm | MS14 1: Inverse Modelling with Applications Location: VG1.104 Session Chair: Daniel Lesnic Session Chair: Karel Van Bockstal |
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Scanning biological tissues using thermal-waves University of Leeds, United Kingdom
Many materials in nature possess properties that are unknown and difficult to measure directly. In such a situation, inverse modelling offers a viable alternative where one is trying to infer those unknown properties from appropriate measurements of the main dependent variable(s) governing the physical process under investigation. Our investigation is driven by the fact that knowledge of the properties of biological tissues is essential in monitoring any abnormalities that may be forming and have a major impact on organs malfunctioning. Therefore, these disorders must be detected and treated early to save lives and improve the general health. Within the framework of thermal therapies, e.g. hyperthermia or cryoablation, the knowledge of the tissue temperature and its properties, as well as the blood perfusion rate are of utmost importance. Motivated by such a significant biomedical application, this study investigates the reconstruction of biological properties in the thermal-wave hyperbolic model of bio-heat transfer.
The support of the EPSRC grant EP/W000873/1 on “Transient Tomography for Defect Detection'' is acknowledged.
Identification of the time-dependent part of a heat source in thermoelasticity 1Ghent University, Belgium; 2University of Bucharest, Romania
The isotropic thermoelasticity system of type-III, describing the mechanical and thermal behaviours of a body occupying a bounded domain with a Lipschitz boundary, is considered. The displacement vector and either the normal heat flux or the temperature are prescribed on the boundary.
This talk deals with the theoretical and numerical reconstruction of a time-dependent heat source from the knowledge of an additional weighted integral measurement of the temperature in the framework mentioned above. Firstly, it is proved that the measurement type depends on the available thermal boundary condition, expressed by different conditions on the associated weight function. Secondly, for both thermal boundary conditions, the existence of a unique weak solution for exact data is proved, which is achieved by employing Rothe's method. This approach has the advantage of including a time-discrete numerical scheme for computations. Hence, for each of the two inverse source problems considered in this talk, a numerical algorithm that builds upon a decoupling technique is proposed, and the convergence of these numerical schemes is proved for exact data. Furthermore, the uniqueness of a solution is obtained by using an energy estimate. Finally, using the finite element method, the numerical results obtained for various numerical examples with noisy measurements are presented to validate the convergence and stability of the proposed algorithms. The noisy data are regularised using the nonlinear least-squares method; hence, they can be used as input for the proposed numerical scheme.
The results presented in this talk are published in [1].
[1] K. Van Bockstal, L. Marin. Finite element method for the reconstruction of a time-dependent heat source in isotropic thermoelasticity systems of type-III, Zeitschrift für angewandte Mathematik und Physik 73, 2022.
Uniqueness of determining a space-dependent source for inverse source problems in thermoelasticity Ghent University, Belgium
A thermoelastic system describes the interaction between the changes in the shape of an object $\mathbf{u}(\mathbf{x},t)$ and the fluctuation in the temperature $\theta(\mathbf{x},t)$. We consider an isotropic thermoelastic system of type-III which links the elastic and thermal behaviors of an isotropic material occupying a bounded domain $\Omega \subset \mathbb{R}^d$ with Lipschitz continuous boundary.
In this contribution, we will study and discuss uniqueness results for solutions to several inverse source problems (ISPs). Our main assumption is that either the heat source $h$ or load source $\mathbf{p}$ can be decomposed as a product of a given time-dependent and an unknown space-dependent function. The main goal is to find the spatial component given some measurement of the function(s) $\mathbf{u}(\mathbf{x},t)$ and/or $\theta(\mathbf{x},t).$
More specifically, the first ISP under consideration deals with the determination of the spatial component $\mathbf{f}(\mathbf{x})$ of the load source $\mathbf{p}(\mathbf{x},t) = g(t)\mathbf{f}(\mathbf{x})$ from the final in time measurement $\mathbf{u}(\mathbf{x},T),$ or from the time-average measurement $\int_0^T \mathbf{u}(\mathbf{x},t)\,\mathrm{d}t,$ where $T$ denotes the final time. The second ISP concerns finding $f(\mathbf{x})$ in the heat source $h(\mathbf{x},t) = g(t) f(\mathbf{x}) $ from the time-average measurement $\int_0^T \theta(\mathbf{x},t)\,\mathrm{d}t.$ The uniqueness results are formulated under suitable assumptions on the temporal component $g(t)$ and its derivative. Some examples will be provided showing the necessity of these (sign) conditions on $g.$ The results holds for (homogeneous) Dirichlet boundary conditions on $\mathbf{u}$ and $\theta$ as well as in the case a (homogeneous) Neumann boundary condition for $\theta$ is used. Finally, the in last ISP, we discuss the problem of finding both $\mathbf{f}$ and $f$ simultaneously when a combination of different measurements is available.
