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: VG3.103 |
| Date: Monday, 04/Sept/2023 | |
| 1:30pm - 3:30pm | MS34 1: Learned reconstructions for nonlinear inverse problems Location: VG3.103 Session Chair: Simon Robert Arridge Session Chair: Andreas Selmar Hauptmann |
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Continuous generative models for nonlinear inverse problems 1University of Genoa, Italy; 2Technische Universität Berlin, Germany
Generative models are a large class of deep learning architectures, trained to describe a subset of a high dimensional space with a small number of parameters. Popular models include variational autoencoders, generative adversarial networks, normalizing flows and, more recently, score-based diffusion models. In the context of inverse problems, generative models can be used to model prior information on the unknown with a higher level of accuracy than classical regularization methods.
In this talk we will present a new data-driven approach to solve inverse problems based on generative models. Taking inspiration from well-known convolutional architectures, we construct and explicitly characterize a class of injective generative models defined on infinite dimensional functions spaces. The construction is based on wavelet multi resolution analysis: one of the key theoretical novelties is the generalization of the strided convolution between discrete signals to an infinite dimensional setting. After an off-line training of the generative model, the proposed reconstruction method consists in an iterative scheme in the low-dimensional latent space. The main advantages are the faster iterations and the reduced ill-posedness, which is shown with new Lipschitz stability estimates. We also present numerical simulations validating the theoretical findings for linear and nonlinear inverse problems such as electrical impedance tomography.
Data-driven quantitative photoacoustic imaging University of Cambridge, United Kingdom
Photoacoustic imaging faces the challenge of accurately quantifying measurements to accurately reconstruct chromophore concentrations and thus improve patient outcomes in clinical applications. Proposed approaches to solve the quantification problem are often limited in scope or only applicable to simulated data. We use a collection of well-characterised imaging targets (phantoms) as well as simulated data to enable supervised training and validation of quantification methods and train a U-Net on the data set. Our experiments demonstrate that phantoms can serve as reliable calibration objects and that deep learning methods can generalize to estimate the optical properties of previously unseen test images. Application of the trained model to a blood flow phantom and a mouse model highlights the strengths and weaknesses of the proposed approach.
Mapping properties of neural networks and inverse problems 1University of Helsinki, Finland; 2South Dakota State University, USA; 3University of Basel, Switzerland; 4Rice University, USA
We will consider mapping properties of neural networks, in particular, injectivity
of neural networks, universal approximation property of neural networks and
the properties which the ranges of neural networks need to have. Also, we
study approximation of probability measures using neural networks composed
of invertible flows and injective layer and applications of these results in inverse problems.
Data-driven regularization theory of invertible ResNets for solving inverse problems University of Bremen, Germany
Data-driven solution techniques for inverse problems, typically based on specific learning strategies, exhibit remarkable performance in image reconstruction tasks. These learning-based reconstruction strategies often follow a two-step scheme. First, one uses a given dataset to train the reconstruction scheme, which one often parametrizes via a neural network. Second, the reconstruction scheme is applied to a new measurement to obtain a reconstruction. We follow these steps but specifically parametrize the reconstruction scheme with invertible residual networks (iResNets). We demonstrate that the invertibility opens the door to new investigations into the influence of the training and the architecture on the resulting reconstruction scheme. To be more precise, we analyze the effect of different iResNet architectures, loss functions, and prior distributions on the trained network. The investigations reveal a formal link to the regularization theory of linear inverse problems for shallow network architectures and connections to MAP estimation with Gaussian noise models. Moreover, we analytically optimize the parameters of specific classes of architectures in the context of Bayesian inversion, revealing the influence of the prior and noise distribution on the solution.
