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A computational inverse random source problem for elastic waves

Hao Gu, Tianjiao Wang, Xiang Xu, Yue Zhao

TL;DR

The paper tackles the inverse random source problem for three-dimensional elastic waves driven by additive white noise, aiming to reconstruct the variance of the random source from single-frequency boundary correlation data. It develops a non-iterative method that uses pairs of complex exponential solutions and Itô isometry to link correlation data to Fourier coefficients of the variance, forming a 3×3 linear system to recover the diagonal variance components for each Fourier mode. A comprehensive error analysis accounts for high-frequency truncation, linear-system conditioning, and Monte Carlo data noise, providing an explicit bound that guides regularization. Numerical experiments in 3D validate the approach, showing accurate variance reconstructions with manageable error growth and demonstrating the method’s stability across frequencies and regularization parameters; the framework also suggests straightforward extensions to stochastic Maxwell equations and to inhomogeneous media.

Abstract

This paper investigates the inverse random source problem for elastic waves in three dimensions, where the source is assumed to be driven by an additive white noise. A novel computational method is proposed for reconstructing the variance of the random source from the correlation boundary measurement of the wave field. Compared with existing multi-frequency iterative approaches, our method is non-iterative and requires data at only a single frequency. As a result, the computational cost is significantly reduced. Furthermore, rigorous error analysis is conducted for the proposed method, which gives a quantitative error estimate. Numerical examples are presented to demonstrate effectiveness of the proposed method. Moreover, this method can to be directly applied to stochastic Maxwell equations.

A computational inverse random source problem for elastic waves

TL;DR

The paper tackles the inverse random source problem for three-dimensional elastic waves driven by additive white noise, aiming to reconstruct the variance of the random source from single-frequency boundary correlation data. It develops a non-iterative method that uses pairs of complex exponential solutions and Itô isometry to link correlation data to Fourier coefficients of the variance, forming a 3×3 linear system to recover the diagonal variance components for each Fourier mode. A comprehensive error analysis accounts for high-frequency truncation, linear-system conditioning, and Monte Carlo data noise, providing an explicit bound that guides regularization. Numerical experiments in 3D validate the approach, showing accurate variance reconstructions with manageable error growth and demonstrating the method’s stability across frequencies and regularization parameters; the framework also suggests straightforward extensions to stochastic Maxwell equations and to inhomogeneous media.

Abstract

This paper investigates the inverse random source problem for elastic waves in three dimensions, where the source is assumed to be driven by an additive white noise. A novel computational method is proposed for reconstructing the variance of the random source from the correlation boundary measurement of the wave field. Compared with existing multi-frequency iterative approaches, our method is non-iterative and requires data at only a single frequency. As a result, the computational cost is significantly reduced. Furthermore, rigorous error analysis is conducted for the proposed method, which gives a quantitative error estimate. Numerical examples are presented to demonstrate effectiveness of the proposed method. Moreover, this method can to be directly applied to stochastic Maxwell equations.

Paper Structure

This paper contains 10 sections, 1 theorem, 53 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model problem
  3. Reconstruction method
  4. Error estimate for the algorithm
  5. Truncation of the high-frequency part in the Fourier transform
  6. Error from solving the linear system
  7. Error for using the Monte Carlo method to calculate expectation
  8. Integral product error for each Monte Carlo sample
  9. Numerical examples
  10. Conclusion

Key Result

Theorem 1

Denote the reconstructed variance by $\boldsymbol \sigma^2_r$. Assume $\| \sigma_j^2\|_{H^1(D)}\leq M, \|\boldsymbol u^{\delta}-\boldsymbol u\|_{L^1(\partial B_R)} \leq\epsilon,\|D\boldsymbol u^{\delta}-D\boldsymbol u\|_{L^1(\partial B_R)} \leq\epsilon$, and the Monte Carlo sample size is $N_s$. Let where $C_1$ depends solely on $\beta$, and $C_2,C_3$ are constants depending on $M, \kappa_{s}$.

Figures (7)

  • Figure 1: Condition number of the coefficient matrices. For each value of $|\boldsymbol\xi|$, we select 256 $\boldsymbol\xi$ uniformly distributed on the sphere with radius $R=|\boldsymbol\xi|$ and determine the corresponding coefficient matrices either randomly or by our proposed approach. For each set of data (each set consists of 256 condition numbers), we compute its mean, maximum, and median.
  • Figure 2: Synthetic observation data of a sample.
  • Figure 3: Reconstructed variance. From left to right: reconstructed $\sigma_1^2,\sigma_2^2,\sigma_3^2$.
  • Figure 4: Slice of the reconstructed variance on the plane $x=0$. Top row: reconstruction; middle row: ground truth; bottom row: difference.
  • Figure 5: Slice of the reconstructed variance on the plane $y=0$. Top row: reconstruction; middle row: ground truth; bottom row: difference.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1: Error estimate for Algorithm 1
  • Remark 1