The presented work is based on joint work with Dr. Karel Van Bockstal [1].
[1] F. Maes, K. Van Bockstal. Uniqueness for inverse source problems of determining a space-dependent source in thermoeleastic systems, J. Inverse Ill-Posed Probl. 30(6): 845-856, 2022.
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| 4:00pm - 6:00pm | MS14 2: Inverse Modelling with Applications Location: VG1.104 Session Chair: Daniel Lesnic Session Chair: Karel Van Bockstal |
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Boundary identification in cantilever beam equation related to the atomic force microscopy 1University of Malta, Malta; 2Kocaeli University, Turkey; 3University of São Paulo, Brasil
In this work, identification of the shear force in the Atomic Force Microscopy cantilever tip-sample interaction is considered. This interaction is governed by the following dynamic Euler-Bernoulli beam equation.
$$
\left\{ \begin{array}{ll}
\rho_A(x) u_{tt}+\mu(x)u_{t}+ (r(x)u_{xx}+\kappa(x)u_{xxt})_{xx} =0,\, (x,t)\in \Omega_{T},\\ [1pt]
u(x,0)=u_{t}(x,0)=0, ~x \in (0,\ell ), \\ [1pt]
u(0,t)=u_x(0,t)=0,~ \left (r(x)u_{xx}+\kappa(x)u_{xxt}\right)_{x=\ell}=M(t),\\
\qquad \qquad \qquad \left (-(r(x)u_{xx}+\kappa(x)u_{xxt})_x \right )_{x=\ell}=g(t),~t \in [0,T],
\end{array} \right.
$$
where the momentum $M(t)$ is correlated with the transverse shear force $g(t)$ by a certain formula. For the identification of $g(t)$ the deflection on the right hand tip is used as an measured data to minimize the corresponding objective functional by an explicit gradient formula. As a next step, the conjugate gradient algorithm (CGA) is designed for the reconstruction process to have numerical solution of the considered inverse problem. This algorithm is based on the weak solution theory, adjoint problem approach and method of lines combined with Hermite finite elements. Computational results, obtained for noisy output data, are illustrated to show an efficiency and accuracy of the proposed approach, for typical classes of shear force functions with realistic problem parameters.
Recent developments on integral equation approaches for Electrical Impedance Tomography University of Malta, Malta
The talk is focused on recent developments of reconstruction algorithms that can be used to approximate admittivity distributions in Electrical Impedance Tomography. The algorithms are non-iterative and are based on linearized integral equation formulations [1,2] which have been extended to reconstruct the conductivity and/or permittivity distributions of two and three-dimensional domains from boundary measurements of both low and high-frequency alternating input currents and induced potentials [3]. The linearized approaches rely on the solutions to the Laplace equation on a disk and a hemispherical domain subject to appropriate idealized Neumann boundary conditions corresponding to applied spatial varying trigonometric current patterns. Reconstructions from noisy simulated data are obtained from single-time, time-difference and multiple-times data. Moreover, a proposed design of a prototype for a novel integrated circuit based electrical impedance mammographic system embedded in a brassiere will be presented.
[1] C. Hähnlein, K. Schilcher, C. Sebu, H. Spiesberger. Conductivity imaging with interior potential measurements, Inverse Problems in Science and Engineering 19(5): 729-750, 2011.
[2] K-H. Georgi, C. Hähnlein, K. Schilcher, C. Sebu, H. Spiesberger. Conductivity Reconstructions Using Real Data from a New Planar Electrical Impedance Device, Inverse Problems in Science and Engineering, Inverse Problems in Science and Engineering 21(5): 801-822, 2013.
[3] C. Sebu, A. Amaira, J. Curmi. A linearized integral equation reconstruction method of admittivity distributions using Electrical Impedance Tomography, Engineering Analysis with Boundary Elements 150: 103-110, 2023.
