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: VG1.103 |
| Date: Monday, 04/Sept/2023 | |
| 1:30pm - 3:30pm | MS18 1: Inverse problems for fractional and nonlocal equations Location: VG1.103 Session Chair: Yi-Hsuan Lin Session Chair: Jesse Railo Session Chair: Mikko Salo |
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On the determination of a coefficient in a space-fractional equation with operators of Abel type 1University of Klagenfurt, Austria; 2Texas A&M University
We consider the inverse problem of recovering an unknown,
spatially-dependent coefficient $q(x)$ from the fractional order equation
$\mathbb{L}_\alpha u = f$ defined
in a region of $\mathbb{R}^2$ from boundary information.
Here $\mathbb{L_\alpha} ={D}^{\alpha_x}_x +{D}^{\alpha_y}_y +q(x)$
where the operators ${D}^{\alpha_x}_x$, ${D}^{\alpha_y}_y$ denote
fractional derivative operators based on the Abel fractional integral.
In the classical case this reduces to $-\triangle u + q(x)u = f$
and this has been a well-studied problem.
We develop both uniqueness and reconstruction results and show how the
ill-conditioning of this inverse problem depends on the geometry of the region
and the fractional powers $\alpha_x$ and $\alpha_y$.
Nonlocality Helps University of Washington and HKUST, United States of America
We give several examples of solutions inverse problems involving lon-range interactions whosecorresponding local problem is not solved.
Fractional p-Calderón problems ETH Zürich, Switzerland
The main purpose of this talk is to discuss two different nonlocal variants of the $p$-Calderón problem.
In the first model the nonlocal operator under consideration is a weighted fractional $p$-Laplacian and, similarly as for the $p$-Laplacian in dimensions $n\geq 3$, it is an open problem, whether it satisfies a unique continuation principle (UCP). However, it will be explained that the variational structure of the problem is still sufficiently nice that one can explicitly reconstruct the weight $\sigma(x,y)$ on the diagonal $D=\{(x,x): x\in W\}$ of the measurement set $W$. This reconstruction formula establishes a global uniqueness result for separable, real analytic coefficients [1].
In the second model, we consider the (anisotropic) fractional $p$-biharmonic operator, which naturally appears in the variational characterization of the optimal fractional Poincaré constant in Bessel potential spaces $H^{s,p}$. In contrast to the one above, this operator satisfies the UCP and so heuristically corresponds to the $p$-Laplacian in dimension $n=2$. Finally, we explain how this can be used to establish a global uniqueness result of the related inverse problem under a monotonicity condition [2].
[1] M. Kar, Y. Lin, and P. Zimmermann. Determining coefficients for a fractional $ p $-Laplace equation from exterior measurements, arXiv:2212.03057, 2022.
[2] M. Kar, J. Railo, and P. Zimmermann. The fractional $ p\, $-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems, arXiv:2208.09528, 2022.
Uniqueness in an inverse problem for the anisotropic fractional conductivity equation University of Bonn, Germany
We study an inverse problem for the fractional anisotropic conductivity equation. Our nonlocal operator is based on the well-developed theory of nonlocal vector calculus, and
differs substantially from other generalizations of the classical anisotropic conductivity operator obtained spectrally. We show that the anisotropic conductivity matrix can be recovered uniquely from fractional Dirichlet-to-Neumann data up to a natural gauge. Our analysis makes use of techniques recently developed for the study of the isotropic fractional elasticity equation, and generalizes them to the case of non-separable, anisotropic conductivities. The motivation for our study stems from its relation to the classical anisotropic Calderòn problem, which is one of the main open problems in the field.
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| 4:00pm - 6:00pm | MS18 2: Inverse problems for fractional and nonlocal equations Location: VG1.103 Session Chair: Yi-Hsuan Lin Session Chair: Jesse Railo Session Chair: Mikko Salo |
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The Calderón problem for directionally antilocal operators Universitat de Barcelona, Spain
The Calderón problem for the fractional Schrödinger equation, introduced by T. Ghosh, M. Salo and G. Uhlmann, satisfies global uniqueness with only one single measurement. This result exploits the antilocality property of the fractional Laplacian, that is, if a function and its fractional Laplacian vanish in a subset, then the function is zero everywhere.
