# American Institute of Mathematical Sciences

• Previous Article
Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems
• IPI Home
• This Issue
• Next Article
A regularization operator for source identification for elliptic PDEs
August  2021, 15(4): 619-639. doi: 10.3934/ipi.2021008

## 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

Received  February 2020 Revised  July 2020 Published  January 2021

Fund Project: 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.

Citation: Shuli Chen, Zewen Wang, Guolin Chen. Cauchy problem of non-homogenous stochastic heat equation and application to inverse random source problem. Inverse Problems & Imaging, 2021, 15 (4) : 619-639. doi: 10.3934/ipi.2021008
##### References:

show all references

##### References:
the decaying property of singular values: (A) for Eq.(12); (B) for Eq.(13)
The statistical properties of the exact source
The statistical properties of the inverse source for $\mu = 10^{-4}, \epsilon = 0$
The statistical properties of the inverse source for $\mu = 10^{-3}, \epsilon = 0.03$
The statistical properties of the exact source
The statistical properties of the inverse source for $\mu = 10^{-4}, \epsilon = 0$
The statistical properties of the inverse source for $\mu = 10^{-3}, \epsilon = 0.03$
The statistical properties of the random source for $\mu = 5\times 10^{-3}, \epsilon = 0.05$
The statistical properties of the random source for $\mu = 5\times 10^{-3}, \epsilon = 0.05$
 [1] Yuxuan Gong, Xiang Xu. Inverse random source problem for biharmonic equation in two dimensions. Inverse Problems & Imaging, 2019, 13 (3) : 635-652. doi: 10.3934/ipi.2019029 [2] Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002 [3] Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032 [4] Masaru Ikehata, Mishio Kawashita. An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method. Inverse Problems & Imaging, 2014, 8 (4) : 1073-1116. doi: 10.3934/ipi.2014.8.1073 [5] Jong-Shenq Guo, Bei Hu. Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 927-937. doi: 10.3934/dcds.2008.20.927 [6] Guirong Liu, Yuanwei Qi. Sign-changing solutions of a quasilinear heat equation with a source term. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1389-1414. doi: 10.3934/dcdsb.2013.18.1389 [7] Z. K. Eshkuvatov, M. Kammuji, Bachok M. Taib, N. M. A. Nik Long. Effective approximation method for solving linear Fredholm-Volterra integral equations. Numerical Algebra, Control & Optimization, 2017, 7 (1) : 77-88. doi: 10.3934/naco.2017004 [8] Yueling Li, Yingchao Xie, Xicheng Zhang. Large deviation principle for stochastic heat equation with memory. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5221-5237. doi: 10.3934/dcds.2015.35.5221 [9] Fulvia Confortola, Elisa Mastrogiacomo. Optimal control for stochastic heat equation with memory. Evolution Equations & Control Theory, 2014, 3 (1) : 35-58. doi: 10.3934/eect.2014.3.35 [10] Kazuhiro Ishige, Tatsuki Kawakami, Kanako Kobayashi. Global solutions for a nonlinear integral equation with a generalized heat kernel. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 767-783. doi: 10.3934/dcdss.2014.7.767 [11] Kenichi Sakamoto, Masahiro Yamamoto. Inverse source problem with a final overdetermination for a fractional diffusion equation. Mathematical Control & Related Fields, 2011, 1 (4) : 509-518. doi: 10.3934/mcrf.2011.1.509 [12] Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367 [13] Liu Liu. Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling. Kinetic & Related Models, 2018, 11 (5) : 1139-1156. doi: 10.3934/krm.2018044 [14] Galina Kurina, Vladimir Zadorozhniy. Mean periodic solutions of a inhomogeneous heat equation with random coefficients. Discrete & Continuous Dynamical Systems - S, 2020, 13 (5) : 1543-1551. doi: 10.3934/dcdss.2020087 [15] Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176 [16] Chan Liu, Jin Wen, Zhidong Zhang. Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation. Inverse Problems & Imaging, 2020, 14 (6) : 1001-1024. doi: 10.3934/ipi.2020053 [17] Giuseppe Da Prato. An integral inequality for the invariant measure of some finite dimensional stochastic differential equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3015-3027. doi: 10.3934/dcdsb.2016085 [18] Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432 [19] Noui Djaidja, Mostefa Nadir. Comparison between Taylor and perturbed method for Volterra integral equation of the first kind. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020039 [20] Beatrice Bugert, Gunther Schmidt. Analytical investigation of an integral equation method for electromagnetic scattering by biperiodic structures. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 435-473. doi: 10.3934/dcdss.2015.8.435

2019 Impact Factor: 1.373

## Tools

Article outline

Figures and Tables