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| 4:00pm - 6:00pm | MS34 2: Learned reconstructions for nonlinear inverse problems Location: VG3.103 Session Chair: Simon Robert Arridge Session Chair: Andreas Selmar Hauptmann |
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Learned iterative model-based approaches in quantitative photoacoustic tomography 1University of Oulu, Finland; 2Centrum Wiskunde & Informatica
Quantitative photoacoustic tomography (QPAT) is an upsurging imaging modality which can provide high-resolution tissue images based on optical absorption. Classical reconstruction methods rely on sufficient prior information to overcome noisy and imperfect data. As these methods utilise computationally expensive forward models, the computation becomes slow, delimiting the possibilities of QPAT in time-critical applications. As an alternative approach, deep learning-based reconstruction methods have been proposed to allow fast computation of accurate reconstructions. In our work, we adopt the model-based learned iterative approach to solve the nonlinear optical problem of QPAT. In the learned iterative model-based approach, the forward operator and its derivative are iteratively evaluated to compute an update step direction, which is then fed to the network. The learning task is formulated as greedy, requiring iterate-wise optimality, as well as in an end-to-end manner, where all updating networks are trained jointly. We formulated these training schemes and evaluated the performances when the step direction was computed with gradient descent and with the Gauss-Newton method.
Autocorrelation analysis for cryo-EM with sparsifying priors Princeton University, United States of America
Cryo-electron microscopy is a non-linear inverse problem that aims to reconstruct 3-D molecular structures from randomly oriented tomographic projection images, taken at extremely low signal-to-noise-ratio.
This talk presents new results for using the method of moments to reconstruct sparse molecular structures. We prove that molecular structures modeled as sparse sums of Gaussians can be uniquely recovered from the autocorrelations of the images, which significantly lowers the sample complexity of the problem compared to previous results. Moreover, we provide practical reconstruction algorithms inspired by crystallographic phase retrieval.
The full reconstruction pipeline includes estimating autocorrelations from projection images, using rotation-invariant principal component analysis made possible by recent improvements to approximation algorithms into the Fourier-Bessel basis.
Model corrections in linear and nonlinear inverse problems 1University of Oulu, Finland; 2University College London, UK
Solving inverse problems in a variational formulation requires repeated evaluation of the forward operator and its derivative. This can lead to a severe computational burden, especially so for nonlinear inverse problems, where the derivative has to be recomputed at every iteration. This motivates the use of faster approximate models to make computations feasible, but due to an arising approximation error the need to introduce a designated correction arises.
In this talk we first discuss the concept of learned model corrections applied to linear inverse problems, when computationally fast but approximate forward models are used. We then proceed to examine the possibility to approximate nonlinear models with a linear one and then solve the linear problem instead, avoiding differentiation of the nonlinear model. To correct for the arising approximation errors, we sequentially estimate the error between linear and nonlinear model and update a correction term in the variational formulation. In both cases we discuss convergence properties to solutions of the variational problem given the accurate models.
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| Date: Tuesday, 05/Sept/2023 | |
| 1:30pm - 3:30pm | MS19 1: Theory and algorithms of super-resolution in imaging and inverse problems Location: VG3.103 Session Chair: Habib Ammari Session Chair: Ping Liu |
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Stability and super-resolution of MUSIC and ESPRIT for multi-snapshot spectral estimation CUNY City College, United States of America
We study the spectral estimation problem of estimating the locations of a fixed number of point sources given multiple snapshots of Fourier measurements collected by a uniform array of sensors. We prove novel stability bounds for MUSIC and ESPRIT as a function of the noise standard deviation, number of snapshots, source amplitudes, and support. When the point sources are located in several clumps, we provide explicit upper bounds for MUSIC and ESPRIT in terms of a Super-Resolution Factor (SRF). We also derive a new Cramér-Rao lower bound for the clumps model, and as a result, implies that ESPRIT is comparable to that of the optimal unbiased estimator(s) in terms of the dependence on noise, number of snapshots and SRF. As a byproduct of our analysis, we discover several fundamental differences between the single-snapshot and multi-snapshot problems.
A Mathematical Theory of Computational Resolution Limit and Super-resolution ETH Zurich, Switzerland
Due to the physical nature of wave propagation and diffraction, there is a fundamental diffraction barrier in optical imaging systems which is called the diffraction limit or resolution limit. Rayleigh investigated this problem and formulated the well-known Rayleigh limit. However, the Rayleigh limit is empirical and only considers the resolving ability of the human visual system. On the other hand, resolving sources separated below the Rayleigh limit to achieve so-called “super-resolution” has been demonstrated in many numerical experiments.
