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

• * Corresponding author
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
• 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.

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

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

Figure 2.  The statistical properties of the exact source

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

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

Figure 5.  The statistical properties of the exact source

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

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

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

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

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