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: VG2.102 |
| Date: Monday, 04/Sept/2023 | |
| 1:30pm - 3:30pm | MS12 1: Fast optimization-based methods for inverse problems Location: VG2.102 Session Chair: Tuomo Valkonen |
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Parameter-Robust Preconditioning for Oseen Iteration Applied to Navier–Stokes Control Problems 1Scuola Normale Superiore, Pisa (Italy); 2The University of Edinburgh, Edinburgh (UK)
Optimal control problems with PDEs as constraints arise very often in scientific and industrial problems. Due to the difficulties arising in their numerical solution, researchers have put a great effort into devising robust solvers for this class of problems. An example of a highly challenging problem attracting significant attention is the (distributed) control of incompressible viscous fluid flow problems. In this case, the physics may be described, for very viscous flow, by the (linear) incompressible Stokes equations, or, in case the convection of the fluid plays a non-negligible role in the physics, by the (non-linear) incompressible Navier–Stokes equations. In particular, as the PDEs given in the constraints are non-linear, in order to obtain a solution of Navier–Stokes control problems one has to iteratively solve linearizations of the problems until a prescribed tolerance on the non-linear residual is achieved.
In this talk, we present novel, fast, and parameter-robust preconditioned iterative methods for the solution of the distributed time-dependent Navier–Stokes control problems with Crank-Nicolson discretization in time. The key ingredients of the solver are a saddle-point type approximation for the linear systems, an inner iteration for the $(1, 1)$-block accelerated by a generalization of the preconditioner for convection–diffusion control derived in [2], and an approximation to the Schur complement based on a potent commutator argument applied to an appropriate block matrix. The flexibility of the commutator argument, which is a generalization of the technique derived in [1], allows one to alternatively apply a backward Euler scheme in time, as well as to solve the stationary Navier–Stokes control problem. We show the effectiveness and robustness of our approach through a range of numerical experiments.
This talk is based on the work in [3].
[1] D. Kay, D. Loghin, A.J. Wathen. A Preconditioner for the Steady-State Navier–Stokes Equations, SIAM Journal on Scientific Computing 24: 237–256, 2002.
[2] S. Leveque, J.W. Pearson. Fast Iterative Solver for the Optimal Control of Time-Dependent PDEs with Crank–Nicolson Discretization in Time, Numerical Linear Algebra with Applications 29: e2419, 2022.
[3] S. Leveque, J.W. Pearson. Parameter-Robust Preconditioning for Oseen Iteration Applied to Stationary and Instationary Navier–Stokes Control, SIAM Journal on Scientific Computing 44: B694–B722, 2022.
Sparse Bayesian Inference with Regularized Gaussian Distributions Technical University of Denmark, Denmark
In this talk, we will present a method for Bayesian inference by implicitly defining a posterior distribution as the solution to a regularized linear least-squares problem with randomized data. This method combines Gaussian distributions with the deterministic effects of sparsity-inducing regularization like $l_1$ norms, total variation and/or constraints. The resulting posterior distributions assign positive probability to various low-dimensional subspaces and therefore promote sparsity. Samples from this distribution can be generated by repeatedly solving regularized linear least-squares problems with properly chosen data perturbations, thus, existing tools from optimization theory can be used for sampling. We will discuss some properties of the methodology and discuss an efficient algorithm for sampling from a Bayesian hierarchical model with sparsity structure.
An Accelerated Level-Set Method for Inverse Scattering Problems 1EDF R&D PRISME, 78400, Chatou, France; 2INRIA, Center of Saclay Ile de France and UMA, ENSTA Paris Tech, Palaiseau Cedex, FRANCE; 3School of Mathematical Sciences, Beihang University, Beijing, 100191, CHINA
We propose a rapid and robust iterative algorithm to solve inverse acoustic scattering problems formulated as a PDE constrained shape optimization problem. We use a level-set method to represent the obstacle geometry and propose a new scheme for updating the geometry based on an adaptation of accelerated gradient descent methods. The resulting algorithm aims at reducing the number of iterations and improving the accuracy of reconstructions. To cope with regularization issues, we propose a smoothing to the shape gradient using a single layer potential associated with $ik$ where $k$ is the wave number. Numerical experiments are given for several data types (full aperture, backscattering, phaseless, multiple frequencies) and show that our method outperforms a nonaccelerated approach in terms of convergence speed, accuracy, and sensitivity to initial guesses.