In this talk, we will propose a new concept “computational resolution limit” which reveals the fundamental limits in superresolving the number and locations of point sources from a data-processing point of view. We will quantitatively characterize the computational resolution limits by the signal-to-noise ratio, the sparsity of sources, and the cutoff frequency of the imaging system. As a direct consequence, it is demonstrated that $l_0$ optimization achieves the optimal order resolution in solving super-resolution problems. For the case of resolving two point sources, the resolution estimate is improved to an exact formula, which answers the long-standing question of diffraction limit in a general circumstance. We will also propose an optimal algorithm to distinguish images generated by single or multiple point sources. Generalization of our results to the imaging of positive sources and imaging in multi-dimensional spaces will be briefly discussed as well.
Total variation regularized problems: a support stability result 1Inria, France; 2CEREMADE, Université Paris-Dauphine, PSL University; 3Universita di Genova; 4Institut Camille Jordan; 5Ecole Centrale Lyon
The total (gradient) variation has been used in many imaging applications following the seminal work of Rudin, Osher and Fatemi.[1]
In this talk, I will describe a "support stability'' result for total-variation regularized inverse problems: under some assumptions, the solutions at low noise and low regularization have the same number of values as the unknown image, and their level sets converge to those of the unknown image.
It is a joint work with Romain Petit and Yohann De Castro.
[1] L. I. Rudin, S. Osher, E. Fatemi. Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, Volume 60, Issues 1–4, 1 November 1992, Pages 259-268.
Theoretical and numerical off-the-grid curve reconstruction 1Morpheme (Inria/CNRS), France; 2LJAD (CNRS), France
Recent years have seen the development of super-resolution variational optimisation
optimisation in measure spaces. These so-called off-the-grid approaches offer both theoretical (uniqueness, reconstruction guarantees) and numerical results, with very convincing results in biomedical imaging. However, the gridless variational optimisation is formulated for reconstruction of point sources, which is not always suitable for biomedical imaging applications: more realistic biological structures such as curves should also be reconstructed, to represent blood vessels or filaments for instance.
We propose a new strategy for the reconstruction of curves in an image through an off-the-grid variational framework, inspired by the reconstruction of spikes in the literature. We introduce a new functional CROC on the space of 2-dimensional Radon measures with finite divergence denoted V. Our main contribution lies in the sharp characterisation of the extreme points of the unit ball of the V -norm: there are exactly measures supported on 1-rectifiable oriented simple Lipschitz curves, thus enabling a precise characterisation of our functional minimisers and further opening the avenue for the algorithmic implementation.
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| 4:00pm - 6:00pm | MS19 2: Theory and algorithms of super-resolution in imaging and inverse problems Location: VG3.103 Session Chair: Habib Ammari Session Chair: Ping Liu |
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IFF: A Super-resolution Algorithm for Multiple Measurements The Hong Kong University of Science and Technology, Hong Kong S.A.R. (China)
The problem of reconstructing one-dimensional point sources from their Fourier measurements in a bounded interval $[-\Omega, \Omega]$ is known to be challenging in the regime where the spacing of the sources is below the Rayleigh length $\frac{\pi}{\Omega}$. In this talk, we present a super-resolution algorithm, called Iterative Focusing-localization and Filtering (IFF), to resolve closely spaced point sources from their multiple measurements that are obtained by using multiple unknown illumination patterns. The new proposed algorithm requires no prior information about the source numbers and allows for a subsampling strategy that can circumvent the computation of singular-value decomposition for large matrices as in the usual subspace methods.
In the talk, we will also discuss the theoretical results of the methods behind the algorithm. The derived results imply a phase transition phenomenon. Numerical results show that the algorithm can achieve a stable reconstruction for point sources with a minimum separation distance that is close to the theoretical limit as well as the phase transition phenomenon predicted by the theoretical analysis.
Vectorized Hankel Lift: A Convex Approach for Blind Super-Resolution of Point Sources Fudan University, China, People's Republic of
Blind super-resolution is the problem of estimating high-resolution information about a signal from its low-resolution measurements when the point spread functions (PSFs) are unknown. It is a common problem in many scientific and engineering research areas, such as machine learning, signal processing, and computer vision. Blind super-resolution can be cast as a low-rank matrix recovery problem by exploiting the inherent simplicity of the signal and the low-dimensional structure of the PSFs.
