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A data-assisted two-stage method for the inverse random source problem

Peijun Li, Ying Liang, Yuliang Wang

TL;DR

Numerical experiments demonstrate that the data-assisted two-stage method provides satisfactory reconstruction for both homogeneous and inhomogeneous media with fewer realizations.

Abstract

We propose a data-assisted two-stage method for solving an inverse random source problem of the Helmholtz equation. In the first stage, the regularized Kaczmarz method is employed to generate initial approximations of the mean and variance based on the mild solution of the stochastic Helmholtz equation. A dataset is then obtained by sampling the approximate and corresponding true profiles from a certain a-priori criterion. The second stage is formulated as an image-to-image translation problem, and several data-assisted approaches are utilized to handle the dataset and obtain enhanced reconstructions. Numerical experiments demonstrate that the data-assisted two-stage method provides satisfactory reconstruction for both homogeneous and inhomogeneous media with fewer realizations.

A data-assisted two-stage method for the inverse random source problem

TL;DR

Numerical experiments demonstrate that the data-assisted two-stage method provides satisfactory reconstruction for both homogeneous and inhomogeneous media with fewer realizations.

Abstract

We propose a data-assisted two-stage method for solving an inverse random source problem of the Helmholtz equation. In the first stage, the regularized Kaczmarz method is employed to generate initial approximations of the mean and variance based on the mild solution of the stochastic Helmholtz equation. A dataset is then obtained by sampling the approximate and corresponding true profiles from a certain a-priori criterion. The second stage is formulated as an image-to-image translation problem, and several data-assisted approaches are utilized to handle the dataset and obtain enhanced reconstructions. Numerical experiments demonstrate that the data-assisted two-stage method provides satisfactory reconstruction for both homogeneous and inhomogeneous media with fewer realizations.
Paper Structure (16 sections, 4 theorems, 57 equations, 10 figures, 6 tables)

This paper contains 16 sections, 4 theorems, 57 equations, 10 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Stage one: the Kaczmarz method
  4. Integral equations
  5. Error analysis
  6. Stage two: machine learning techniques
  7. Principal component analysis
  8. Dynamic mode decomposition
  9. U-Net based network
  10. Pix2pix neural network
  11. Numerical experiments
  12. Preparation for training
  13. Stage one configuration
  14. Stage two configuration
  15. Numerical experiments
  16. ...and 1 more sections

Key Result

Lemma 3.1

Let $\sigma_\gamma$ be the smallest nonzero singular value of $R_\gamma$, then it holds that $\sigma_\gamma \leq 1$, and

Figures (10)

  • Figure 1: Reconstructions with different numbers of trials. (A) the exact mean of a random source; (B/C/D) The reconstruction of mean with measurements of 10/100/1000 realizations.
  • Figure 2: The U-Net architecture.
  • Figure 3: The pix2pix algorithm. The generator $\mathcal{G}$ learns to fool the discriminator $\mathcal{D}$, while the discriminator $\mathcal{D}$ learns to distinguish the fake $(X, \mathcal{G}(X))$ and real $(X,Y)$ tuples in the training set.
  • Figure 4: (A) Pseudocolor plot of the function $h$; (B) Pseudocolor plot of the function $\eta$; (C) Computational domain for the direct problems.
  • Figure 5: Loss versus epoches for Example 1.
  • ...and 5 more figures

Theorems & Definitions (14)

  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • Remark 3.4
  • Corollary 3.5
  • Remark 4.1
  • Example 1
  • ...and 4 more