Nonlocal operators which only see the functions in some directions and not on the whole space cannot satisfy an analogous antilocality property. In theses cases, only directional antilocality conditions may be expected.
In this talk, we will consider antilocality in cones, introduced by Y. Ishikawa in the 80s, and its possible implications in the corresponding Calderón problem. In particular, we will see that uniqueness for the associated Calderón problem holds even with a singe measurement, but new geometric conditions are required.
This is a joint work with G. Covi and A. Rüland.
An Inverse Problem for Nonlinear Fractional Magnetic Schrodinger Equation University of Minnesota, United States of America
The study of nonlinear equations arises in many physical phenomena and is an active direction in the field of inverse problems. In this talk, we will discuss inverse problems for the fractional magnetic Schrodinger equation with nonlinear potential and address the crucial techniques that are applied to reconstruct the unknown coefficient from measurements.
Properties of solutions for anisotropic viscoelastic systems 1Rice Univ., U.S.; 2Kanazawa Univ., Japan; 3NCKU, Taiwan; 4Hokkaido Univ., Japan
In this talk I will consider two kinds of anisotropic viscoelastic systems. One is an anisotropic viscoelastic systems which is described as an integro-differential system (ID system) and the other is the so-called extended Maxwell model (EM system) which is schematically described using springs and dashpots. There is a relation between them but they are different systems. I will discuss about their relation and the large time behavior of their solutions. Further, for the EM system, I will discuss about a generation of semigroup and the limiting amplitude principle. The method of proof for the ID system is an energy estimate, and that for the EM system is mainly based on several resolvent estimates. The limiting amplitude principle could be useful for setting up the inverse problem when the measurements may use a sequence of several different time harmonic inputs such as the magnetic resonance elastography.
This is a joint work with Maarten de Hoop, Ching-Lung Lin for the EM system and also including Masato Kimura for the EM system.
Fractional anisotropic Calderon problem on Riemannian manifolds University of California, Irvine, United States of America
We shall discuss some recent progress on the fractional anisotropic Calderon problem on closed Riemannian manifolds of dimensions two and higher. Specifically, we show that the knowledge of the local source-to-solution map for the fractional Laplacian, given on an arbitrary small open nonempty a priori known subset of a smooth closed Riemannian manifold, determines the Riemannian manifold up to an isometry. This can be viewed as a nonlocal analog of the anisotropic Calderon problem in the setting of closed Riemannian manifolds, which is wide open in dimensions three and higher. This is joint work with Ali Feizmohammadi, Tuhin Ghosh, and Gunther Uhlmann.
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| Date: Tuesday, 05/Sept/2023 | |
| 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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| 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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| Date: Wednesday, 06/Sept/2023 | |
| 9:00am - 11:00am | MS23 2: 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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Inverse medium scattering for a nonlinear Helmholtz equation Karlsruher Institut für Technologie, Germany
The linear Helmholtz equation is used to model the propagation of
sound waves or electromagnetic waves of small amplitude in
inhomogeneous isotropic media in the time-harmonic regime.
However, if the amplitudes are large then intensity-dependent
material laws are required and nonlinear Helmholtz equations are more
appropriate.
A prominent example are Kerr-type nonlinear media.
In this talk we discuss an inverse medium scattering problem for a
class of nonlinear Helmholtz equations
\begin{equation*}
\Delta u + k^2 u
\,=\, - k^2 q(x,|u|)u \,, \qquad x\in\mathbb{R}^d \,, \;d=2,3 \,,
\end{equation*}
that covers generalized Kerr-type nonlinear media of the form
\begin{equation*}
q(x,|z|)
\,=\, q_0(x) + \sum_{l=1}^L q_l(x)|z|^{\alpha_l} \,,
\qquad x\in\mathbb{R}^d \,,\; z\in\mathbb{C} \,,
\end{equation*}
where $q_0,\ldots,q_L\in L^\infty(\mathbb{R}^d)$ with support in some
bounded open set $D\subset\mathbb{R}^d$, the lowest order term
satisfies $\mathrm{essinf} q_0>-1$ in $\mathbb{R}^d$, and the
exponents fulfill $0<\alpha_1<\cdots<\alpha_L<\infty$.