A first-order optimization method with simultaneous adaptive pde constraint solver 1University of Jyväskylä, Finland; 2University of Helsinki, Finland
We consider a pde-constrained optimization problem and based on the nonlinear primal dual proximal splitting method, a nonconvex generalization of the well-known Chambolle-Pock algorithm, we develop a new iterative algorithmic approach to the problem by splitting the inner problem of solving the pde in each step over the outer iterations. In our work we split our pde-problem in a fashion similar to the classical Gauss-Seidel and Jacobi methods, though other iterative schemes may be fruitful too. We show through numerical experiments that significant speed ups can be attained compared to a naive full pde-solve in each step, and we prove convergence under sufficients second-order growth conditions.
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| 4:00pm - 6:00pm | MS12 2: Fast optimization-based methods for inverse problems Location: VG2.102 Session Chair: Bjørn Christian Skov Jensen |
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Limited memory restarted $\ell^p-\ell^q$ minimization methods using generalized Krylov subspaces 1University of Cagliari, Cagliari, Italy; 2Kent State University, Kent, Ohio
Regularization of certain linear discrete ill-posed problems, as well as of certain regression problems, can be formulated as large-scale, possibly nonconvex, minimization problems, whose objective function is the sum of the $p$-th power of the $\ell^p$-norm of a fidelity term and the $q$-th power of the $\ell^q$-norm of a regularization term, with $0< p,q \leq 2$. We describe new restarted iterative solution methods that require less computer storage and execution time than the methods described by [1]. The reduction in computer storage and execution time is achieved by periodic restarts of the method. Computed examples illustrate that restarting does not reduce the quality of the computed solutions.
[1] G.-X. Huang, A. Lanza, S. Morigi, L. Reichel and F. Sgallari. Majorization–minimization generalized Krylov subspace methods for $\ell_p-\ell_q$ optimization applied to image restoration, BIT Numerical Mathematics 57: 351-378, 2017.
A high order PDE-constrained optimization for the image denoising problem 1University Sultan Moulay Slimane, Morocco; 2University Ibn Zohr, Morroco
In the present work, we investigate the inverse problem of identifying simultaneously the denoised image and the weighting parameter that controls the balance between two diffusion operators for an evolutionary partial differential equation (PDE). The problem is formulated as a non-smooth PDE-constrained optimization model. This PDE is constructed by second- and fourth-order diffusive tensors that combines the benefits from the diffusion model of Perona-Malik in the homogeneous regions, the Weickert model near sharp edges and the fourth-order term in reducing staircasing. The existence and uniqueness of solutions for the proposed PDE-constrained optimization
system are provided in a suitable Sobolev space. Also, an optimization problem for the determination of the weighting parameter is introduced based on the Primal-Dual algorithm. Finally, simulation results show that the obtained parameter usually coincides with the better choice related to the best restoration quality of the image.
A primal dual projection algorithm for efficient constraint preconditioning 1Universität Bayreuth, Germany; 2Zuse Institute Berlin, Germany
We consider a linear iterative solver for large scale linearly constrained quadratic minimization problems that arise, for example, in optimization with partial differential equations (PDEs). By a primal-dual projection (PDP) iteration, which can be interpreted and analysed as a gradient method on a quotient space, the given problem can be solved by computing sulutions for a sequence of constrained surrogate problems, projections onto the feasible subspaces, and Lagrange multiplier updates. As a major application we consider a class of optimization problems with PDEs, where PDP can be applied together with a projected cg method using a block triangular constraint preconditioner. Numerical experiments show reliable and competitive performance for an optimal control problem in elasticity.
An Inexact Trust-Region Algorithm for Nonsmooth Nonconvex Optimization Sandia National Laboratories, United States of America
In this talk, we develop a new trust-region method to minimize the sum of a smooth nonconvex function and a nonsmooth convex function. Our method is unique in that it permits and systematically controls the use of inexact objective function and derivative evaluations. We prove global convergence of our method in Hilbert space and analyze the worst-case complexity to reach a prescribed tolerance. Our method employs the proximal mapping of the nonsmooth objective function and is simple to implement. Moreover, when using a quadratic Taylor model, our algorithm represents a matrix-free proximal Newton-type method that permits indefinite Hessians. We additionally elaborate on potential trust-region sub-problem solvers and discuss local convergence guarantees. We demonstrate the efficacy of our algorithm on various examples from PDE-constrained optimization.