Inverse problem of determining time-dependent diffusion coefficient in the time-fractional heat equation 1Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan; 2Ghent University, Belgium
Let $\mathcal{H}$ be a separable Hilbert space
and let $\mathcal{L}$ be operator with a discrete spectrum on $\mathcal{H}.$ For
$$\begin{cases}
&\mathcal{D}_t^\alpha u(t)+a(t)\mathcal{L}u(t)=f(t) \,\, \hbox{in} \,\, \mathcal{H},\,0<t \leq T,\\
&u (0) =h \; \text{in}\; \mathcal{H},
\end{cases}\;\;\;\;(1)
$$
we study
Coefficient inverse problem: Given $f(t), h$ and $E(t), $ find a pair of functions $\{a(t),u(t)\}$ satisfying the problem (1) and the additional condition
$$
F[u(t)]=E(t),\; t\in[0,T],
$$
where $F$ is the linear bounded functional.
As for this kind of inverse problem for parabolic equation, see [1].
Under suitable restrictions on the given data, we prove the existence, uniqueness and continuous dependence of the solution on the data.
[1] Z. Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation, Inverse Problems and Imaging. 11: 875–900, 2017.
Determining an Iwatsuka Hamiltonian through quantum velocity measurement Aix Marseille université, France
The main purpose of this talk is to explain how quantum currents induced by either the classical or the time-fractional Schrödinger equation, associated with an Iwatsuka Hamiltonian, uniquely determine the magnetic potential.
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| Date: Tuesday, 05/Sept/2023 | |
| 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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| 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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| Date: Wednesday, 06/Sept/2023 | |
| 9:00am - 11:00am | MS32 1: Parameter identification in time dependent partial differential equations Location: VG1.104 Session Chair: Barbara Kaltenbacher Session Chair: William Rundell |
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Spacetime finite element methods for inverse and control problems subject to the wave equation 1University of Helsinki, Finland; 2University of Toronto, Canada; 3Fields Institute, Canada
There is a well-known duality between inverse initial source problems and control problems for the wave equation, and analysis of both these boils down to the so-called observability estimates. I will present recent results on numerical analysis of these problems. The inverse initial source problem gives a model for the acoustic step of Photoacoustic tomography.
Mathematical challenges in Full Waveform inversion Karlsruhe Institute of Technology, Germany
Full Waveform inversion (FWI) is a state-of-the-art geophysical imaging method that exploits seismic measurements to reconstruct shallow earth parameters. Mathematically, this translates into a non-linear inverse problem where the seismic measurements are modeled as solutions to a time-dependent wave-type system and the searched-for parameters are (some of) the coefficients. In order for numerical solution to be successful, both the parameter and measurement spaces need to be selected carefully: For instance, the reconstruction of sharp material interfaces requires non-smooth parameter spaces which are numerically difficult to cope with. Further, to minimize artifacts and spurious reconstructions, the resulting non-linear objective functional should be as convex as possible, which thus constraints the choice of compatible metrics for the seismic measurements. In this talk, we present novel ideas and solutions regarding these challenges in FWI.
Optimality of pulse energy for photoacoustic tomography University of Klagenfurt, Austria
Photoacoustic tomography (PAT) is a rapidly evolving imaging technique that combines high contrast of optical imaging with high resolution of ultrasound imaging. Using typically noisy measurement data, one is interested in identifying some parameters in the governing PDEs for the photoacoustic tomography system. Hence, an essential factor in estimating these parameters is the design of the system, which typically involves multiple factors that can impact the accuracy of reconstruction. In this work, employing a Bayesian approach to a PAT inverse problem we are interested in optimizing the laser pulse of the PAT system in order to minimize the uncertainty of the reconstructed parameter. Additionally, we take into account wave propagation attenuation for the inverse problem of PAT, which is governed by a fractionally damped wave equation. Finally, we illustrate the effectiveness of our proposed method using a numerical simulation.