Assuming the knowledge of a nonlinear far field operator, which maps
Herglotz incident waves to the far field patterns of the
corresponding unique small solutions of the nonlinear scattering
problem, we show that the nonlinear index of refraction is uniquely
determined.
This is joint work with Marvin Knöller and Rainer Mandel (KIT).
Linearised inverse conductivity problem: reconstruction and Lipschitz stability for infinite-dimensional spaces of perturbations 1Aalto University, Finland; 2Aarhus University, Denmark
The linearised inverse conductivity problem is investigated in a two-dimensional bounded simply connected domain with a smooth enough boundary. After extending the linearised problem for square integrable perturbations, the space of perturbations is orthogonally decomposed and Lipschitz stability, with explicit Lipschitz constants, is proven for each of the infinite-dimensional subspaces. The stability estimates are based on using the Hilbert-Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fréchet derivative with respect to the conductivity coefficient. A direct reconstruction method that inductively yields the orthogonal projections of a conductivity coefficient onto the aforementioned subspaces is devised and numerically tested with data simulated by solving the original nonlinear forward problem.
Optimizing electrode positions in electrical impedance tomography for head imaging Aalto University, Finland
Electrical impedance tomography is an imaging modality for deducing information about the conductivity inside a physical body from boundary measurements of current and voltage by a finite number of contact electrodes. This work applies techniques of Bayesian experimental design to the linearized forward model of impedance tomography in order to select optimal positions for the available electrodes. The aim is to place the electrodes so that the conditional probability distribution of the (discretized) conductivity given the electrode measurements is as localized as possible in the sense of the A- or D-optimality criterion of Bayesian experimental design. The focus is on difference imaging of a human head under the assumption that an MRI or CT image of the patient in question is available. The algorithm is developed in the computational framework introduced in [1].
[1] V. Candiani, A. Hannukainen, N. Hyvönen. Computational framework for applying electrical impedance tomography to head imaging. SIAM J. Sci. Comput. 41 (2019), no. 5, B1034–B1060. https://doi.org/10.1137/19M1245098
Immersed boundary method for electrical impedance tomography in the frame of electrocardiography 1Université de Toulouse, Institut de Mathématiques de Toulouse; 2Univ. Bordeaux, CNRS, INRIA, Bordeaux INP, IHU-LIRYC, IMB; 3Univ. Bordeaux, CNRS, INRIA, Bordeaux INP, IHU-LIRYC, IMB; 4Univ Toulouse, Institut de mathématiques de Toulouse,UMR 5219
EIT is a non-invasive imaging technique that aims, to reconstruct the electrical conductivity distribution inside a domain by imposing electrical currents on the boundary
of this domain, and measuring the resulting voltages on the same boundary. It has several applications
in the medical field, in particular in lung monitoring and stroke detection. Mathematically,
the problem, known as Calderón’s problem or inverse conductivity problem, is a severely ill-posed inverse problem.
In practical experiments, the currents are driven inside the body of interest through a collection
of surface electrodes, no current being driven between the electrodes. For each current pattern, the
potential differences between the electrodes are measured. This practical setting is accurately modeled
by the Complete Electrode Model (CEM). It takes into account the shape of the electrodes
as well as the shunting effect, that is the thin resistive layer that appears at the interface between
the electrodes and the object during the measurements. The CEM is known to correctly predict
experimental data, and therefore is widely used in the numerical resolution of both direct and inverse
problems related to EIT.
The CEM is as follows :
Find the potentials $u\in H^1(\Omega)$ and $U \in \mathbb{R}^M_\diamond$ such that,
\begin{equation*}
\left\lbrace
\begin{array}{rcl}
\nabla \cdot (\sigma \nabla u) & = & 0 \text{ in} \ \Omega \\
u + z_m \sigma \partial_\nu u & = & U_m \text{ on } E_m, \\
\sigma \partial_\nu u & = & 0 \text{ on } \partial \Omega \setminus \overline{E}, \\
\displaystyle \int_{E_m} \sigma \partial_\nu u \, ds & = & I_m, \\%\text{ for all } m \text{ in } \left\lbrace 1 ,\ldots , M\right\rbrace.