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| Date: Tuesday, 05/Sept/2023 | |
| 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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| 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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| Date: Wednesday, 06/Sept/2023 | |
| 9:00am - 11:00am | MS05 1: Numerical meet statistical methods in inverse problems Location: VG2.102 Session Chair: Martin Hanke Session Chair: Markus Reiß Session Chair: Frank Werner |
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Aggregation by the Linear Functional Strategy in Regularized Domain Adaptation The Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austria
In this talk we are going to discuss the problem of hyperparameters tuning in
the context of learning from different domains known also as domain adaptation. The domain adaptation scenario arises when one studies two input-output relationships governed by probabilistic laws with respect to different probability measures, and uses the data drawn from one of them to minimize the expected prediction risk over the other measure.
The problem of domain adaptation has been tackled by many approaches, and most domain adaptation algorithms depend on the so-called hyperparameters that change the performance of the algorithm and need to be tuned. Usually, algorithm performance variation can be attributed to just a few hyperparameters, such as a regularization parameter in kernel ridge regression, or batch size and number of iterations in stochastic gradient descent training.
In spite of its importance, the question of selecting these parameters has not been much studied in the context of domain adaptation. In this talk, we are going to shed light on this issue. In particular, we discuss how a regularization of the Radon-Nikodym differentiation can be employed in hyperparameters tuning. Theoretical results will be illustrated by application to stenosis detection in different types of arteries.
The talk is based on the recent joint work [1] performed within COMET-Module project S3AI funded by the Austrian Research Promotion Agency (FFG).
[1] E.R. Gizewski, L. Mayer, B.A. Moser, D.H. Nguyen, S. Pereverzyev Jr.,
S.V. Pereverzyev, N. Shepeleva, W. Zellinger. On a regularization of unsupervised domain adaptation in RKHS. Appl. Comput. Harmon. Anal. 57: 201-227, 2022.
The Henderson problem and the relative entropy functional Johannes Gutenberg Universität Mainz, Germany
The inverse Henderson problem of statistical mechanics is the theoretical foundation for many bottom-up coarse-graining techniques for the numerical simulation of complex soft matter physics.
This inverse problem concerns classical particles in continuous space which interact
according to a pair potential depending on the distance of the particles.
Roughly stated, it asks for the interaction potential given the equilibrium pair correlation function of the system.
In 1974 Henderson proved that this potential is uniquely determined in
a canonical ensemble and recently it has been argued by Rosenberger et al. that
this potential minimises a relative entropy. Here we provide a rigorous extension of these results for the thermodynamical limit and define a corresponding relative entropy density for this. We investigate further properties of this functional for suitable classes of pair potentials.
Early stopping for $L^{2}$-boosting in high-dimensional linear models Bocconi University Milano, Italy
We consider $ L^{2} $-boosting in a sparse high-dimensional linear model via orthogonal matching pursuit (OMP). For this greedy, nonlinear subspace selection procedure, we analyze a data-driven early stopping time $ \tau $, which is sequential in the sense that its computation is based on the first $ \tau $ iterations only. Our approach is substantially less costly than established model selection criteria, which require the computation of the full boosting path.
We prove that sequential early stopping preserves statistical optimality in this setting in terms of a general oracle inequality for the empirical risk and recently established optimal convergence rates for the population risk. The proofs include a subtle $ \omega $-pointwise analysis of a stochastic bias-variance trade-off, which is induced by the greedy optimization procedure at the core of OMP. Simulation studies show that, at a significantly reduced computational cost, these types of methods match or even exceed the performance of other state of the art algorithms such as the cross-validated Lasso or model selection via a high-dimensional Akaike criterion based on the full boosting path.
Early stopping for conjugate gradients in statistical inverse problems Humboldt-Universität zu Berlin, Germany
We consider estimators obtained by applying the conjugate gradient algorithm to the normal equation of a prototypical statistical inverse problem. For such iterative procedures, it is necessary to choose a suitable iteration index to avoid under- and overfitting. Unfortunately, classical model selection criteria can be prohibitively expensive in high dimensions. In contrast, it has been shown for several methods that sequential early stopping can achieve statistical and computational efficiency by halting at a data-driven index depending on previous iterates only. Residual-based stopping rules, similar to the discrepancy principle for deterministic problems, are well understood for linear regularization methods. However, in the case of conjugate gradients, the estimator depends nonlinearly on the observations, allowing for greater flexibility. This significantly complicates the error analysis. We establish adaptation results in this setting.
