Determination of forcing functions in the wave equation. Part I: the space-dependent case

TL;DR

This paper investigates an inverse problem for the wave equation to determine unknown space-dependent forces from boundary data, employing regularization techniques to ensure stable solutions, with numerical validation demonstrating effectiveness.

Contribution

It introduces a regularization-based numerical method for identifying space-dependent forces in the wave equation from boundary measurements, addressing ill-posedness.

Findings

Stable solutions with exact data

Robustness to noisy data

Effective regularization parameter selection

Abstract

We consider the inverse problem for the wave equation which consists of determining an unknown space-dependent force function acting on a vibrating structure from Cauchy boundary data. Since only boundary data are used as measurements, the study has importance and significance to non-intrusive and non-destructive testing of materials. This inverse force problem is linear, the solution is unique, but the problem is still ill-posed since, in general, the solution does not exist and, even if it exists, it does not depend continuously upon the input data. Numerically, the finite difference method combined with the Tikhonov regularization are employed in order to obtain a stable solution. Several orders of regularization are investigated. The choice of the regularization parameter is based on the L-curve method. Numerical results show that the solution is accurate for exact data and stable for noisy data. An extension to the case of multiple additive forces is also addressed. In a companion paper, in Part II, the time-dependent force identification will be undertaken.