\end{array}
\right.
\end{equation*}
with $E_m$ the $m$th electrode , $z_m$ the associated contact impedance, $I \in \mathbb{R}^M_\diamond$ the current pattern. Where $$
\mathbb{R}^M_\diamond = \left\lbrace I \in \mathbb{R}^M, \sum_{k=1}^M I_k = 0 \right\rbrace.
$$
We propose an immersed boundary method (IBM) for the numerical resolution of the CEM in Electrical Impedance Tomography, that we use as a main ingredient in the
resolution of inverse problems in medical imaging. Such method allows to use a Cartesian mesh
without accurate discretization of the boundary, which is useful in situations where the boundary is
complicated and/or changing. We prove the convergence of our method, and illustrate its efficiency
with two dimensional direct and inverse problems.
[1] J. Dardé, N. Nasr, L. Weynans. Immersed boundary method for the complete electrode model in electrical impedance tomography. 2022.
[2] M. Cisternino, L. Weynans. A parallel second order Cartesian
method for elliptic interface problems. Addison Wesley, Massachusetts, 2nd
ed.
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| Date: Thursday, 07/Sept/2023 | |
| 1:30pm - 3:30pm | MS10 1: Optimization in Inverse Scattering: from Acoustics to X-rays Location: VG1.103 Session Chair: Radu Ioan Bot Session Chair: Russell Luke |
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Numerical Linear Algebra Networks for Solving Linear Inverse Problems University Vienna, Austria
We consider solving a probably ill-conditioned linear operator equation, where the operator is not modeled, but given indirectly via training samples.
The method de- and encoding is very useful for such applications and will be analyzed from a point of regularization theory. This analysis in particular shows that preprocessing of the image data is required, which is implemented by linear algebra networks.
Regularization by Randomization: The Case of Partially Separable Optimization University of Göttingen, Germany
We present a Markov-chain analysis of blockwise-stochastic algorithms for solving partially
block-separable optimization problems. Our main contributions to the extensive literature
on these methods are statements about the Markov operators and distributions behind the iterates of stochastic algorithms, and in particular the regularity of Markov operators and rates of convergence of the distributions of the corresponding Markov chains. This provides a detailed characterization of the moments of the sequences beyond just the expected behavior. This also serves as a case study of how randomization restores favorable properties to algorithms that iterations of only partial information destroys. We demonstrate this on stochastic blockwise implementations of the forward-backward and Douglas-Rachford algorithms for nonconvex
(and, as a special case, convex), nonsmooth optimization.
Fast convex optimization via closed-loop time scaling of gradient dynamics 1Univ. Montpellier, France; 2University of Vienna, Austria
In a Hilbert setting, for convex differentiable optimization, we develop a general framework for adaptive accelerated gradient methods. They are based on damped inertial dynamics where the coefficients are designed in a closed-loop way. Specifically, the damping is a feedback control of the velocity, or of the gradient of the objective function. For this, we develop a closed-loop version of the time scaling and averaging technique introduced by the authors. We thus obtain autonomous inertial dynamics which involve vanishing viscous damping and implicit Hessian driven damping. By simply using the convergence rates for the continuous steepest descent and Jensen's inequality, without the need for further Lyapunov analysis, we show that the trajectories have several remarkable properties at once: they ensure fast convergence of values, fast convergence of the gradients towards zero, and they converge to optimal solutions. Our approach leads to parallel algorithmic results, that we study in the case of proximal algorithms. These are among the very first general results of this type obtained using autonomous dynamics.
[1] H. Attouch, R.I. Bot, D.-K. Nguyen. Fast convex optimization via time scale and averaging of the steepest descent, arXiv:2208.08260, 2022
[2] H. Attouch, R.I. Bot, D.-K. Nguyen. Fast convex optimization via closed-loop time scaling of gradient dynamics, arXiv:2301.00701, 2023
Fast Optimistic Gradient Descent Ascent method in continuous and discrete time University of Vienna, Austria
In this talk we address continuous in time dynamics as well as numerical algorithms for the problem of approaching the set of zeros of a single-valued monotone and continuous operator $V$ ([1,2]). The starting point of our investigations is a second order dynamical system that combines a vanishing damping term with the time derivative of $V$ along the trajectory, which can be seen as an analogous of the Hessian-driven damping in case the operator is originating from a potential. Our method exhibits fast convergence rates for the norm of the operator along the trajectory and also for the restricted gap function, which is a measure of optimality for variational inequalities. We also prove the weak convergence of the trajectory to a zero of $V$.
