Domain Uncertainty Quantification for the Lippmann–Schwinger Volume Integral Equation

Abstract. In this work, we consider the propagation of acoustic waves in unbounded domains characterized by a constant wavenumber, except possibly within a bounded region. The geometry of this inhomogeneity is assumed to be uncertain, and we are particularly interested in studying how this uncertainty propagates throughout the physical model under consideration. A key step in our analysis involves recasting the physical model—originally defined in an unbounded domain—into a computationally tractable formulation based on volume integral equations, specifically 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 inhomogeneity. This property, known as shape holomorphy, is crucial for the analysis and implementation of various methods used in computational uncertainty quantification (UQ). We explore the implications of this result for both forward and inverse UQ and provide numerical experiments that illustrate and confirm the theoretical predictions.