A Survey on Inverse Problems for Applied Sciences

Abstract

The aim of this paper is to introduce inversion-based engineering applications and to investigate some of the important ones from mathematical point of view. To do this we employ acoustic, electromagnetic, and elastic waves for presenting different types of inverse problems. More specifically, we first study location, shape, and boundary parameter reconstruction algorithms for the inaccessible targets in acoustics. The inverse problems for the time-dependent differential equations of isotropic and anisotropic elasticity are reviewed in the following section of the paper. These problems were the objects of the study by many authors in the last several decades. The physical interpretations for almost all of these problems are given, and the geophysical applications for some of them are described. In our last section, an introduction with many links into the literature is given for modern algorithms which combine techniques from classical inverse problems with stochastic tools into ensemble methods both for data assimilation as well as for forecasting.

An Efficient Technique for Solving Inhomogeneous Electromagnetic Inverse Scattering Problems

“…Forward problem techniques are always based on a previously selected configuration, and the effectiveness of the forward configuration strongly influences the efficiency of the inversion approach. Approaches used for modeling electromagnetic forward problems fall into two main categories: mesh-based and meshless approaches [15].…”

Approximate Controllability of Semilinear Control System Using Tikhonov Regularization

“…For more details on ill-posed problems and regularization methods one can refer to [14][15][16][17][18][19][20].…”

Point cloud‐based scatterer approximation and affine invariant sampling in the inverse scattering problem

“…Despite that there exists a wealth of theoretical and applied results for the inverse acoustic scattering problem ranging across several areas of physics and mathematics [7][8][9][10][11][12][13][14], the development of methods for the reliable solution of this classical inverse problem remains an active research area (see [15] and references therein for a review). Many applications have been found for inverse scattering problem, including echolocation, geophysical survey, nondestructive testing, and medical imaging.…”