A variational Lippmann-Schwinger-type approach for the Helmholtz impedance problem on bounded domains
Andreas Tataris, Alexander V. Mamonov
TL;DR
The paper addresses bounded-domain Helmholtz impedance problems where the classical Lippmann–Schwinger equation is unavailable, by deriving a variational Lippmann–Schwinger-type equation from the weak form of $(-\Delta - k^2 m)u = f$ with impedance boundary conditions. The core idea is to define $u_i=(\mathcal{S}-k^2\mathcal{M}_1-ik\mathcal{B})^{-1}f$ and $\mathcal{V}_q=(\mathcal{S}-k^2\mathcal{M}_1-ik\mathcal{B})^{-1}\mathcal{M}_q$ to obtain the LS equation $u-k^2\mathcal{V}_q u=u_i$, with $\mathcal{V}_q$ compact on $H^1(\Omega)$ and $I-k^2\mathcal{V}_q$ invertible. The authors establish weak-to-strong sequential continuity of the parameter-to-wavefield map $q\mapsto u(q)$ via collectively compact operator theory and demonstrate existence of minimizers for both data-driven ROM-based waveform inversion and conventional FWI under $L^p$ regularization on bounded domains. This provides a rigorous foundation for stable, ROM-informed inversion in bounded-domain scattering problems and paves the way for extensions to broader parameter spaces. Key formulas include $u-k^2\widetilde{\mathcal{V}_q}u=u_i$ and $\widetilde{\mathcal{V}_q}=(\mathcal{S}-k^2\mathcal{M}_1-ik\mathcal{B})^{-1}\mathcal{M}_q$, with $\mathcal{V}_q$ compact and $I-k^2\mathcal{V}_q$ invertible.
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.