Inverse Source Problem Research Articles

Direct and inverse scattering in an optical waveguide*

Abstract In this paper, we study the direct and inverse scattering of the Schrödinger equation in a three-dimensional optical planar waveguide. For the direct problem, we derive a resonance-free region and resolvent estimates for the resolvent of the Schrödinger operator in such a geometry. Based on the analysis of the resolvent, several inverse problems are investigated. First, given the potential function, we prove the uniqueness of the inverse source problem with multi-frequency data. We also develop a Fourier-based method to reconstruct the source function. The capability of this method is numerically illustrated by examples. Second, the uniqueness and increased stability of an inverse potential problem from data generated by incident waves are achieved in the absence of the source function. To derive the stability estimate, we use an argument of quantitative analytic continuation in complex theory. Third, we prove the uniqueness of simultaneously determining the source and potential by active boundary data generated by incident waves. In these inverse problems, we only use the limited lateral Dirichlet boundary data at multiple wavenumbers within a finite interval.

Inverse source problem for the subdiffusion equation with edge-dependent order of time-fractional derivative on the metric star graph

Inverse source problem for the subdiffusion equation with edge-dependent order of time-fractional derivative on the metric star graph

Fractional Tikhonov regularization and error estimation in inverse source problems for Biharmonic equations: a priori and a posteriori analysis under deterministic and random perturbations

Fractional Tikhonov regularization and error estimation in inverse source problems for Biharmonic equations: a priori and a posteriori analysis under deterministic and random perturbations

Analysis and computation of an inverse source problem for the biharmonic wave equation

Abstract In this paper, we study the inverse source problem for the biharmonic wave equation. Mathematically, we characterize the radiating and non-radiating sources at a fixed wavenumber. We show that a general source can be decomposed into a radiating source and a non-radiating source. The radiating source can be uniquely determined by Dirichlet boundary measurements at a fixed wavenumber. Moreover, we derive a Lipschitz stability estimate for determining the radiating source. On the other hand, the non-radiating source does not produce any scattered fields outside the support of the source function. Numerically, we propose a novel source reconstruction method based on the Fourier series expansion by multi-wavenumber boundary measurements. The stability of the proposed method is analyzed and numerical experiments are presented to verify the accuracy and efficiency.

Identification Source Term of the Distributed Order Time-Space Fractional Diffusion Equation

Abstract The inverse source problem for time-distributed order spatial fractional diffusion equations is examined in this study. The forward problem is solved by combining finite difference methods with matrix transformation techniques. The Tikhonov regularization approach is then applied to convert the inverse problem into a variational problem. The optimal perturbation approach is used to find the minimizer of the variational problem, leading to an approximation solution that depends solely on the time-dependent source term. The algorithm’s effectiveness and stability are demonstrated using numerical examples in one-dimensional domains.

Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation and its applications

Abstract In this paper, we study discrete Carleman estimates for space semi-discrete approximations of one-dimensional stochastic parabolic equation. We then apply these Carleman estimates to investigate two inverse problems for the space semi-discrete stochastic parabolic equations, including a discrete inverse random source problem and a discrete Cauchy problem. We firstly establish two Carleman estimates for a one-dimensional semi-discrete stochastic parabolic equation, one for homogeneous boundary and the other for non-homogeneous boundary. Then we apply these two estimates separately to derive two stability results. The first one is the Lipschitz stability for the discrete inverse random source problem. The second one is the H"{o}lder stability for the discrete Cauchy problem.

Fourier method for inverse source problem using correlation of passive measurements*

Abstract We consider the inverse source problem for the time-dependent, constant-coefficient wave equation with Cauchy data and passive cross-correlation data.We propose to consider the cross-correlation as a wave equation itself and reconstruct the cross-correlation in the support of the source for the original Cauchy wave equation. Having access to the cross-correlation in the support of the source, we show that the cross-correlation solves a wave equation, and we reconstruct the cross-correlation from boundary data to recover the source in the original Cauchy wave equation. In addition, we show the inverse source problem is ill-posed and suffers from non-uniqueness when the mean of the source is zero and provide a uniqueness result and stability estimate in case of non-zero mean sources.

Inverse Source Problem of the Biharmonic Equation from Multifrequency Phaseless Data

Inverse Source Problem of the Biharmonic Equation from Multifrequency Phaseless Data

Inverse problem of reconstructing source term for a class of non‐divergence parabolic equations

This paper explores an inverse problem pertaining to the determination of a source function in non‐divergence parabolic equations, where the solution is known at a discrete set of points. Being different from other ordinary inverse source problems, which are often dependent on only one variable, the unknown coefficient in this paper not only depends on the space variable but also depends on the time . On the basis of the optimal control framework, the existence of the optimal solution of the control function is proved. The necessary conditions to be satisfied by the optimal solution are given. The convergence of the optimal solution when the mesh parameters tend to zero is obtained. The conjugate gradient method is applied to the inverse problem and some numerical results are presented for various typical test examples.