Temporal discretizations of the dynamical system generate implicit and explicit numerical algorithms, which can be both seen as accelerated versions of the Optimistic Gradient Descent Ascent (OGDA) method for monotone operators, for which we prove that they share the asymptotic features of the continuous dynamics. All convergence rate statements are last iterate convergence results. Numerical experiments indicate the that explicit numerical algorithm outperform other methods designed to solve equations governed by monotone operators.
[1] R. I. Bot, E. R. Csetnek, D.-K. Nguyen. Fast OGDA in continuous and discrete time, arXiv:2203.10947, 2022.
[2] R. I. Bot, D.-K. Nguyen. Fast Krasnosel'skii-Mann algorithm with a convergence rate of the fixed point iteration of $o(1/k)$, arXiv:2206.09462, 2022
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| 4:00pm - 6:00pm | MS10 2: Optimization in Inverse Scattering: from Acoustics to X-rays Location: VG1.103 Session Chair: Radu Ioan Bot Session Chair: Russell Luke |
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PAC-Bayesian Learning of Optimization Algorithms Saarland University, Germany
The change of paradigm from purely model driven to data driven (learning based) approaches has tremendously altered the picture in many applications in Machine Learning, Computer Vision, Signal Processing, Inverse Problems, Statistics and so on. There is no need to mention the significant boost in performance for many specific applications, thanks to the advent of large scale Deep Learning. In this talk, we open the area of optimization algorithms for this data driven paradigm, for which theoretical guarantees are indispensable. The expectations about an optimization algorithm are clearly beyond empirical evidence, as there may be a whole processing pipeline depending on a reliable output of the optimization algorithm, and application domains of algorithms can vary significantly. While there is already a vast literature on "learning to optimize", there is no theoretical guarantees associated with these algorithms that meet these expectations from an optimization point of view. We develop the first framework to learn optimization algorithms with provable generalization guarantees to certain classes of optimization problems, while the learning based backbone enables the algorithms' functioning far beyond the limitations of classical (deterministic) worst case bounds. Our results rely on PAC-Bayes bounds for general, unbounded loss-functions based on exponential families. We learn optimization algorithms with provable generalization guarantees (PAC-bounds) and explicit trade-off between a high probability of convergence and a high convergence speed.
Accelerated Griffin-Lim algorithm: A fast and provably convergent numerical method for phase retrieval 1Austrian Academy of Sciences, Austria; 2University of Vienna
The recovery of a signal from the magnitudes of its transformation, like the Fourier transform, is known as the phase retrieval problem and is of big relevance in various fields of engineering and applied physics. The Griffin-Lim algorithm is a staple method commonly used for the phase retrieval problem, which is based on alternating projections. In this talk, we introduce and motivate a fast inertial/momentum modification of the Griffin-Lim Algorithm for the phase retrieval problem and we present a convergence guarantee for the new algorithm.
Audio Inpainting 1Austrian Academy of Sciences, Austria; 2Technical University of Vienna
The goal of audio inpainting is to fill missing data, i.e., gaps, in an acoustical signal. Depending on the length of the gap this procedure should either recreate the original signal, or at least provide a perceptually pleasant and meaningful solution.
We give an overview of existing methods for different gap lengths, and discuss details of our own method [1] for gaps of medium duration. This approach is based on promoting sparsity in the time-frequency domain, combined with a convexifaction using ADMM with a dictionary learning technique that perturbs the time-frequency atoms around the gap, using an optimization technique originally developed in the context of channel estimation.
[1] G. Tauböck, S. Rajbamshi, P. Balazs. Dictionary learning for sparse audio inpainting, IEEE Journal of Selected Topics in Signal Processing 15 no. 1: 104–119, 2021.
