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Efficient Numerical Reconstruction of Wave Equation Sources via Droplet-Induced Asymptotics

Shutong Hou, Mourad Sini, Haibing Wang

TL;DR

The paper tackles the ill-posed 3D inverse source problem for the acoustic wave equation by introducing a high-contrast, small droplet to perturb the medium and extract information from measurements at a single exterior point. It develops a theoretical droplet-induced asymptotic expansion of the perturbed field in terms of the Newtonian operator's eigenpairs, and formulates a practical inversion using a Riesz-basis expansion with spectral truncation. A forward Lippmann-Schwinger solver based on a convolution-spline time discretization and weakly singular volume integrals enables efficient data simulation, while a mollification-based differentiation pipeline reconstructs the source from the recovered field. Numerical experiments in three dimensions demonstrate accurate source recovery under noise and provide explicit error and truncation guidelines tied to the droplet size $a$ and spectral truncation $N$, with single-point measurements offering a practical advantage for imaging and non-destructive testing.

Abstract

In this paper, we develop and numerically implement a novel approach for solving the inverse source problem of the acoustic wave equation in three dimensions. By injecting a small high-contrast droplet into the medium, we exploit the resulting wave field perturbation measured at a single external point over time. The method enables stable source reconstructions where conventional approaches fail due to ill-posedness, with potential applications in medical imaging and non-destructive testing. Key contributions include: 1. Implementation of a theoretically justified asymptotic expansion, from [33], using the eigensystem of the Newtonian operator, with error analysis for the spectral truncation. 2. Novel numerical schemes for solving the time-domain Lippmann-Schwinger equation and reconstructing the source via Riesz basis expansions and mollification-based numerical differentiations. 3. Reconstruction requiring only single-point measurements, overcoming traditional spatial data limitations. 4. 3D numerical experiments demonstrating accurate source recovery under noise (SNR of the order $1/a$), with error analysis for the droplet size (of the order $a$) and the number of spectral modes $N$.

Efficient Numerical Reconstruction of Wave Equation Sources via Droplet-Induced Asymptotics

TL;DR

The paper tackles the ill-posed 3D inverse source problem for the acoustic wave equation by introducing a high-contrast, small droplet to perturb the medium and extract information from measurements at a single exterior point. It develops a theoretical droplet-induced asymptotic expansion of the perturbed field in terms of the Newtonian operator's eigenpairs, and formulates a practical inversion using a Riesz-basis expansion with spectral truncation. A forward Lippmann-Schwinger solver based on a convolution-spline time discretization and weakly singular volume integrals enables efficient data simulation, while a mollification-based differentiation pipeline reconstructs the source from the recovered field. Numerical experiments in three dimensions demonstrate accurate source recovery under noise and provide explicit error and truncation guidelines tied to the droplet size and spectral truncation , with single-point measurements offering a practical advantage for imaging and non-destructive testing.

Abstract

In this paper, we develop and numerically implement a novel approach for solving the inverse source problem of the acoustic wave equation in three dimensions. By injecting a small high-contrast droplet into the medium, we exploit the resulting wave field perturbation measured at a single external point over time. The method enables stable source reconstructions where conventional approaches fail due to ill-posedness, with potential applications in medical imaging and non-destructive testing. Key contributions include: 1. Implementation of a theoretically justified asymptotic expansion, from [33], using the eigensystem of the Newtonian operator, with error analysis for the spectral truncation. 2. Novel numerical schemes for solving the time-domain Lippmann-Schwinger equation and reconstructing the source via Riesz basis expansions and mollification-based numerical differentiations. 3. Reconstruction requiring only single-point measurements, overcoming traditional spatial data limitations. 4. 3D numerical experiments demonstrating accurate source recovery under noise (SNR of the order ), with error analysis for the droplet size (of the order ) and the number of spectral modes .
Paper Structure (12 sections, 7 theorems, 92 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 12 sections, 7 theorems, 92 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theoretical results
  3. The droplet-induced asymptotics of the field
  4. Inversion formulas using the droplet-induced asymptotics
  5. Stability estimates of the inversion formulas
  6. Numerical algorithms
  7. An efficient algorithm for solving the Lippmann-Schwinger equation
  8. An efficient algorithm for reconstructing the source
  9. Numerical experiments
  10. Asymptotic expansion of the wave field
  11. Reconstruction of the source
  12. Conclusions

