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Domain Uncertainty Quantification for the Lippmann-Schwinger Volume Integral Equation
Numerical Analysis · Submitted 16 Jul 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | In this work, we consider the propagation of acoustic waves in unbounded domains characterized by a constant wavenumber, except possibly in a bounded region. | background 0.98 | 0.14 | 0.02 | asserted 0.82 |
| s1 | 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. | background 0.81 | 0.16 | 0.03 | asserted 0.96 |
| s2 | 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. | method 0.94 | 0.82 | 0.05 | asserted 0.97 |
| s3 | 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. | result 0.98 | 0.93 | 0.61 | asserted 1.00 |
| s4 | This property, known as shape holomorphy, is crucial in the analysis and implementation of various methods used in computational Uncertainty Quantification (UQ). | interpretation 0.95 | 0.21 | 0.13 | asserted 1.00 |
| s5 | We explore the implications of this result in forward and inverse UQ and provide numerical experiments illustrating and confirming the theoretical predictions. | contribution 0.96 | 0.94 | 0.23 | asserted 0.94 |
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The paper
Original 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.
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