Inverse problems for PDEs: Models, computations and applications

Abstract

Inverse problems for partial differential equations (PDEs) are of great importance in the areas of applied mathematics, whichcover different mathematical branches including PDEs, functional analysis, nonlinear analysis,optimizations, regularization and numerical analysis. These problems have found wide applicationsin many important engineering areas such as media imaging, remote sensing and image processing.Due to the nature of ill-posedness of such kinds of problems, the techniques of regularizationshould be applied for efficiently solving these problems. However, it is very hard to establisha unified framework for inverse problems of PDEs, due to the variety and complexity of the problems.This paper aims to give an overview on several important inverse problems of PDEs models.Based on the systematic recalls on the origins and specialities of inverse problems for PDEs,we focus on three kinds of PDEs models for inverse problems: electrical impedance tomograph, inverse wave scattering, thermal imaging. The crucial problems of fundamental interests, existing resultsand methods as well as further possible research directions are reviewed. Moreover, we also give asystematic analysis of numerical methods for solving inverse problems for PDEs.