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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  August 2021 Early access  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 and Imaging, 2021, 15 (4) : 619-639. doi: 10.3934/ipi.2021008
##### References:
 [1] G. Bao, C. Chen and P. Li, Inverse random source scattering problems in several dimensions, SIAM/ASA J. Uncertainty Quantification, 4 (2016), 1263-1287.  doi: 10.1137/16M1067470. [2] G. Bao, C. Chen and P. Li, Inverse random source scattering for elastic waves, SIAM J. Numer. Anal., 55 (2017), 2616-2643.  doi: 10.1137/16M1088922. [3] G. Bao, S. Chow, P. Li and H. Zhou, An inverse random source problem for the Helmholtz equation, Math. Comput., 83 (2014), 215-233.  doi: 10.1090/S0025-5718-2013-02730-5. [4] G. Bao and X. Xu, An inverse random source problem in quantifying the elastic modulus of nanomaterials, Inverse Problems, 29 (2012), 015006, 16pp. doi: 10.1088/0266-5611/29/1/015006. [5] F. Dou, C. Fu and F. Yang, Identifying an unknown source term in a heat equation, Inverse Probl. Sci. Eng., 17 (2009), 901-913.  doi: 10.1080/17415970902916870. [6] R. Dalang, D. Khoshnevisan, C. Mueller, D. Nualalart and Y. Xiao, A Minicourse on Stochastic Partial Differential Equations, Springer, Heidelberg, Berlin, 2009. [7] H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Springer Science & Business Media, 1996. [8] L. Evans, An Introduction to Stochastic Differential Equations, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/mbk/082. [9] L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Rhode Island, 1998. [10] A. Hasanov and M. Slodička, An analysis of inverse source problems with final time measured output data for the heat conduction equation: A semigroup approach, Appl. Math. Lett., 26 (2013), 207-214.  doi: 10.1016/j.aml.2012.08.013. [11] T. Johansson and D. Lesnic, Determination of a spacewise dependent heat source, J. Comput. Appl. Math., 209 (2007), 66-80.  doi: 10.1016/j.cam.2006.10.026. [12] J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160. Springer-Verlag, New York, 2005. [13] P. Kazimierczyk, On the stochastic inverse problem for the heat conduction equation, Reports on Mathematical Physics, 26 (1988), 245-259.  doi: 10.1016/0034-4877(88)90027-4. [14] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer Science & Business Media, 2011. doi: 10.1007/978-1-4419-8474-6. [15] G. Li, Data compatibility and conditional stability for an inverse source problem in the heat equation, Appl. Math. Comput., 173 (2006), 566-581.  doi: 10.1016/j.amc.2005.04.053. [16] M. Li, C. Chen and P. Li, Inverse random source scattering for the Helmholtz equation in inhomogeneous media, Inverse Problems, 34 (2017), 015003, 19pp. doi: 10.1088/1361-6420/aa99d2. [17] P. Li, An inverse random source scattering problem in inhomogeneous media, Inverse Problems, 27 (2011), 035004, 22pp. doi: 10.1088/0266-5611/27/3/035004. [18] P. Li and G. Yuan, Stability on the inverse random source scattering problem for the one-dimensional Helmholtz equation, J. Math. Anal. Appl., 450 (2017), 872-887.  doi: 10.1016/j.jmaa.2017.01.074. [19] Q. Lü, Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems, Inverse Problems, 28 (2012), 045008, 18pp. doi: 10.1088/0266-5611/28/4/045008. [20] Y. Ma, C. Fu and Y. Zhang, Identification of an unknown source depending on both time and space variables by a variational method, Appl. Math. Model., 36 (2012), 5080-5090.  doi: 10.1016/j.apm.2011.12.046. [21] F. Natterer, The Mathematics of Computerized Tomography, Teubner, Stuttgart, 1986. [22] P. Niu, T. Helin and Z. Zhang, An inverse random source problem in a stochastic fractional diffusion equation, Inverse Problems, 36 (2020), 045002, 23pp. doi: 10.1088/1361-6420/ab532c. [23] J. Nolen and G. Papanicolaou, Fine scale uncertainty in parameter estimation for elliptic equations, Inverse Problems, 25 (2009), 115021, 22pp. doi: 10.1088/0266-5611/25/11/115021. [24] A. Prilepko, V. Kamynin and A. Kostin, Inverse source problem for parabolic equation with the condition of integral observation in time, J. Inverse Ill-posed Probl., 26 (2018), 523-539.  doi: 10.1515/jiip-2017-0049. [25] E. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford University Press, London, 1939. [26] J. Walsh, An introduction to stochastic partial differential equations, École d'été de Probabilités de Saint-Flour, XIV–1984, 265–439, Lecture Notes in Math., 1180, Springer, Berlin, 1986. doi: 10.1007/BFb0074920. [27] Z. Wang and J. Liu, Identification of the pollution source from one-dimensional parabolic equation models, Appl. Math. Comput., 219 (2012), 3403-3413.  doi: 10.1016/j.amc.2008.03.014. [28] Z. Wang and D. Xu, On the linear model function method for choosing Tikhonov regularization parameters in linear ill-posed problems, Chinese J. Eng. Math., 30 (2013), 451-466. [29] T. Wei and J. Wang, A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl. Numer. Math., 78 (2014), 95-111.  doi: 10.1016/j.apnum.2013.12.002. [30] F. Yang and C. Fu, A simplified Tikhonov regularization method for determining the heat source, Appl. Math. Model., 34 (2010), 3286-3299.  doi: 10.1016/j.apm.2010.02.020.

