Cauchy problem of non-homogenous stochastic heat equation and application to inverse random source problem

Shuli Chen; Zewen Wang; Guolin Chen

doi:10.3934/ipi.2021008

Article Contents

2021, Volume 15, Issue 4: 619-639. Doi: 10.3934/ipi.2021008

This issue Previous Article A regularization operator for source identification for elliptic PDEs Next Article Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems

Cauchy problem of non-homogenous stochastic heat equation and application to inverse random source problem

1.
School of Mathematics, Southeast University, Nanjing, Jiangsu, 210096, China
2.
School of Science, East China University of Technology, Nanchang, Jiangxi, 330013, China

^* Corresponding author
^* Corresponding author

Received: February 2020

Revised: July 2020

Early access: January 2021

Published: August 2021

The first author is supported by National Natural Science Foundation of China grant 11761007; the second author is supported by National Natural Science Foundation of China grant 11961002; Natural Science Foundation of Jiangxi Province and Foundation of Academic and Technical Leaders Program for Major Subjects in Jiangxi Province grant 20172BCB22019

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Abstract

In this paper, a Cauchy problem of non-homogenous stochastic heat equation is considered together with its inverse source problem, where the source term is assumed to be driven by an additive white noise. The Cauchy problem (direct problem) is to determine the displacement of random temperature field, while the considered inverse problem is to reconstruct the statistical properties of the random source, i.e. the mean and variance of the random source. It is proved constructively that the Cauchy problem has a unique mild solution, which is expressed an integral form. Then the inverse random source problem is formulated into two Fredholm integral equations of the first kind, which are typically ill-posed. To obtain stable inverse solutions, the regularized block Kaczmarz method is introduced to solve the two Fredholm integral equations. Finally, numerical experiments are given to show that the proposed method is efficient and robust for reconstructing the statistical properties of the random source.

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Mathematics Subject Classification: Primary: 35R60; Secondary: 35R30.

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Figure 1. the decaying property of singular values: (A) for Eq.(12); (B) for Eq.(13)

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Figure 2. The statistical properties of the exact source

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Figure 3. The statistical properties of the inverse source for $ \mu = 10^{-4}, \epsilon = 0 $

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Figure 4. The statistical properties of the inverse source for $ \mu = 10^{-3}, \epsilon = 0.03 $

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Figure 5. The statistical properties of the exact source

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Figure 6. The statistical properties of the inverse source for $ \mu = 10^{-4}, \epsilon = 0 $

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Figure 7. The statistical properties of the inverse source for $ \mu = 10^{-3}, \epsilon = 0.03 $

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Figure 8. The statistical properties of the random source for $ \mu = 5\times 10^{-3}, \epsilon = 0.05 $

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Figure 9. The statistical properties of the random source for $ \mu = 5\times 10^{-3}, \epsilon = 0.05 $

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