In this talk, we will discuss the low-rank matrix recovery problem for blind super-resolution of point sources. The target matrices associated with these problems are not only low rank but also highly structured. Convex approaches are proposed for the corresponding low-rank matrix recovery problems. Theoretical guarantees are established showing that near-optimal sample complexity is sufficient for successful recovery.
Super-resolved Lasso University of Bath, United Kingdom
The behaviour of sparse regularization using the Lasso method is well understood when dealing with discretized linear models. However, the behaviour of Lasso is poor when dealing with models with very large parameter spaces and exact localisation of the sparse support is often not possible due to discretization (gridding) issues. We introduced a new optimization problem known as the super-resolved Lasso, by considering a higher order expansion of the continuous operator, we show that we can precisely recover the support when the 'true' signal lies up to a fraction off the grid. This is joint work with Gabriel Peyre.
Approximate inverse scattering via convex programming MaLGa center, department of Mathematics, University of Genoa
In this work, we propose to apply and adapt known results on convex variational methods for inverse problems to the inverse scattering problem. We rely on approximations to circumvent its nonlinearity, and discuss recovery guarantees and numerical methods.
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| Date: Wednesday, 06/Sept/2023 | |
| 9:00am - 11:00am | MS19 3: Theory and algorithms of super-resolution in imaging and inverse problems Location: VG3.103 Session Chair: Habib Ammari Session Chair: Ping Liu |
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On Beurling-Selberg Approximations and the Stability of Super-Resolution CentraleSupélec, Université Paris-Saclay, France
Of particular interest for super-resolution is the line spectrum estimation problem, which consists in recovering a stream of spikes (point sources) from the noisy observation of a few number of its first trigonometric moments weighted by the ones of the point-spread function (PSF). The empirical feasibility of this problem has been known since the work of Rayleigh on diffraction to be essentially driven by the separation between the spikes to recover.
We present a novel statistical framework based on the spectrum of the Fisher information matrix (FIM) to quantify the stability limit of super-resolution as a function of the PSF. In the regime where the minimal separation is inversely proportional to the number of acquired moments, we show the existence of a separation constant above which the minimal eigenvalue of the FIM is not asymptotically vanishing—defining a statistical resolution limit. Notably, a relationship between the total variation of the autocorrelation function of the PSF and its association resolution limit is highlighted. Those novel bounds are derived by relating the extremal eigenvalues of the FIM with a higher-order Beurling–Selberg type extremal approximation problem over the functions of bounded variation, for which we provide solutions.
A Parameter Identification Algorithm for Gaussian Mixture Models Hong Kong University of Science and Technology, Hong Kong S.A.R. (China)
In this talk, we consider the problem of learning the parameters from the Fourier measurements of the one-dimensional Gaussian mixture models(GMM). Unlike most algorithms requires the number of Gaussians a prior, our method only need to know the number of different variances as prior information. We also illustrate that for stably recovering all the components under certain noise level, a separation condition for the variances is necessary. Our method can be generalized into high dimensional cases.
Super-localisation of a point-like emitter in a resonant environment : correction of the mirage effect CNRS, France
In this paper, we show that it is possible to overcome one of the fundamental limitations of super-resolution microscopy techniques: the necessity to be in an $\text{optically homogeneous}$ environment. Using recent modal approximation results we show as a proof of concept that it is possible to recover the position of a single point-like emitter in a $\text{known resonant environment}$ from far-field measurements with a precision two orders of magnitude below the classical Rayleigh limit. The procedure does not involve solving any partial differential equation, is computationally light (optimisation in $\mathbb{R}^d$ with $d$ of the order of 10) and therefore suited for the recovery of a very large number of single emitters.
Optimal super-resolution of close point sources and stability of Prony's method Tel Aviv Universtiy, Israel
We consider the problem of recovering a linear combination of Dirac masses from noisy Fourier samples, also known as the problem of super-resolution. Following recent derivation of min-max bounds for this problem when some of the sources collide, we develop an optimal algorithm which provably achieves these bounds in such a challenging scenario. Our method is based on the well-known Prony's method for exponential fitting, and a novel analysis of its stability in the near-colliding regime, combined with the decimation technique for improving the conditioning of the problem.