Inverse source problem for a one-dimensional degenerate hyperbolic problem

Inverse source problem for a one-dimensional degenerate hyperbolic problem

Constrained Pseudoinverses for the Electromagnetic Inverse Source Problem

Constrained Pseudoinverses for the Electromagnetic Inverse Source Problem

Inverse source problem for discrete Helmholtz equation

Abstract We consider multi-frequency inverse source problem for the discrete Helmholtz operator on the square lattice Z d , d ⩾ 1 . We consider this problem for the cases with and without phase information. We prove uniqueness results and present examples of non-uniqueness for this problem for the case of compactly supported source function, and a Lipshitz stability estimate for the phased case is established. Relations with inverse scattering problem for the discrete Schrödinger operators in the Born approximation are also provided.

Increasing stability of the acoustic and elastic inverse source problems in multi-layered media

Abstract This paper investigates inverse source problems for the Helmholtz and Navier equations in multi-layered media, considering both two and three-dimensional cases respectively. The study reveals a consistent increase in stability for each scenario, characterized by two main terms: a Hölder-type term associated with data discrepancy, and a logarithmic-type term that diminishes as more frequencies are considered. In the two-dimensional case, measurements on interfaces and far-field data are essential. By employing the fundamental solution in free-space as the test function and utilizing the asymptotic behavior of the solution and continuation principle, stability results are obtained. In the three-dimensional case, measurements on interfaces and artificial boundaries are taken, and the stability result can be derived by applying the arguments for inverse source problems in homogeneous media.

A strong positivity property and a related inverse source problem for multi-term time-fractional diffusion equations

A strong positivity property and a related inverse source problem for multi-term time-fractional diffusion equations

Regularization of an inverse source problem for fractional diffusion-wave equations under a general noise assumption

Regularization of an inverse source problem for fractional diffusion-wave equations under a general noise assumption

Shape and parameter identification by the linear sampling method for a restricted Fourier integral operator

In this paper we provide a new linear sampling method based on the same data but a different definition of the data operator for two inverse problems: the multi-frequency inverse source problem for a fixed observation direction and the Born inverse scattering problems. We show that the associated regularized linear sampling indicator converges to the average of the unknown in a small neighborhood as the regularization parameter approaches to zero. We develop both a shape identification theory and a parameter identification theory which are stimulated, analyzed, and implemented with the help of the prolate spheroidal wave functions and their generalizations. We further propose a prolate-based implementation of the linear sampling method and provide numerical experiments to demonstrate how this linear sampling method is capable of reconstructing both the shape and the parameter.

Sine Transform Based Preconditioning for an Inverse Source Problem of Time-Space Fractional Diffusion Equations

Sine Transform Based Preconditioning for an Inverse Source Problem of Time-Space Fractional Diffusion Equations

Nonlocal active sound control for composite regions

In the active sound control problem, additional sound sources, known as control sources, are implemented at the perimeter of a protected domain. The control sources can generate secondary sound inside the shielded domain, which attenuates incoming noise. This leads to an inverse source problem. Such a problem becomes much more complicated if some desired sound is generated inside the protected area. The algorithm, known as nonlocal active sound control, successfully tackles this problem due to the projection property of the nonlocal operators involved. With the application of nonlocal control, noise generated outside the shielded domain is attenuated while the internal component of sound remains unaffected. In the present paper, the approach of nonlocal active sound control is significantly modified to be applicable to composite domains. In the test cases, three different sound propagation patterns are realized such as sound shielded from peers, free propagation, and selective sound cancelation. Numerical simulation is carried out in both frequency and time domains for broadband regimes. The results show that, even with a relatively small number of sensors and control sources, significant noise attenuation can still be achieved for a predetermined communication pattern among individual subdomains.

A Learned-SVD Approach to the Electromagnetic Inverse Source Problem.

We propose an artificial intelligence approach based on deep neural networks to tackle a canonical 2D scalar inverse source problem. The learned singular value decomposition (L-SVD) based on hybrid autoencoding is considered. We compare the reconstruction performance of L-SVD to the Truncated SVD (TSVD) regularized inversion, which is a canonical regularization scheme, to solve an ill-posed linear inverse problem. Numerical tests referring to far-field acquisitions show that L-SVD provides, with proper training on a well-organized dataset, superior performance in terms of reconstruction errors as compared to TSVD, allowing for the retrieval of faster spatial variations of the source. Indeed, L-SVD accommodates a priori information on the set of relevant unknown current distributions. Different from TSVD, which performs linear processing on a linear problem, L-SVD operates non-linearly on the data. A numerical analysis also underlines how the performance of the L-SVD degrades when the unknown source does not match the training dataset.

Increasing stability for inverse source problem with limited-aperture far field data at multi-frequencies

We study the increasing stability of an inverse source problem for the Helmholtz equation from limited-aperture far field data at multiple wave numbers. The measurement data are given by the far field patterns u∞(xˆ,k) for all observation directions xˆ in some neighborhood of a fixed direction xˆ0 and for all wave numbers k belonging to a finite interval (0,K). In this paper, we discuss the increasing stability with respect to the width of the wavenumber interval K>1. In three dimensions we establish stability estimates of the L2-norm and H−1-norm of the source function from the far field data. The ill-posedness of the inverse source problem turns out to be of Hölder type while increasing the wavenumber band K. We also discuss an analytic continuation argument of the far-field data with respect to the wavenumbers at a fixed direction.