Domain Uncertainty Quantification for the Lippmann-Schwinger Volume Integral Equation

TL;DR

This paper investigates how uncertainties in the shape of inhomogeneities affect acoustic wave propagation modeled by the Lippmann-Schwinger volume integral equation, emphasizing shape holomorphy for uncertainty quantification.

Contribution

It demonstrates that the Lippmann-Schwinger operator and its solution depend holomorphically on shape variations, enabling advanced uncertainty quantification methods.

Findings

Shape holomorphy of the operator and solution established

Numerical experiments confirm theoretical predictions

Implications for forward and inverse UQ analyzed

Abstract

In this work, we consider the propagation of acoustic waves in unbounded domains characterized by a constant wavenumber, except possibly in a bounded region. The geometry of this inhomogeneity is assumed to be uncertain, and we are particularly interested in studying the propagation of this behavior throughout the physical model considered. A key step in our analysis consists of recasting the physical model-originally set in an unbounded domain-into a computationally manageable formulation based on Volume Integral Equations (VIEs), particularly the Lippmann-Schwinger equation. We show that both the leading operator in this volume integral formulation and its solution depend holomorphically on shape variations of the support of the aforementioned inhomogeneity. This property, known as shape holomorphy, is crucial in the analysis and implementation of various methods used in computational Uncertainty Quantification (UQ). We explore the implications of this result in forward and inverse UQ and provide numerical experiments illustrating and confirming the theoretical predictions.