An inverse source problem for the stochastic wave equation

Xiaoli Feng; Meixia Zhao; Peijun Li; Xu Wang

doi:10.3934/ipi.2021055

Article Contents

2022, Volume 16, Issue 2: 397-415. Doi: 10.3934/ipi.2021055

This issue Previous Article A fuzzy edge detector driven telegraph total variation model for image despeckling Next Article Small defects reconstruction in waveguides from multifrequency one-side scattering data

An inverse source problem for the stochastic wave equation

1.
School of Mathematics and Statistics, Xidian University, Xi'an, 713200, China
2.
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907, USA

^* Corresponding author: Xu Wang
^* Corresponding author: Xu Wang

Received: January 2021

Revised: May 2021

Early access: September 2021

Published: April 2022

The research of XF is supported partially by the CSC fund (No. 201806965033) and the Fundamental Research Funds for the Central Universities (No. JB210706). The research of PL is supported in part by the NSF grant DMS-1912704.

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Abstract

This paper is concerned with an inverse source problem for the stochastic wave equation driven by a fractional Brownian motion. Given the random source, the direct problem is to study the solution of the stochastic wave equation. The inverse problem is to determine the statistical properties of the source from the expectation and covariance of the final-time data. For the direct problem, it is shown to be well-posed with a unique mild solution. For the inverse problem, the uniqueness is proved for a certain class of functions and the instability is characterized. Numerical experiments are presented to illustrate the reconstructions by using a truncation-based regularization method.

Keywords:

Mathematics Subject Classification: 35R30, 35R60, 65M32.

Citation:

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Figure 1. The relative errors of the reconstruction for $ f $ (left) and $ g^2 $ (right) with respect to $ N $ ($ H = 0.9, \delta = 0.001 $)

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Figure 2. The reconstruction for $ f $ (left column) and $ g^2 $ (right column) for different $ N = 50, 100 $ ($ H = 0.9, \delta = 0.001 $)

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Figure 3. The exact solution is plotted against the reconstructed solutions for $ f(x) $ (left column) and $ g^2(x) $ (right column) with $ H = 0.2, 0.5 $ and $ \delta = 0.1, 0.001 $ ($ N = 9 $)

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Table 1. The relative errors of the reconstruction for $ f $ and $ g^2 $ with respect to $ \delta $ ($ N = 9, H = 0.9 $)

$ \delta $	$ 0.001 $	$ 0.005 $	$ 0.01 $	$ 0.05 $	$ 0.1 $
$ f $	$ 0.0260 $	$ 0.0261 $	$ 0.0261 $	$ 0.0286 $	$ 0.0837 $
$ g^2 $	$ 0.0141 $	$ 0.0147 $	$ 0.0237 $	$ 0.0729 $	$ 0.0683 $

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