A mollifier approach to regularize a Cauchy problem for the inhomogeneous Helmholtz equation

TL;DR

This paper introduces a mollifier-based regularization method for the ill-posed Cauchy problem associated with the inhomogeneous Helmholtz equation, demonstrating theoretical consistency and promising numerical results.

Contribution

It proposes a novel mollifier approach to regularize the Cauchy problem for the inhomogeneous Helmholtz equation, addressing ill-posedness with a variational mollification technique.

Findings

Method is theoretically consistent.

Numerical simulations show promising results.

Addresses ill-posedness in Helmholtz problems.

Abstract

The Cauchy problem for the inhomogeneous Helmholtz equation with non-uniform refraction index is considered. The ill-posedness of this problem is tackled by means of the variational form of mollification. This approach is proved to be consistent, and the proposed numerical simulations are quite promising.