Damage detection by guided ultrasonic waves and uncertainty quantification TU Braunschweig, Germany
New materials like fibre metal laminates (FML) call for new methods when it comes to structural health monitoring (SHM). In this talk we describe an approach to SHM in FML based on guided ultrasonic waves that travel through plates as lamb waves. By the controlled emission of such waves and the measurement of the displacement at a few position, we aim to detect if a damage in the material is present. We approach this inverse problem by an analytical model of the forward propoagation and a simple damage model that is (nonlinearly) parameterized by a small number of parameters. To identify the damage parameters we employ Bayesian methods (namely a Markov Chain Monte-Carlo Metropolis-Hastings method and the ensemble Kalman filter). To make these computationally tractable, we use parametric model reduction to speed up the forward evaluations of the model.
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| Date: Friday, 08/Sept/2023 | |
| 1:30pm - 3:30pm | MS10 3: Optimization in Inverse Scattering: from Acoustics to X-rays Location: VG1.103 Session Chair: Radu Ioan Bot Session Chair: Russell Luke |
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Automated tight Lyapunov analysis for first-order methods 1Lund University, Sweden; 2INRIA Paris, France
We present a methodology for establishing the existence of quadratic Lyapunov inequalities for a wide range of first-order methods used to solve convex optimization problems. In particular, we consider
i) classes of optimization problems of finite-sum form with (possibly strongly) convex and possibly smooth functional components,
ii) first-order methods that can be written as a linear system on state-space form in feedback interconnection with the subdifferentials of the functional components of the objective function, and
iii) quadratic Lyapunov inequalities that can be used to draw convergence conclusions.
We provide a necessary and sufficient condition for the existence of a quadratic Lyapunov inequality that amounts to solving a small-sized semidefinite program. We showcase our methodology on several first-order methods that fit the framework. Most notably, our methodology allows us to significantly extend the region of parameter choices that allow for duality gap convergence in the Chambolle–Pock method when the linear operator is the identity mapping.
Learned SVD for limited data inversion in PAT and X-ray CT Universität Innsbruck, Austria
We present a data-driven regularization method for inverse problems introduced in [1,2]. Our approach consists of two steps. In the first step, an intermediate reconstruction is performed by applying the truncated singular value decomposition (SVD). To prevent noise amplification, only coefficients corresponding to sufficiently large singular values are used, while the remaining coefficients are set to zero. In a second step, a trained deep neural network is used to recover the truncated SVD coefficients. We show that the proposed scheme yields a convergent regularization method. Numerical results are presented for limited data problems in PAT (photacoustic tomography) and X-ray CT (computed tomography), showing that learned SVD regularization significantly improves pure truncated SVD regularization.
[1] J. Schwab, S. Antholzer, M. Haltmeier. Big in Japan: Regularizing networks for solving inverse problems, Journal of Mathematical Imaging and Vision, 62(3): 445-455, 2020.
[2] J. Schwab, S. Antholzer, R. Nuster, G. Paltauf, M. Haltmeier. Deep learning of truncated singular values for limited view photoacoustic tomography, Photons Plus Ultrasound: Imaging and Sensing 10878: 254-262, 2019.
Multiscale hierarchical decomposition methods for ill-posed problems 1University of Klagenfurt, Austria; 2Johannes Kepler University Linz, Austria
The Multiscale Hierarchical Decomposition Method (MHDM) is a popular method originating from mathematical imaging. In its original context, it is very well suited to recover fine details of solutions to denoising and deblurring problems. The main idea is to iteratively solve a ROF minimization problem. In every iteration, the data for the ROF functional will be the residual from the previous step, and the approximation to the true data will consist of the sum of all minimizers up to that step. Thus, one obtains iterates that represent a decomposition of the ground truth into multiple levels of detail at different scales. We consider the method in a more general framework, by replacing the total variation seminorm in the ROF functional by more general penalty terms in appropriate settings. We expand existing convergence results for the residual of the iterates in the case when some classes of convex and nonconvex penalties are employed. Moreover, we propose a necessary and sufficient condition under which the iterates of the MHDM agree with Tikhonov regularizers corresponding to suitable regularization parameters. We discuss the results on several examples, including 1- and 2-dimensional TV-denoising.
