Inverse medium problems involve the reconstruction of a spatially varying unknown medium from available observations by exploring a restricted search space of possible solutions. Standard grid-based representations are very general but all too often computationally prohibitive due to the high dimension of the search space. Adaptive spectral (AS) decompositions instead expand the unknown medium in a basis of eigenfunctions of a judicious elliptic operator, which depends itself on the medium. Here the AS decomposition is combined with a standard inexact Newton-type method for the solution of time-harmonic scattering problems governed by the Helmholtz equation. By repeatedly adapting both the eigenfunction basis and its dimension, the resulting adaptive spectral inversion (ASI) method substantially reduces the dimension of the search space during the nonlinear optimization. Rigorous estimates of the AS decomposition are proved for a general piecewise constant medium. Numerical results illustrate the accuracy and efficiency of the ASI method for time-harmonic inverse scattering problems, including a salt dome model from geophysics.
A Proximal Decomposition Method for Solving Convex Variational Inverse Problems
UPMC Université Paris 06（巴黎第六大学）；Laboratoire Jacques-Louis Lions – UMR 7598（雅克-路易·利翁斯实验室）；Université Paris-Est（巴黎东大学）；Institut Gaspard Monge and UMR CNRS 8049（加斯帕尔·蒙日研究所）
We propose domain decomposition preconditioners for the solution of an integral equation formulation of forward and inverse acoustic scattering problems with point scatterers. We study both forward and inverse problems and propose preconditioning techniques to accelerate the iterative solvers. For the forward scattering problem, we extend the domain decomposition based preconditioning techniques presented for partial differential equations in {\em "A restricted additive Schwarz preconditioner for general sparse linear systems", SIAM Journal on Scientific Computing, 21 (1999), pp. 792--797}, to…
A primary interest in dynamic inverse problems is to identify the underlying temporal behaviour of the system from outside measurements. In this work we consider the case, where the target can be represented by a decomposition of spatial and temporal basis functions and hence can be efficiently represented by a low-rank decomposition. We then propose a joint reconstruction and low-rank decomposition method based on the Nonnegative Matrix Factorisation to obtain the unknown from highly undersampled dynamic measurement data. The proposed framework allows for flexible incorporation of separate re…
The characteristic feature of inverse problems is their instability with respect to data perturbations. In order to stabilize the inversion process, regularization methods have to be developed and applied. In this work we introduce and analyze the concept of filtered diagonal frame decomposition which extends the standard filtered singular value decomposition to the frame case. Frames as generalized singular system allows to better adapt to a given class of potential solutions. In this paper, we show that filtered diagonal frame decomposition yield a convergent regularization method. Moreover,…
We propose a stochastic gradient descent approach with partitioned-truncated singular value decomposition for large-scale inverse problems of magnetic modulus data. Motivated by a uniqueness theorem in gravity inverse problem and realizing the similarity between gravity and magnetic inverse problems, we propose to solve the level-set function modeling the volume susceptibility distribution from the nonlinear magnetic modulus data. To deal with large-scale data, we employ a mini-batch stochastic gradient descent approach with random reshuffling when solving the optimization problem of the inver…
Two-level space-time domain decomposition methods for unsteady inverse problems
Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences（中国科学院深圳先进技术研究院）；University of Colorado Boulder（科罗拉多大学博尔德分校）；The Chinese University of Hong Kong（香港中文大学）
In this work, we are interested in the determination of the shape of the scatterer for the two dimensional time harmonic inverse medium scattering problems in acoustics. The scatterer is assumed to be a piecewise constant function with a known value inside inhomogeneities, and its shape is represented by the level set functions for which we investigate the information using the Bayesian method. In the Bayesian framework, the solution of the geometric inverse problem is defined as a posterior probability distribution. The well-posedness of the posterior distribution would be discussed, and the …
We present a two-stage least-squares method to inverse medium problems of reconstructing multiple unknown coefficients simultaneously from noisy data. A direct sampling method is applied to detect the location of the inhomogeneity in the first stage, while a total least-squares method with mixed regularization is used to recover the medium profile in the second stage. The total least-squares method is designed to minimize the residual of the model equation and the data fitting, along with an appropriate regularization, in an attempt to significantly improve the accuracy of the approximation ob…
Acceleration techniques for regularized Newton methods applied to electromagnetic inverse medium scattering problems
Institut für Numerische und Angewandte Mathematik, Universität Göttingen（哥廷根大学数值与应用数学研究所）；Deutsches Zentrum für Luft- und Raumfahrt e. V. in der Helmholtz-Gemeinschaft, Institut für Aerodynamik und Strömungstechnik（德国航空航天中心（戴姆勒-克莱斯勒-阿利安特-马瑟里基金会），空气动力学与流动技术研究所）