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| Date: Thursday, 07/Sept/2023 | |
| 1:30pm - 3:30pm | MS05 2: Numerical meet statistical methods in inverse problems Location: VG2.102 Session Chair: Martin Hanke Session Chair: Markus Reiß Session Chair: Frank Werner |
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Utilising Monte Carlo method for light transport in the inverse problem of quantitative photoacoustic tomography 1University of Eastern Finland, Finland; 2University College London, United Kingdom
We study the inverse problem of quantitative photoacoustic tomography in a situation where the forward operator is stochastic. In the approach, Monte Carlo method for light transport is used to simulate light propagation in the imaged target. Monte Carlo method is based on random sampling of photon paths as they propagate in the medium. In the inverse problem, MAP estimates for absorption and scattering are computed, and the reliability of the estimates is evaluated. Now, due to the stochastic nature of the forward operator, also the search direction of the optimization algorithm for solving the MAP estimates is stochastic. An adaptive approach for controlling the number of simulated photons during the iteration is studied.
Discretisation-adaptive regularisation via frame decompositions University of Bonn, Germany
We consider linear inverse problems under white (non-Gaussian) noise. We introduce a discretisation scheme to apply the discrepancy principle and the heuristic discrepancy principle, which require bounded data norm. Choosing the discretisation dimension in an adaptive fashion yields convergence, without further restrictions for the operator, the distribution of the white noise or the unknown ground solution. We discuss connections to Lepski's method and apply the technique to ill-posed integral equations with noisy point evaluations and show that here discretisation-adaptive regularisation can be used in order to reduce the numerical complexity. Finally, we apply the technique to methods based on the frame decomposition, tailored for applications in atmospheric tomography.
Operator Learning Meets Inverse Problems 1Caltech, USA; 2Rice University, USA; 3NVIDIA, USA
This talk introduces two connections between operator learning and inverse problems. The first involves framing the supervised learning of a linear operator between function spaces as a Bayesian inverse problem. The resulting analysis of this inverse problem establishes posterior contraction rates and generalization error bounds in the large data limit. These results provide practical insights on how to reduce sample complexity. The second connection is about solving inverse problems with operator learning. This work focuses on the inverse problem of electrical impedance tomography (EIT). Classical methods for EIT tend to be iterative (hence slow) or lack sufficient accuracy. Instead, a new type of neural operator is trained to directly map the data (the Neumann-to-Dirichlet boundary map, a linear operator) to the unknown parameter of the inverse problem (the conductivity, a function). Theory based on emulating the D-bar method for direct EIT shows that the EIT solution map is well-approximated by the proposed architecture. Numerical evidence supports the findings in both settings.
UNLIMITED: The UNiversal Lepskii-Inspired MInimax Tuning mEthoD 1Georg-August-Universität Göttingen, Germany; 2Universität Würzburg
In this talk we consider statistical linear inverse problems in separable Hilbert spaces. They arise in applications spanning from astronomy over medical imaging to engineering. We study the (ordered) filter-based regularization methods, including e.g. spectral cutoff, Tikhonov, iterated Tikhonov, Landweber, and Showalter. The proper choice of regularization parameter is always crucial, and often relies on the (unknown) structure assumptions of the true signal. Aiming at a fully automatic procedure, we investigate a specific a posteriori parameter choice rule, which we call UNiversal Lepskii-Inspired MInimax Tuning method (UNLIMITED). We show that the UNLIMTED rule leads to adaptively minimax optimal rates over various smoothness function classes in mildly and severely ill-posed problems. In particular, our results reveal that the “common sense” that one typically loses a log-factor for Lepskii-type methods is actually wrong! In addition, the empirical performance of UNLIMITED is examined in simulations.
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| 4:00pm - 6:00pm | MS05 3: Numerical meet statistical methods in inverse problems Location: VG2.102 Session Chair: Martin Hanke Session Chair: Markus Reiß Session Chair: Frank Werner |
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Bayesian hypothesis testing in statistical inverse problems Institute of Mathematics, University of Würzburg, Germany
In many inverse problems, one is not primarily interested in the whole solution $u^\dagger$, but in specific features of it that can be described by a family of linear functionals of $u^\dagger$. We perform statistical inference for such features by means of hypothesis testing.
This problem has previously been treated by multiscale methods based upon unbiased estimates of those functionals [1]. Constructing hypothesis tests using unbiased estimators, however, has two severe drawbacks: Firstly, unbiased estimators only exist for sufficiently smooth linear functionals, and secondly, they suffer from a huge variance due to the ill-posedness of the problem, so that the corresponding tests have bad detection properties.
