Reconstruction of the electric field of the Helmholtz equation in 3D

TL;DR

This paper rigorously analyzes a truncation regularization method for solving the 3D Helmholtz equation's inverse problem, demonstrating stability, convergence, and providing error estimates with practical examples.

Contribution

It offers a detailed theoretical framework for the regularization of the Helmholtz equation's inverse problem, including stability, convergence, and parameter selection.

Findings

Proved stability and convergence of the regularized solutions.

Derived error estimates in the L^2 norm.

Provided illustrative examples confirming the theoretical results.

Abstract

In this paper, we rigorously investigate the truncation method for the Cauchy problem of Helmholtz equations which is widely used to model propagation phenomena in physical applications. The method is a well-known approach to the regularization of several types of ill-posed problems, including the model postulated by Regi\' nska and Regi\' nski \cite{RR06}. Under certain specific assumptions, we examine the ill-posedness of the non-homogeneous problem by exploring the representation of solutions based on Fourier mode. Then the so-called regularized solution is established with respect to a frequency bounded by an appropriate regularization parameter. Furthermore, we provide a short analysis of the nonlinear forcing term. The main results show the stability as well as the strong convergence confirmed by the error estimates in $L^2$-norm of such regularized solutions. Besides, the regularization parameters are formulated properly. Finally, some illustrative examples are provided to corroborate our qualitative analysis.