Bi-level iterative regularization for inverse problems in nonlinear PDEs Max Planck Institute for Solar System Research, Germany
We investigate the ill-posed inverse problem of recovering spatially dependent parameters in nonlinear evolution PDEs from linear measurements. We propose a bi-level Landweber scheme, where the upper-level parameter reconstruction embeds a lower-level state approximation. This can be seen as combining the classical reduced setting and the newer all-at-once setting, allowing us to, respectively, utilize well-definedness of the parameter-to-state map, and to bypass having to solve nonlinear PDEs. Using this, we derive stopping rules for lower- and upper-level iterations and convergence of the bi-level method.
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| Date: Thursday, 07/Sept/2023 | |
| 1:30pm - 3:30pm | MS32 2: Parameter identification in time dependent partial differential equations Location: VG1.104 Session Chair: Barbara Kaltenbacher Session Chair: William Rundell |
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On the identification of cavities in a nonlinear diffusion-reaction model arising from cardiac electrophysiology 1New York University Abu Dhabi, United Arab Emirates; 2Università degli Studi di Milano, Italy; 3Università di Firenze, Italy; 4Politecnico di Milano, Italy
Detecting ischemic regions from noninvasive (or minimally invasive) measurements is of primary importance to prevent lethal ventricular ischemic tachycardia.
This is usually performed by recording the electrical activity of the heart, by
means of either body surface or intracardiac measurements.
Mathematical and numerical models of cardiac electrophysiology can be used to
shed light on the potentialities of electrical measurements in detecting ischemia. More
specifically, the goal is to combine boundary measurements of (body-surface or intracavitary) potentials and a mathematical description of the electrical activity of the heart in order to possibly identify the position, shape, and size of heart ischemias and/or infarctions. The ischemic region is a non-excitable tissue that can be modeled as an electrical insulator (cavity) and the cardiac electrical activity can be comprehensively described in terms of the monodomain model, consisting of a boundary value problem for the nonlinear reaction-diffusion monodomain system.
In my talk, I will illustrate some recent results concerning the inverse problem of detecting the cavity from boundary measurements.
Identification of the electric potential of the time-fractional Schrödinger équation by boundary measurement Aix Marseille université, France
This talk deals with the inverse problem of identifying the real valued electric potential of the time-fractional Schrödinger equation, by boundary observation of its solution. Its main purpose is to establish that the Dirichlet-to-Neumann map computed at one fixed arbitrary time uniquely determines the time-independent potential.
The Recovery of Coefficients in Wave Equations from Time-trace Data Texas A&M University, United States of America
The Westervelt equation is a common formulation used in nonlinear optics
and several of its coefficients are meaningful as imaging parameters of physical consequence.
We look at the recovery of some of these from both an analytic and a reconstruction perspective.
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| 4:00pm - 6:00pm | MS46 1: Inverse problems for nonlinear equations Location: VG1.104 Session Chair: Lauri Oksanen Session Chair: Teemu Kristian Tyni |
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Weakly nonlinear geometric optics and inverse problems for hyperbolic nonlinear PDEs Purdue University, United States of America
We review recent results by the presenter, Antônio Sá Barreto, and Nikolas Eptaminitakis about inverse problems for the semilinear wave equation and the quasilinear Westervelt wave equation modeling nonlinear acoustic. We study them in a regime in which the solutions are not "small" so that we can linearize; when the nonlinear effects are strong and correspond to the observed phyisical effects. We show that a propagating high frequency pulse recovers the nonlinearity uniquely by recovering its X-ray transform, and we will show numerical simulations.
Identification of nonlinear effects in X-ray tomography Emory University, United States of America
Due to beam-hardening effects, metal objects in X-ray CT often produce streaking artefacts which cause degradation in image reconstruction. It is known that the nature of the phenomena is nonlinear. An outstanding inverse problem is to identify the nonlinearity which is crucial for reduction of the artefacts. In this talk, we show how to use microlocal techniques to extract information of the nonlinearity from the artefacts. An interesting aspect of our analysis is to explore the connection of the artefacts and the geometry of metal objects.
Inverse problems for non-linear hyperbolic equations and many-to-one scattering relations University of Helsinki, Finland
In the talk we give an overview on inverse problems for Lorentzian manifolds. We also discuss how inverse problems for partial differential equations can be solved using non-linear interaction of solutions. In the talk we concentrate on the geometric tools used to solve these problems, for instance to the k-to-1 scattering relation associated to the $k$-th order interactions and the observation time functions on Lorentzian manifolds.
