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.

翻译：近年来，降阶建模方法已被应用于求解频域散射理论中出现的反边值问题。基于投影的降阶模型方法的关键步骤是利用与正边值问题相关的半双线性形式。然而，与在$\\mathbb{R}^d$中提出的散射问题不同，边值公式会失去某些结构特性，最显著的是经典的Lippmann-Schwinger积分方程不再适用。本文推导了一个Lippmann-Schwinger型方程，旨在研究具有可变折射率和阻抗边界条件的亥姆霍兹边值问题的解。具体而言，我们从边值问题的变分公式出发，得到一个等价的算子方程，该方程可视为经典Lippmann-Schwinger方程在有界域上的类比。我们首先建立了变分型Lippmann-Schwinger算子的解析性质。基于这些结果，我们证明了参数-状态映射（将折射率映射到相应的波场）在限制于指数大于2的Lebesgue空间中的折射率时，能将弱收敛序列映射为强收敛序列。最后，我们利用导出的弱到强序列连续性，证明了基于降阶模型的优化方法（旨在求解反边值问题）以及基于常规数据失配的波形反演方法中极小值的存在性。