Based on joint works with N.Diab and R.Katz:
[1] R. Katz, N. Diab, D. Batenkov. Decimated Prony's Method for Stable Super-resolution. 2022. http://arxiv.org/abs/2210.13329
[2] R. Katz, N. Diab, D. Batenkov. On the accuracy of Prony's method for recovery of exponential sums with closely spaced exponents. 2023. http://arxiv.org/abs/2302.05883
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| Date: Thursday, 07/Sept/2023 | |
| 1:30pm - 3:30pm | MS06 2: Inverse Acoustic and Electromagnetic Scattering Theory - 30 years later Location: VG3.103 Session Chair: Fioralba Cakoni Session Chair: Houssem Haddar |
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Nonlinearity parameter imaging in the frequency domain texas A&M University, United States of America
Nonlinear parameter tomography is a technique for enhancing ultrasound imaging and amounts to identifying the spatially varying coefficient $\eta=\eta(x)$
in the Westervelt equation
$ p_{tt}-c^2\triangle p - b\triangle p_t = \eta(p^2)_{tt} + h$
in a domain $(0,T)\times\Omega$.
Here $p$ is the acoustic pressure, $c$ the speed of sound,
$b$ the diffusivity of sound, and $h$ the excitation.
Observations consist of pressure measurements
on some manifold $\Sigma$ immersed in the acoustic domain $\Omega$.
Our imaging goal is to show unique recovery when $\eta(x)$
is a finite set $\{a_i\chi(D_i)\}_i$ and where each $D_i$ is
starlike with respect to its centroid.
Assuming periodic excitations of the form $h(x,t) = A e^{i\omega t}$
for some fixed frequency $\omega$ one can convert this to an
infinite system of coupled linear Helmholtz equations.
We will give both uniqueness and reconstructions results and note that
this work was inspired by a previous paper of one author and Rainer Kress.
The Lippmann-Schwinger Lanczos algorithm for inverse scattering. 1WPI, United States of America; 2Drexel University, United States of America; 3Southern Methodist University, United States of America; 4University of Utah, United States of America
We combine data-driven reduced order models with the Lippmann- Schwinger integral equation to produce a direct nonlinear inversion method. The ROM is viewed as a Galerkin projection and is sparse due to Lanczos orthogonalization. Embedding into the continuous problem, a data-driven internal solution is produced. This internal solution is then used in the Lippmann-Schwinger equation, in a direct or iterative framework. The approach also allows us to process more general transfer functions, i.e., to remove the main limitation of the earlier versions of the ROM based inversion algorithms. We describe how the generation of internal solutions simplifies in the time domain, and show how Lanczos orthogonalization in the spectral domain relates to time stepping. We give examples of its use given mono static data, targeting synthetic aperture radar.
Analysis of topological derivative for qualitative defect imaging using elastic waves ENSTA Paris, France
The concept of topological derivative has proved effective as a qualitative inversion tool for wave-based identification of finite-sized objects. This approach is often based on a heuristic interpretation of the topological derivative. Its mathematical justification has however also been studied, in particular in cases where the true obstacle is small enough for asymptotic approximations of wave scattering to be applicable, and also for finite-sized objects in the scalar wave framework. This work extends our previous efforts in the latter direction to the identification of elastic inhomogeneities embedded in elastic media interrogated by elastic waves. The data used for identification, assumed to be of near-field nature (i.e. no far-field approximation is introduced), is introduced through a misfit functional $J$. The imaging functional that reveals embedded inhomogeneities then consists of the topological derivative $\mathcal{T}_J$ of $J$ (in particular, the actual minimization of $J$ is not performed, making the procedure significantly faster than standard inversion based on PDE-constrained minimization). Our main contribution consists in an analysis of $\mathcal{T}_J$ using a suitable factorization of the near fields, achievable thanks to a convenient reformulation of the volume integral equation formulation of the forward elastodynamic scattering problem established earlier. Our results include justification of both the sign heuristics for $\mathbf{z}\mapsto\mathcal{T}_J(\mathbf{z})$ (which is expected to be most negative at points $\mathbf{z}$ inside, or close to, the support of the sought flaw) and the spatial decay of $\mathcal{T}_J(\mathbf{z})$ as $\mathbf{z}$ moves away from the flaw support. This result, subject to a limitation on the strength of the inhomogeneity to be identified, provides a theoretical conditional validation of the usual heuristic interpretation of $\mathcal{T}_J$ as an imaging functional. Our findings are demonstrated on 3D computational experiments.
