A variational Lippmann-Schwinger-type approach for the Helmholtz impedance problem on bounded domains

TL;DR

This paper introduces a variational Lippmann-Schwinger-type equation for Helmholtz boundary value problems with impedance conditions, enabling reduced order modeling and inverse problem solutions on bounded domains.

Contribution

It derives a new operator equation analogous to the classical Lippmann-Schwinger equation for bounded domains, facilitating analysis and solution of inverse Helmholtz problems.

Findings

Established analytical properties of the variational Lippmann-Schwinger operator.

Proved the parameter-to-state map is weakly to strongly continuous in certain function spaces.

Demonstrated existence of minimizers for inverse problem optimization methods.

Abstract

Recently, reduced order modeling methods have been applied to solving inverse boundary value problems arising in frequency domain scattering theory. A key step in projection-based reduced order model methods is the use of a sesquilinear form associated with the forward boundary value problem. However, in contrast to scattering problems posed in $\mathbb{R}^d$, boundary value formulations lose certain structural properties, most notably the classical Lippmann-Schwinger integral equation is no longer available. In this paper we derive a Lippmann-Schwinger type equation aimed at studying the solution of a Helmholtz boundary value problem with a variable refractive index and impedance boundary conditions. In particular, we start from the variational formulation of the boundary value problem and we obtain an equivalent operator equation which can be viewed as a bounded domain analogue of the classical Lippmann-Schwinger equation. We first establish analytical properties of our variational Lippmann-Schwinger type operator. Based on these results, we then show that the parameter-to-state map, which maps a refractive index to the corresponding wavefield, maps weakly convergent sequences to strongly convergent ones when restricted to refractive indices in Lebesgue spaces with exponent greater than 2. Finally, we use the derived weak to strong sequential continuity to show existence of minimizers for a reduced order model based optimization methods aimed at solving the inverse boundary value problem as well as for a conventional data misfit based waveform inversion method.