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Inverse source problem of the biharmonic equation from multi-frequency phaseless data

Yan Chang, Yukun Guo, Yue Zhao

TL;DR

The paper tackles the inverse source problem for the biharmonic equation $\Delta^2 u - k^4 u = S$ from multifrequency phaseless data. It develops a two-stage approach: first, phase retrieval using auxiliary reference sources to convert phaseless data into phased measurements, and then a Fourier-based reconstruction of $S$ from the phased data. The authors provide a detailed stability analysis for the phase retrieval, leveraging properties of Bessel-type functions, and demonstrate through numerical tests that the method yields accurate source reconstructions even under substantial noise. The work extends Fourier-based inverse source techniques to the biharmonic setting with phaseless data and offers a practical, solver-free reconstruction framework.

Abstract

This work deals with an inverse source problem for the biharmonic wave equation. A two-stage numerical method is proposed to identify the unknown source from the multi-frequency phaseless data. In the first stage, we introduce some artificially auxiliary point sources to the inverse source system and establish a phase retrieval formula. Theoretically, we point out that the phase can be uniquely determined and estimate the stability of this phase retrieval approach. Once the phase information is retrieved, the Fourier method is adopted to reconstruct the source function from the phased multi-frequency data. The proposed method is easy-to-implement and there is no forward solver involved in the reconstruction. Numerical experiments are conducted to verify the performance of the proposed method.

Inverse source problem of the biharmonic equation from multi-frequency phaseless data

TL;DR

The paper tackles the inverse source problem for the biharmonic equation from multifrequency phaseless data. It develops a two-stage approach: first, phase retrieval using auxiliary reference sources to convert phaseless data into phased measurements, and then a Fourier-based reconstruction of from the phased data. The authors provide a detailed stability analysis for the phase retrieval, leveraging properties of Bessel-type functions, and demonstrate through numerical tests that the method yields accurate source reconstructions even under substantial noise. The work extends Fourier-based inverse source techniques to the biharmonic setting with phaseless data and offers a practical, solver-free reconstruction framework.

Abstract

This work deals with an inverse source problem for the biharmonic wave equation. A two-stage numerical method is proposed to identify the unknown source from the multi-frequency phaseless data. In the first stage, we introduce some artificially auxiliary point sources to the inverse source system and establish a phase retrieval formula. Theoretically, we point out that the phase can be uniquely determined and estimate the stability of this phase retrieval approach. Once the phase information is retrieved, the Fourier method is adopted to reconstruct the source function from the phased multi-frequency data. The proposed method is easy-to-implement and there is no forward solver involved in the reconstruction. Numerical experiments are conducted to verify the performance of the proposed method.
Paper Structure (9 sections, 5 theorems, 86 equations, 3 figures, 5 tables, 2 algorithms)

This paper contains 9 sections, 5 theorems, 86 equations, 3 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Model problem
  4. Fourier method
  5. Phase retrieval by auxiliary sources
  6. Stability of the phase retrieval
  7. Numerical experiments
  8. Phase retrieval
  9. Reconstruct the source

Key Result

Lemma 4.1

For $x>0,$ it holds that

Figures (3)

  • Figure 1: The exact source function $S$. (a) Surface plot; (b) contour plot.
  • Figure 2: Geometry setup for the phase retrieval technique. The thick black line plots the boundary of $B_1$. The small red points denote the reference source points. (a) $k=k_0=\pi/90;$ (b) $k=\pi.$
  • Figure 3: The reconstruction of source function $S$ with $\varepsilon=1\%$. (a) Surface plot; (b) contour plot.

Theorems & Definitions (9)

  • Definition 2.1: Admissible wave numbers,
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 4.3