We develop hybrid projection methods for computing solutions to large-scale inverse problems, where the solution represents a sum of different stochastic components. Such scenarios arise in many imaging applications (e.g., anomaly detection in atmospheric emissions tomography) where the reconstructed solution can be represented as a combination of two or more components and each component contains different smoothness or stochastic properties. In a deterministic inversion or inverse modeling framework, these assumptions correspond to different regularization terms for each solution in the sum.…
A theoretical and computational framework for three dimensional inverse medium scattering using the linearized low-rank structure
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；PRISME, EDF R&D（EDF研发部PRISME）；UMA, Inria, ENSTA Paris, Institut Polytechnique de Paris（巴黎理工学院ENSTA巴黎大学应用数学实验室，法国国家数字科学研究所）；Department of Mathematical Sciences, The University of Texas at Dallas（德克萨斯大学达拉斯分校数学系）
This paper is concerned with the inverse medium problem of determining the location and shape of penetrable scattering objects from measurements of the scattered field. We study a sampling indicator function for recovering the scattering object in a fast and robust way. A flexibility of this indicator function is that it is applicable to data measured in near-field regime or far-field regime. The implementation of the function is simple and does not involve solving any ill-posed problems. The resolution analysis and stability estimate of the indicator function are investigated using the factor…
This paper deals with the reconstruction of small-amplitude perturbations in the electric properties (permittivity and conductivity) of a medium from boundary measurements of the electric field at a fixed frequency. The underlying model is the three-dimensional time-harmonic Maxwell equations in the electric field. Sensitivity analysis with respect to the parameters is performed, and explicit relations between the boundary measurements and the characteristics of the perturbations are found from an appropriate integral equation and extensive numerical simulations in 3D. The resulting non-iterat…
Globally strictly convex cost functional for a 1-D inverse medium scattering problem with experimental data
University of North Carolina at Charlotte（北卡罗来纳大学夏洛特分校）；Institute of Mathematics and Information Science, North-Eastern Federal University（东北联邦大学数学与信息科学研究所）
We develop innovative algorithms for solving the strong-constraint formulation of four-dimensional variational data assimilation in large-scale applications. We present a space-time decomposition approach that employs domain decomposition along both the spatial and temporal directions in the overlapping case and involves partitioning of both the solution and the operators. Starting from the global functional defined on the entire domain, we obtain a type of regularized local functionals on the set of subdomains providing the order reduction of both the predictive and the data assimilation mode…
We study the inverse medium scattering problem to reconstruct the unknown inhomogeneous medium from the far-field patterns of scattered waves. The inverse scattering problem is generally ill-posed and nonlinear, and the iterative optimization method is often adapted. A natural iterative approach to this problem is to place all available measurements and mappings into one long vector and mapping, respectively, and to iteratively solve the linearized large system equation using the Tikhonov regularization method, which is called the Levenberg-Marquardt scheme. However, this is computationally ex…
In this paper, we introduce the tensor similar transforation and propose the T-Jordan canonical form and its properties. The concept of T-minimal polynomial and T-characteristic polynomial are raised. As a special case, we present properties when two tensors commutes via the tensor T-product. The Cayley-Hamilton theorem also holds for tensor cases. Then we focus on the tensor decomposition theory. T-polar, T-LU, T-QR and T-Schur decomposition of tensors are obtained. When a F-square tensor is not invertible via the T-product, we give the T-group inverse and T-Drazin inverse which can be viewed…
Convergent and efficient iteratively regularized contrast source inversion-type methods for inverse medium scattering problems
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
Multi-frequency iterative methods for the inverse medium scattering problems in elasticity
School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）；College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）；Department of Computing & Mathematical Sciences, California Institute of Technology（加州理工学院计算与数学系）
Recursive linearization method for inverse medium scattering problems with complex mixture Gaussian error learning
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；School of Electronic and Information Engineering, Xi’an Jiaotong University（西安交通大学电子与信息工程学院）
In this paper, we propose a direct probing method for the inverse problem based on the Eikonal equation. For the Eikonal equation with a point source, the viscosity solution represents the least travel time of wave fields from the source to the point at the high-frequency limit. The corresponding inverse problem is to determine the inhomogeneous wave-speed distribution from the first-arrival time data at the measurement surfaces corresponding to distributed point sources, which is called transmission travel-time tomography. At the low-frequency regime, the reconstruction approximates the frequ…
We describe an algorithm for the application of the forward and inverse spherical harmonic transforms. It is based on a new method for rapidly computing the forward and inverse associated Legendre transforms by hierarchically applying the interpolative decomposition butterfly factorization (IDBF). Experimental evidence suggests that the total running time of our method -- including all necessary precomputations -- is $\mathcal{O}(N^2 \log^3(N))$, where $N$ is the order of the transform. This is nearly asymptotically optimal. Moreover, unlike existing algorithms which are asymptotically optimal…