Scalable moment relaxations for graph-structured problems with values in a manifold: An optimal transport approach KULeuven, Belgium
In this paper we consider a moment relaxation for large-scale nonsmooth optimization problems with graphical structure and manifold constraints. In the context of probabilistic inference this can be interpreted as MAP-inference in a continuous graphical model.
In contrast to classical moment relaxations for global polynomial optimization we exploit the partially separable structure of the optimization problem and leverage Kantorovich–Rubinstein duality for optimal transport to decouple the problem. The proposed formulation is obtained via a dual subspace approximation which allows us to tackle possibly nonpolynomial optimization problems with manifold constraints and geodesic coupling terms.
We show that the duality gap vanishes in the limit by proving that a Lipschitz continuous dual multiplier on a unit sphere can be approximated as closely as desired in terms of a Lipschitz continuous polynomial. This is closely related to spherical harmonics and the eigenfunctions of the Laplace–Beltrami operator.
The formulation is applied to manifold-valued imaging problems with total variation regularization and graph-based SLAM. In imaging tasks our approach achieves small duality gaps for a moderate degree.
In graph-based SLAM our approach often finds solutions which after refinement with a local method are near the ground truth solution.
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| 4:00pm - 6:00pm | MS10 4: Optimization in Inverse Scattering: from Acoustics to X-rays Location: VG1.103 Session Chair: Radu Ioan Bot Session Chair: Russell Luke |
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Phase retrieval from overexposed PSFs: theory and practice 1TU Delft, the Netherlands; 2Flexible Optical B.V., the Netherlands
In industrial applications, phase retrieval algorithms can be used to obtain information on optical system misalignment. Because of the specific wavelengths used, often the input data for such algorithms are affected by a high level of noise and quantized with a low bit resolution, and the traditional methods fail to restore the phase accurately. The restoration accuracy can be increased with the presented method of phase retrieval from a (single) overexposed measurement of a point-spread function. We demonstrate that under certain conditions, any projection-based phase retrieval method can be adjusted to accept the input data with the saturated pixels. The modification uses a concept of the clipped set which is able to represent and restore the information lost due to overexposure correctly. With moderate levels of overexposure, the phase restoration accuracy is increased due to the improved signal-to-noise ratio of the PSF. The presentation describes the concept of a clipped set and the procedure of calculating the projection on it and demonstrates the application on the simulated and experimental data.
Tensor-free algorithms for lifted quadratic and bilinear inverse problems 1University of Graz, Austria; 2Technische Universität Berlin, Germany
We present a class of novel algorithms that aim at solving bilinear and quadratic inverse problems. It bases on first-order proximal algorithms for minimizing a Tikhonov functional associated with the respective tensorial lifted problem with nuclear norm regularization [1]. It is well known, however, that a direct application of such algorithms involves computations in the tensor-product space, in particular, singular-value thresholding. Due to the prohibitively high dimension of the latter space, such algorithms are infeasible without appropriate modification. Thus, to overcome this limitation, we show that all computational steps can be adapted to perform on low-rank representations of the iterates, yielding feasible, memory and computationally efficient tensor-free algorithms [2]. We present and discuss the numerical performance of these methods for the two-dimensional Fourier phase retrieval problem. In particular, we show that the incorporation of smoothness constraints within the framework greatly improve image recovery results.
[1] R. Beinert, K. Bredies. Non-convex regularization of bilinear and quadratic inverse problems by tensorial lifting, Inverse Problems 35(1): 015002, 2019.
[2] R. Beinert, K. Bredies. Tensor-free proximal methods for lifted bilinear/quadratic inverse problems with applications to phase retrieval, Foundations of Computational Mathematics 21(5): 1181-1232, 2021.
Implicit regularization via re-parametrization UniGe, Italy
Recently, the success of optimization is related to re- and over-parametrization, that are widely used - for instance - in neural networks applications. However, there is still an open question of how to find systematically what is the inductive bias hidden behind the model for a particular optimization scheme. The goal of this talk is taking a step in this direction, studying extensively many reparametrization used in the state of the art and providing a common structure to analyze the problem in a unified way. We show that gradient descent on the objective function for many reparametrization is equivalent to mirror descent on the original problem. The mirror function depends on the reparametrization and introduces an inductive bias, which plays the role of the regularizer. Our theoretical results provide asymptotic behavior and convergence results.
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