show all references

##### References:
 [1] G. Bao, C. Chen and P. Li, Inverse random source scattering problems in several dimensions, SIAM/ASA J. Uncertainty Quantification, 4 (2016), 1263-1287.  doi: 10.1137/16M1067470. [2] G. Bao, C. Chen and P. Li, Inverse random source scattering for elastic waves, SIAM J. Numer. Anal., 55 (2017), 2616-2643.  doi: 10.1137/16M1088922. [3] G. Bao, S. Chow, P. Li and H. Zhou, An inverse random source problem for the Helmholtz equation, Math. Comput., 83 (2014), 215-233.  doi: 10.1090/S0025-5718-2013-02730-5. [4] G. Bao and X. Xu, An inverse random source problem in quantifying the elastic modulus of nanomaterials, Inverse Problems, 29 (2012), 015006, 16pp. doi: 10.1088/0266-5611/29/1/015006. [5] F. Dou, C. Fu and F. Yang, Identifying an unknown source term in a heat equation, Inverse Probl. Sci. Eng., 17 (2009), 901-913.  doi: 10.1080/17415970902916870. [6] R. Dalang, D. Khoshnevisan, C. Mueller, D. Nualalart and Y. Xiao, A Minicourse on Stochastic Partial Differential Equations, Springer, Heidelberg, Berlin, 2009. [7] H. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Springer Science & Business Media, 1996. [8] L. Evans, An Introduction to Stochastic Differential Equations, American Mathematical Society, Providence, RI, 2013. doi: 10.1090/mbk/082. [9] L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Rhode Island, 1998. [10] A. Hasanov and M. Slodička, An analysis of inverse source problems with final time measured output data for the heat conduction equation: A semigroup approach, Appl. Math. Lett., 26 (2013), 207-214.  doi: 10.1016/j.aml.2012.08.013. [11] T. Johansson and D. Lesnic, Determination of a spacewise dependent heat source, J. Comput. Appl. Math., 209 (2007), 66-80.  doi: 10.1016/j.cam.2006.10.026. [12] J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160. Springer-Verlag, New York, 2005. [13] P. Kazimierczyk, On the stochastic inverse problem for the heat conduction equation, Reports on Mathematical Physics, 26 (1988), 245-259.  doi: 10.1016/0034-4877(88)90027-4. [14] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer Science & Business Media, 2011. doi: 10.1007/978-1-4419-8474-6. [15] G. Li, Data compatibility and conditional stability for an inverse source problem in the heat equation, Appl. Math. Comput., 173 (2006), 566-581.  doi: 10.1016/j.amc.2005.04.053. [16] M. Li, C. Chen and P. Li, Inverse random source scattering for the Helmholtz equation in inhomogeneous media, Inverse Problems, 34 (2017), 015003, 19pp. doi: 10.1088/1361-6420/aa99d2. [17] P. Li, An inverse random source scattering problem in inhomogeneous media, Inverse Problems, 27 (2011), 035004, 22pp. doi: 10.1088/0266-5611/27/3/035004. [18] P. Li and G. Yuan, Stability on the inverse random source scattering problem for the one-dimensional Helmholtz equation, J. Math. Anal. Appl., 450 (2017), 872-887.  doi: 10.1016/j.jmaa.2017.01.074. [19] Q. Lü, Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems, Inverse Problems, 28 (2012), 045008, 18pp. doi: 10.1088/0266-5611/28/4/045008. [20] Y. Ma, C. Fu and Y. Zhang, Identification of an unknown source depending on both time and space variables by a variational method, Appl. Math. Model., 36 (2012), 5080-5090.  doi: 10.1016/j.apm.2011.12.046. [21] F. Natterer, The Mathematics of Computerized Tomography, Teubner, Stuttgart, 1986. [22] P. Niu, T. Helin and Z. Zhang, An inverse random source problem in a stochastic fractional diffusion equation, Inverse Problems, 36 (2020), 045002, 23pp. doi: 10.1088/1361-6420/ab532c. [23] J. Nolen and G. Papanicolaou, Fine scale uncertainty in parameter estimation for elliptic equations, Inverse Problems, 25 (2009), 115021, 22pp. doi: 10.1088/0266-5611/25/11/115021. [24] A. Prilepko, V. Kamynin and A. Kostin, Inverse source problem for parabolic equation with the condition of integral observation in time, J. Inverse Ill-posed Probl., 26 (2018), 523-539.  doi: 10.1515/jiip-2017-0049. [25] E. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford University Press, London, 1939. [26] J. Walsh, An introduction to stochastic partial differential equations, École d'été de Probabilités de Saint-Flour, XIV–1984, 265–439, Lecture Notes in Math., 1180, Springer, Berlin, 1986. doi: 10.1007/BFb0074920. [27] Z. Wang and J. Liu, Identification of the pollution source from one-dimensional parabolic equation models, Appl. Math. Comput., 219 (2012), 3403-3413.  doi: 10.1016/j.amc.2008.03.014. [28] Z. Wang and D. Xu, On the linear model function method for choosing Tikhonov regularization parameters in linear ill-posed problems, Chinese J. Eng. Math., 30 (2013), 451-466. [29] T. Wei and J. Wang, A modified quasi-boundary value method for an inverse source problem of the time-fractional diffusion equation, Appl. Numer. Math., 78 (2014), 95-111.  doi: 10.1016/j.apnum.2013.12.002. [30] F. Yang and C. Fu, A simplified Tikhonov regularization method for determining the heat source, Appl. Math. Model., 34 (2010), 3286-3299.  doi: 10.1016/j.apm.2010.02.020.
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 $\mu = 5\times 10^{-3}, \epsilon = 0.05$
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