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| 4:00pm - 6:00pm | MS06 3: Inverse Acoustic and Electromagnetic Scattering Theory - 30 years later Location: VG3.103 Session Chair: Fioralba Cakoni Session Chair: Houssem Haddar |
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The direct and inverse scattering problem of obliquely incident electromagnetic waves by an inhomogeneous infinitely long cylinder 1National Technical University of Athens, Greece; 2Agricultural University of Athens, Greece
We consider the scattering problem of electromagnetic waves by an infinitely
long cylinder in three dimensions.The cylinder is dielectric, isotropic and
inhomogeneous(with respect to the lateral directions). The incoming wave is
time-harmonic and obliquely incident on the scatterer. We examine the
well-posedness of the direct problem (uniqueness and existence of solution)
using a Lippmann- Schwinger integral equation formulation. We prove uniqueness
of the inverse problem to reconstruct the refractive index of an isotropic
circular cross-section using the discreteness of the corresponding transmission
eigenvalue problem and solutions based on separation of variables. We solve
numerically the inverse problem for media with radial symmetric parameters using a Newton - type scheme. The direct problem is also solved numerically to provide us with the necessary
far-field patterns of the scattered fields. We present numerical reconstructions
justifying the applicability of the proposed method.
Transmission Eigenvalues for a Conductive Boundary Purdue University, United States of America
In this talk, we will investigate the acoustic transmission eigenvalue problem associated with an inhomogeneous media with a conductive boundary. These are a new class of eigenvalue problems that is not elliptic, not self-adjoint, and non-linear, which gives the possibility of complex eigenvalues. We will discuss the existence of the eigenvalues as well as their dependence on the material parameters. Due to the fact that this is a non-standard eigenvalue problem, a discussion of the numerical calculations will also be highlighted. This is joint work with: R.-C. Ayala, O. Bondarenko, A. Kleefeld, and N. Pallikarakis.
Generalized Sampling method EDF R&D, France
The Generalized Sampling Method has been introduced to justify the so-called Linear Sampling Method of Colton and Kirsch (1996). It offers a framework that allow more flexibility than the Factorization Method of Kirsch which made it possible to extended a little the theoretical analysis of sampling methods. In this contribution we will point out the remaining difficulties of the Generalized Linear Sampling methods namely the form of the regularization term, the treatment of noisy measurements and some configuration of the sources and the receivers that break the symmetry of the near field operator. We will propose solution to address some of this challenges. Numerical illustrations will be provided on various type of measurements from Electrical Impedance Tomography, acoustics and elasticity scattering.
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| Date: Friday, 08/Sept/2023 | |
| 1:30pm - 3:30pm | MS06 4: Inverse Acoustic and Electromagnetic Scattering Theory - 30 years later Location: VG3.103 Session Chair: Fioralba Cakoni Session Chair: Houssem Haddar |
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Nonlinear integral equations for 3D inverse acoustic and electromagnetic scattering Hartree Centre, Science and Technology Facilities Council, UK
We present two extensions of the method, originally developed by Kress and Rundell in 2005 for a 2D inverse boundary value problem for the Laplace equation. In particular, we consider the reconstruction of a 3D perfectly electric conductor obstacle, and the reconstruction of generalized surface impedance functions for acoustic scattering from the knowledge of far-field measurements of a scattered wave associated with a few incident plane waves. Inverse scattering problems are solved numerically by the approach based on the reformulation of a problem as a system of nonlinear and ill-posed integral equations for the unknown boundary (or boundary condition) and the measurements. The iteratively regularized Gauss-Newton method is applied to the resulting system.
[1] R. Kress, W. Rundell. Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Probl. 21(4): 1207--1223, 2005.
