Economic numerical method of solving coefficient inverse problem for 3D wave equation

TL;DR

This paper introduces a new numerical method for solving a 3D inverse coefficient problem for wave equations, using a linear Fredholm integral equation and fast Fourier transform, enabling efficient reconstruction of medium inhomogeneities from scattered wave data.

Contribution

A novel linear 3D Fredholm integral equation is proposed for inverse coefficient problems, along with an efficient numerical algorithm that requires minimal computational resources.

Findings

Successfully reconstructs the unknown coefficient from simulated data.

The algorithm is computationally efficient and suitable for personal computers.

Demonstrates capability to solve 3D inverse problems with limited resources.

Abstract

An inverse problem of acoustic sounding is under consideration in a form of 3D inverse coefficient problem for wave equation. Unknown coefficient is the local propagation velocity of vibrations, which is associated with inhomogeneities of the medium. We are looking for this coefficient, knowing special time integrals of the scattered wave field. In the article, a new linear 3D Fredholm integral equation of the first kind is introduced, of which it is possible to find the unknown coefficient from these time integrals. We present and substantiate a numerical algorithm for solving this integral equation. The algorithm does not require large computational resources and big-time implementation. It is based on the use of fast Fourier transform under some a priori assumptions about unknown coefficient and observation region of the scattered field. Typical results of solving this 3D inverse problem on a personal computer for simulated data demonstrate the capabilities of the proposed algorithm.