Key Result

Theorem 2.1

Assume that $J$ is compactly supported in $\Omega \times [0,\,T]$ and $J \in H^6_{0,\,\sigma}(0,\,T;\,L^2(\Omega))$. Then we have the expansion where $K$ is any bounded domain in $\mathbb{R}^3$ such that $\overline{D} \subset\subset K$. The expansion holds pointwise in both time and space. Here $\{ \lambda_n, \, e_n \}_{n\in \mathbb{N}^+}$ is the family of the eigenelements of the Newtonian opera

Figures (5)

  • Figure 4.1: The decay behavior of $(n-1/2)^{2} (\int_{B(0,1)} e_n(y) dy)^2$ with respect to $n$.
  • Figure 4.2: The wave field $W(x,\,t)$ evaluated at $x=(0.3,\,0.4,\,0.5)$, for sources with different temporal regularities as described in Example \ref{['exa:temporal regularity on W']}. The left and right panels display the results for $p=3$ and $p=4$ in \ref{['eq:Vxt_example2']}, respectively. The green solid line and the red dashed line represent the simulation results of $W_{\mathrm{LSE}}$ and $W_N$, respectively. The yellow dashed line indicates $t=c_0^{-1}|x-z|$, i.e., the first arrival time of the wave emitted from the source point $z$ to the point $x$.
  • Figure 4.3: The numerical results of $W(x,\,t)$ at $x=(0.3,\,0.4,\,0.5)$ for Example \ref{['exa:temporal regularity on W']} with $p=4$. In the left panel, the green line represents the simulation result of $W_{\rm LSE}$. Meanwhile, the black, blue and red dashed lines represent the simulation results of $W_N$ for $N=2,\,4,\,8$, respectively. The yellow dashed line indicates $t=c_0^{-1}|x-z|$. The middle panel shows the differences $W_{\mathrm{LSE}} -W_N$ for $N=2,\,4,\,8$, respectively. And the right panel presents the wave field $W_N(x,\,t)$ at $x=(0.3,\,0.4,\,0.5),\, t=3.5$ for $N=1,\,2,\,\ldots,\,8$. The top and bottom panels present the results for $a=0.05$ and $a=1.0\mathrm{E}-3$, respectively.
  • Figure 4.4: Comparison between the reconstructions and the exact ones at $t=0.8$ for Example \ref{['exa:rec-J']} with $a=\overline{\delta}=1.0\mathrm{E}-3$. The top, medium and bottom figures show the results of $V,\,V_{tt}$ and $J$, respectively. The left figures display the reconstructions and the right figures show the exact ones. In the top panel, $V$ is evaluated at $2601$ points on the slice $\{x=0\}$. On the snapshot of $V$, the reconstruction is performed in $(-0.25,\,0.25) \times (-0.25,\,0.25)$. In the medium and bottom panels, $V_{tt}$ and $J$ are evaluated at $729$ points on the slice $\{x=0\}$. On the snapshots of $V_{tt}$ and $J$, the reconstructions are performed in $(-0.13,\,0.13) \times (-0.13,\,0.13)$.
  • Figure 4.5: Comparison between the reconstructions and the exact ones at $t=0.8$ for Example \ref{['exa:rec-J']} with $a=\overline{\delta}=1.0\mathrm{E}-4$. The top, medium and bottom figures show the results of $V,\,V_{tt}$ and $J$, respectively. The left figures display the reconstructions and the right figures show the exact ones. In the top panel, $V$ is evaluated at $2601$ points on the slice $\{x=0\}$. On the snapshot of $V$, the reconstruction is performed in $(-0.25,\,0.25) \times (-0.25,\,0.25)$. In the medium and bottom panels, $V_{tt}$ and $J$ are evaluated at $1521$ points on the slice $\{x=0\}$. On the snapshots of $V_{tt}$ and $J$, the reconstructions are performed in $(-0.19,\,0.19) \times (-0.19,\,0.19)$.

Theorems & Definitions (16)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Remark 3.2
  • ...and 6 more