Reconstructing a space-dependent source term via the quasi-reversibility method

TL;DR

This paper introduces a method to reconstruct space-dependent source terms in inverse scattering problems by truncating Fourier series, leading to a system of linear elliptic equations for efficient computation.

Contribution

It proposes a novel Fourier series truncation approach combined with the quasi-reversibility method to solve inverse source problems more effectively.

Findings

Successfully reconstructs source functions from numerical examples

Transforms the inverse problem into a system of linear elliptic equations

Demonstrates the method's accuracy and stability

Abstract

The aim of this paper is to solve an important inverse source problem which arises from the well-known inverse scattering problem. We propose to truncate the Fourier series of the solution to the governing equation with respect to a special basis of L2. By this, we obtain a system of linear elliptic equations. Solutions to this system are the Fourier coefficients of the solution to the governing equation. After computing these Fourier coefficients, we can directly find the desired source function. Numerical examples are presented.