We overcome both of these issues by considering the problem from a Bayesian point of view, assigning a prior distribution to $u^\dagger$, and using the resulting posterior distribution to define Bayesian maximum a posteriori (MAP) tests.
The existence of a hypothesis test with maximal power among a class of tests with prescribed level has recently been shown for all linear functionals of interest under certain a priori assumptions on $u^\dagger$ [2]. We study Bayesian MAP tests bases upon Gaussian priors both analytically and numerically for linear inverse problems and compare them with unregularized as well as optimal regularized hypothesis tests.
[1] K. Proksch, F. Werner, A. Munk. Multiscale scanning in inverse problems. Ann. Statist. 46(6B): 3569--3602. 2018. https://doi.org/10.1214/17-AOS1669
[2] R. Kretschmann, D. Wachsmuth, F. Werner. Optimal regularized hypothesis testing in statistical inverse problems. Preprint, 2022. https://doi.org/10.48550/arXiv.2212.12897
Predictive risk regularization for Gaussian and Poisson inverse problems Università degli Studi di Genova, Italy
In this talk, we present two methods for the choice of the regularization parameter in statistical inverse problems based on the predictive risk estimation, in the case of Gaussian and Poisson noise.
In the first case, the criterion for choosing the regularization parameter in Tikhonov regularization is motivated by stability issue in the case of small sized samples and it minimizes a lower bound of the predictive risk. It is applicable when both data norm and noise variance are known, minimizing a function which depends on the signal-to-noise ratio, and also when they are unknown, using an iterative algorithm which alternates between a minimization step of finding the regularization parameter and an estimation step of estimating signal-to-noise ratio.
In this second case, we introduce a novel estimator of the predictive risk with Poisson data, when the loss function is the Kullback–Leibler divergence, in order to define a regularization parameter's choice rule for the expectation maximization (EM) algorithm. We present a Poisson counterpart of the Stein's Lemma for Gaussian variables, and from this result we derive the proposed estimator which is asymptotically unbiased with increasing number of measured counts, when the EM algorithm for Poisson data is considered.
In both cases we present some numerical tests with synthetic data.
Reconstruction of active forces in actomyosin droplets Georg-August-University Göttingen, Germany
Many processes in cells are driven by the interaction of multiple proteins, for example cell contraction, division or migration. Two important types of proteins are actin filaments and myosin motors. Myosin is able to bind to and move along actin filaments with its two ends, leading to the formation of a dynamic actomyosin network, in which stresses are generated and patterns may form. Droplets containing an actomyosin network serve as a strongly simplified model for a cell, which are used to study elemental mechanisms. We are interested in determining the parameters that characterize this active matter, i.e., active forces that cause the dynamics of an actomyosin network, represented by the flow inside the actomyosin droplet, as well as the local viscosity. This leads to a (deterministic) parameter identification problem for the Stokes equation, where the viscosity inside the droplet can be estimated by means of statistical approaches.
Learning Linear Operators TU Braunschweig, Germany
We consider the problem of learning a linear operator $\theta$ between two Hilbert spaces from empirical observations, which we interpret as least squares regression in infinite dimensions. We show that this goal can be reformulated as an inverse problem for $\theta$ with the undesirable feature that its forward operator is generally non-compact (even if $\theta$ is assumed to be compact or of $p$-Schatten class). However, we prove that, in terms of spectral properties and regularisation theory, this inverse problem is equivalent to the known compact inverse problem associated with scalar response regression. Our framework allows for the elegant derivation of dimension-free rates for generic learning algorithms under H\"{o}lder-type source conditions. The proofs rely on the combination of techniques from kernel regression with recent results on concentration of measure for sub-exponential Hilbertian random variables. The obtained rates hold for a variety of practically-relevant scenarios in functional regression as well as nonlinear regression with operator-valued kernels and match those of classical kernel regression with scalar response.
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| Date: Friday, 08/Sept/2023 | |
| 1:30pm - 3:30pm | MS08 1: Integral Operators in Potential Theory and Applications Location: VG2.102 Session Chair: Doosung Choi Session Chair: Mikyoung Lim Session Chair: Stephen Shipman |
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On the identification of small anomaly via MUSIC algorithm without background information Kookmin University, Korea, Republic of (South Korea)
MUltiple SIgnal Classification (MUSIC) is a promising non-iterative technique for identifying small anomaly in microwave imaging. For a successful application, accurate values of permittivity, permeability, and conductivity of the background must be known. If one of these values is unknown, inaccurate location will inevitably retrieved by using the MUSIC. To explain this phenomenon, we investigate the structure of the imaging function of MUSIC by establishing a relationship with an infinite series of the Bessel functions of integer order, antenna arrangement, and applied values of permittivity, permeability, and conductivity. The revealed structure explains the theoretical reason why inaccurate location of anomaly is retrieved. Simulation results with synthetic data are illustrated to support the theoretical result.