[2] O. Ivanyshyn Yaman, F. Le Lou\"{e}r. Material derivatives of boundary integral operators in electromagnetism and application to inverse scattering problems. Inverse Probl. 32(9): 095003, 2016.
[3] O. Ivanyshyn Yaman. Reconstruction of generalized impedance functions for 3D acoustic scattering. J. Comput. Phys, 392(1): 444--455, 2019.
Inverse scattering in a partially embedded waveguide 1ENSTA Paris/POEMS, France; 2CEA LIST, France
This talk concerns the identification of defects in a closed waveguide which is partially embedded in a surrounding medium, from scattering measurements on the free part of the waveguide. We wish to model for example a NDT experiment on a steel cable embedded in concrete. There are two main issues: the back-scattering situation and the leakage of waves from the closed waveguide to the surrounding medium.
We will first introduce Perfectly Matched Layers in the transverse direction in order to transform the structure into a junction of two closed-half waveguides, one of them being a complex stratified medium.
Then, after discussing the well-posedness of the forward problem and its numerical resolution, we will show how we can solve the inverse problem with the help of a modal formulation of the Linear Sampling Method.
Some 2D numerical experiments will be shown.
Revisiting the Hybrid method for the inverse scattering transmission problem 1Universidade Aberta, Portugal; 2CIBIT, University of Coimbra, Portugal; 3CEMAT, University of Lisbon, Portugal
In this talk we will address the numerical solution of the time-harmonic inverse scattering problem for an obstacle with transmission conditions and with given far-field data. To this end we will revisit the ideas of the hybrid method [1,2,3,4,5] that combines the framework of the Kirsch-Kress decomposition method and the iterative Newton-type method.
Instead of linearizing all the equations at once as in [6,7], we will explore the possibility of in a first ill-posed step reconstructing the scattered exterior field and the interior field by imposing the far-field condition and one of the boundary conditions and then in a second step linearizing on the second boundary condition in order to update the approximation of the boundary of the obstacle. The first and second steps are then iterated until some stopping criteria is achieved.
[1] R. Kress, P. Serranho. A hybrid method for two-dimensional crack reconstruction, Inverse Probl. 21 (2): 773--784, 2005.
[2] P. Serranho. A hybrid method for inverse scattering for shape and impedance, Inverse Probl. 22 (2): 663--680, 2006.
[3] R. Kress, P. Serranho. A hybrid method for sound-hard obstacle reconstruction, J. Comput. Appl. Math. 204 (2): 418--427, 2007.
[4] P. Serranho. A hybrid method for inverse scattering for Sound-soft obstacles in $\mathbb R^{3}$. Inverse Problems and Imaging. 1(4): 691--712, 2007.
[5] O. Ivanyshyn, R. Kress, P. Serranho. Huygens’ principle and iterative methods in inverse obstacle scattering. Adv. Comput. Math. 33 (4): 413--429, 2010.
[6] A. Altundag, R. Kress. An iterative method for a two-dimensional inverse scattering problem for a dielectric. J. Inverse Ill-Posed Probl. 20 (4): 575--590, 2012.
[7] A. Altundag. Inverse obstacle scattering with conductive boundary condition for a coated dielectric cylinder. J. Concr. Appl. Math. 13 ,(1--2): 11--22, 2015.
Single Mode Multi-frequency Factorization Method for the Inverse Source Problem in Acoustic Waveguides Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China, People's Republic of
This talk discusses the inverse source problem with a single propagating mode at multiple frequencies in an acoustic waveguide. The goal is to provide both theoretical justifications and efficient algorithms for imaging extended sources using the sampling methods. In contrast to the existing far/near field operator based on the integral over the space variable in the sampling methods, a multi-frequency far-field operator is introduced based on the integral over the frequency variable. This far-field operator is defined in a way to incorporate the possibly non-linear dispersion relation, a unique feature in waveguides. The factorization method is deployed to establish a rigorous characterization of the range support which is the support of source in the direction of wave propagation. A related factorization-based sampling method is also discussed. These sampling methods are shown to be capable of imaging the range support of the source. Numerical examples are provided to illustrate the performance of the sampling methods, including an example to image a complete sound-soft block.
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