Inverse problems for nonlinear elliptic PDE University of California, Irvine, United States of America
We shall discuss some recent progress for inverse boundary problems for nonlinear elliptic PDE. Our focus will be on inverse problems for isotropic quasilinear conductivity equations, as well as nonlinear Schrodinger and magnetic Schrodinger equations. In particular, we shall see that the presence of a nonlinearity may actually help, allowing one to solve inverse problems in situations where the corresponding linear counterpart is open. This talk is based on joint works with Catalin Carstea, Ali Feizmohammadi, Yavar Kian, and Gunther Uhlmann.
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| Date: Friday, 08/Sept/2023 | |
| 1:30pm - 3:30pm | MS46 2: Inverse problems for nonlinear equations Location: VG1.104 Session Chair: Lauri Oksanen Session Chair: Teemu Kristian Tyni |
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Inverse source problems for nonlinear equations National Yang Ming Chiao Tung University, Taiwan
In this talk, we perform inverse source problems for nonlinear equations. Unlike linear differential equations, which always have gauge invariance. We investigate how the gauge symmetry could be broken for several nonlinear and nonlocal equations, which leads to unique determination results for certain equations.
Inverse problem for the minimal surface equation and nonlinear CGO calculus in dimension 2 University of Helsinki
We present our recent results regarding inverse problems for the minimal surface equation. Applications of the result include generalized boundary rigidity problem and AdS/CFT correspondence in physics. Minimal surfaces are solutions to a quasilinear elliptic equation and we determine the minimal surface up to an isometry from the corresponding Dirichlet-to-Neumann map in dimension 2. For this purpose we develop a nonlinear calculus for complex geometric optics solutions (CGOs) to handle numerous correction terms that appear in our analysis. We expect the calculus to be applicable to inverse problems for other nonlinear elliptic equations in dimension 2. The talk is based on joint works with Catalin Carstea, Matti Lassas and Leo Tzou.
Inverse scattering problems for semi-linear wave equations on manifolds 1University of Toronto, Canada; 2University of Tsukuba, Japan; 3University of Helsinki, Finland
We discuss some recent results on inverse scattering problems for semi-linear wave equations. The inverse scattering problem is formulated on a Lorentzian manifold equipped with a Minkowski type infinity. We show that a scattering functional, which roughly speaking maps measurements of solutions of a semi-linear wave equation at the past infinity to the future infinity, determines the manifold, the conformal class of the metric, and the nonlinear potential function up to a gauge. The main tools we employ are a Penrose-type conformal compactification of the Lorentzian manifold, reduction of the scattering problem to the study of the source-to-solution operator, and the use of higher order linearization method to exploit the nonlinearity of the wave equation.
This is a joint work with S. Alexakis, H. Isozaki, and M. Lassas.
Determining a Lorentzian metric from the source-to-solution map for the relativistic Boltzmann equation 1University of Calgary, Canada; 2The Hong Kong University of Science and Technology; 3University of Helsinki
In this talk, we consider the following inverse problem: Given the source-to-solution map for a relativistic Boltzmann equation on a neighbourhood $V$ of an observer in a Lorentzian spacetime $(M,g)$ and knowledge of $g|_V$, can we determine (up to diffeomorphism) the spacetime metric $g$ on the domain of causal influence for the set $V$?
We will show that the answer is yes for certain cases. We will introduce the relativistic Boltzmann equation and the concept of an inverse problem. We then will highlight the key ideas of the proof of our main result. One such key point is that the nonlinear term in the relativistic Boltzmann equation which describes the behaviour of particle collisions captures information about a source-to-solution map for a related linearized problem. We use this relationship together with an analysis of the behaviour of particle collisions by classical microlocal techniques to determine the set of locations in $V$ where we first receive light particle signals from collisions in the unknown domain. From this data we are able to parametrize the unknown region and determine the metric.
Determining Lorentzian manifold from non-linear wave observation at a single point University of Helsinki, Finland
Our research demonstrates that it is possible to determine the Lorenzian manifold by measuring the source-to-solution map for the semilinear wave equation at a single point. (Joint work with Lauri Oksanen and Leo Tzou).
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| 4:00pm - 6:00pm | MS46 3: Inverse problems for nonlinear equations Location: VG1.104 Session Chair: Lauri Oksanen Session Chair: Teemu Kristian Tyni |
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