[1] W.-K. Park. Application of MUSIC algorithm in real-world microwave imaging of unknown anomalies from scattering matrix, Mech. Syst. Signal Proc. 153: Article No. 107501, 2021.
[2] R. Solimene, G. Ruvio, A. Dell'Aversano, A. Cuccaro, Max J. Ammann, R. Pierri. Detecting point-like sources of unknown frequency spectra, Prog. Electromagn. Res. B 50: 347-364, 2013.
Construction of inclusions with vanishing generalized polarization tensors by imperfect interfaces 1Louisiana State University, United States of America; 2Korea Advanced Institute of Science and Technology, Republic of Korea
We address this question and provide a new construction scheme to find GPT-vanishing structures by imperfect interfaces. In particular, we construct GPT-vanishing structures of general shape with imperfect interfaces, where the inclusions have arbitrarily finite conductivity.
Spectral theory of surface plasmons in the nonlocal hydrodynamic Drude model 1Inha University, South Korea; 2Korea University, South Korea; 3University of Edinburgh, Scotland
We study surface plasmons, which are collective oscillations of electrons at metal-dielectric interfaces that can be excited by light. The local Drude model, which is the standard way to describe surface plasmons, ignores the spatial and quantum variations of the electron gas. These variations matter at the nanoscale and can change how metallic nanostructures interact with light. We use integral operator methods to investigate how the nonlocal hydrodynamic Drude model (HDM), which accounts for these variations, affects the spectral properties of surface plasmons in general shapes with smooth boundaries.
Numerical computation of Laplacian eigenvalues based on the layer potential formulation Korea Advanced Institute of Science and Technology, Korea, Republic of (South Korea)
In this talk, we will present a numerical method that allows for the Laplacian eigenvalues of a planar, simply connected domain by only using the coefficients of the conformal mapping of the domain. We formulate the eigenvalue problems using the layer potential characterization and geometric density basis functions, resulting in an infinite dimensional matrix, where geometric density basis functions are associated with the conformal mapping of the domain. We will discuss how to compute the eigenvalues by using this infinite-dimensional matrix. Additionally, we will provide some convergence analysis for this approach based on the Gohberg-Sigal theory for operator-valued functions.
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| 4:00pm - 6:00pm | MS08 2: Integral Operators in Potential Theory and Applications Location: VG2.102 Session Chair: Doosung Choi Session Chair: Mikyoung Lim Session Chair: Stephen Shipman |
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Recovering an elastic inclusion using the shape derivative of the elastic moment tensors Korea Advanced Institute of Science & Technology, Korea, Republic of (South Korea)
An elastic inclusion embedded in a homogeneous background induces a perturbation for a given far-field loading. This perturbation admits a multipole expansion with coefficients known by Elastic Moment Tensors (EMTs), which contain information on the material and geometric properties of the inclusion. Iterative optimization approaches to recover the shape of the inclusion involving the EMTs have been reported. In this talk, we focus on the shape derivative of the EMTs for planar inclusion. In particular, we derive asymptotic expressions for the shape deformation of inclusion from a disk, based on the complex formulation for the solution to the plane elastostatic problem.
Some aspects of the spectrum of the Neumann-Poincaré operator Louisiana State University, United States of America
I will discuss some applications of the spectrum of the Neumann-Poincaré operator.
Spectrum of the Neumann-Poincaré operator on thin domains Ehime University, Japan
We consider the spectral structure of the Neumann–Poincaré operators defined on the boundaries of thin domains in two and three dimensions. In two dimensions, we consider rectangle-shaped domains. We prove that as the aspect ratio of the domains tends to $\infty$, or equivalently, as the domains get thinner, the spectra of the Neumann–Poincaré operators are densely distributed in $[−\frac{1}{2}, \frac{1}{2} ]$. In three dimensions, we consider two different kinds of thin domains: thin oblate domains and thin cylinders. We show that in the first case the spectra are distributed densely in the interval $[−\frac{1}{2}, \frac{1}{2} ]$ as as the domains get thinner. In the second case, as a partial result, we show that the spectra are distributed densely in the half interval $[−\frac{1}{2}, \frac{1}{2} ]$ as the domains get thinner.
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