Numerical solutions to an inverse problem for a non-linear Helmholtz equation

TL;DR

This paper develops numerical methods to solve an inverse problem for a nonlinear Helmholtz equation in a spherical shell, aiming to recover the nonlinear term from known solution data at multiple points.

Contribution

It introduces a novel numerical approach to determine the nonlinear term's Chebyshev coefficients from solution data in a spherical geometry.

Findings

Successful reconstruction of the nonlinear term from data

Demonstration of the method's accuracy and stability

Application to spherical shell problems

Abstract

In this work, we construct numerical solutions to an inverse problem of a nonlinear Helmholtz equation defined in a spherical shell between two concentric spheres centered at the origin.Assuming that the values of the forward problem are known at sufficiently many points, we would like to determine the form of the non-linear term on the right-hand side of the equation via its Chebyshev coefficients.