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A simple inverse problem for the wave equation requires determination of both the wave velocity in a homogenous acoustic material and the transient waveform of an isotropic point radiator, given the time history of the wavefield at a remote…

Optimization and Control · Mathematics 2022-09-28 William W. Symes

The paper studies inverse problems of determining unknown coefficients in various semi-linear and quasi-linear wave equations. We introduce a method to solve inverse problems for non-linear equations using interaction of three waves, that…

Analysis of PDEs · Mathematics 2023-05-10 Ali Feizmohammadi , Matti Lassas , Lauri Oksanen

Consider the two-dimensional inverse elastic wave scattering by an infinite rough surface with a Dirichlet boundary condition. A non-interative sampling technique is proposed for detecting the rough surface by taking elastic wave…

Analysis of PDEs · Mathematics 2020-12-15 Tielei Zhu , Jiaqing Yang

To address the ill-posedness of the inverse source problem for the one-dimensional stochastic Helmholtz equations without attenuation, this study develops a novel computational framework designed to mitigate this inherent challenge at the…

Numerical Analysis · Mathematics 2025-07-11 Yunqing Huang , Shihan Zhang

This paper is concerned with uniqueness, phase retrieval and shape reconstruction methods for inverse elastic scattering problems with phaseless far field data. Systematically, we study two basic models, i.e., inverse scattering of plane…

Analysis of PDEs · Mathematics 2020-01-08 Xia Ji , Xiaodong Liu

In this paper, we deal with the inverse source problem of determining a source in a time fractional diffusion equation where data are given at a fixed time. This problem is ill-posed, i.e., the solution does not depend continuously on the…

Probability · Mathematics 2016-11-29 Tuan Nguyen Huy , Erkan Nane

This work is dedicated to a novel sampling method for accurately reconstructing elastic and electromagnetic sources from the far field patterns. We show that the proposed indicators in the form of integrals with full far field patterns are…

Numerical Analysis · Mathematics 2025-03-11 Xiaodong Liu , Qingxiang Shi

Study of a simple single-trace transmission example shows how an extended source formulation of full-waveform inversion can produce an optimization problem without spurious local minima ("cycle skipping"), hence efficiently solvable via…

Optimization and Control · Mathematics 2022-09-28 William W. Symes , Huiyi Chen , Susan E. Minkoff

This paper is concerned with the inverse time-harmonic elastic scattering problem of recovering unbounded rough surfaces in two dimensions. We assume that elastic plane waves with different directions are incident onto a rigid rough surface…

Numerical Analysis · Mathematics 2018-04-09 Guanghui Hu , Xiaoli Liu , Bo Zhang , Haiwen Zhang

We consider an inverse problem for the elastic wave of simultaneously reconstructing the impedance and the geometric information of the bounded body that is occupied by a homogeneous and isotropic elastic medium from the measured Cauchy…

Numerical Analysis · Mathematics 2025-06-26 Yao Sun , Yan Chang , Yukun Guo

We are concerned with the reconstruction of inclusions in elastic bodies based on measurements from a laboratory experiment. In doing so, we solve the inverse problem of the time-harmonic elastic wave equation, in contrast to the stationary…

Analysis of PDEs · Mathematics 2026-05-21 Sarah Eberle-Blick , Jochen Moll

We consider the inverse elastic scattering problems using the far field data due to one incident plane wave. A simple method is proposed to reconstruct the location and size of the obstacle using different components of the far field…

Analysis of PDEs · Mathematics 2019-04-09 J Liu , X. Liu , J. Sun

This paper is concerned with the inverse elastic scattering problem for a random potential in three dimensions. Interpreted as a distribution, the potential is assumed to be a microlocally isotropic Gaussian random field whose covariance…

Analysis of PDEs · Mathematics 2021-02-16 Jianliang Li , Peijun Li , Xu Wang

We consider an inverse boundary value problem for a nonlinear elastic wave equation which was studied in [de Hoop, Uhlmann, Wang. Math. Ann. (2019) doi:10.1007/s00208-018-01796-y]. We show that all the parameters appearing in the equation…

Analysis of PDEs · Mathematics 2021-01-15 Gunther Uhlmann , Jian Zhai

Stochastic inverse problems considered in this article consist of estimating the probability distributions of intrinsically random inputs of computer models. These estimations are based on observable outputs affected by model noise, and…

Statistics Theory · Mathematics 2025-03-17 Nicolas Bousquet , Mélanie Blazère , Thomas Cerbelaud

This paper presents a fast and robust numerical method for reconstructing point-like sources in the time-harmonic Maxwell's equations given Cauchy data at a fixed frequency. This is an electromagnetic inverse source problem with broad…

Numerical Analysis · Mathematics 2024-12-17 Isaac Harris , Thu Le , Dinh-Liem Nguyen

In this paper, we consider the problem of identifying a single moving point source for a three-dimensional wave equation from boundary measurements. Precisely, we show that the knowledge of the field generated by the source at six different…

Analysis of PDEs · Mathematics 2022-11-09 Hanin Al Jebawy , Abdellatif El Badia , Faouzi Triki

In this paper, we study both the direct and inverse random source problems associated with the multi-term time-fractional diffusion-wave equation driven by a fractional Brownian motion. Regarding the direct problem, the well-posedness is…

Analysis of PDEs · Mathematics 2023-11-03 Xiaoli Feng , Qiang Yao , Peijun Li , Xu Wang

We study the inverse source problem for a class of viscoelastic systems from a single boundary measurement in a general spatial dimension. We give specific reconstruction formula and stability estimate for the source in terms of the…

Analysis of PDEs · Mathematics 2020-04-24 Walton Green , Shitao Liu

A novel method to solve inverse problems for the wave equation is introduced. The method is a combination of the boundary control method and an iterative time reversal scheme, leading to adaptive imaging of coefficient functions of the wave…

Analysis of PDEs · Mathematics 2007-05-23 Kenrick Bingham , Yaroslav Kurylev , Matti Lassas , Samuli Siltanen