# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2021055
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## 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

Received  January 2021 Revised  May 2021 Early access September 2021

Fund Project: 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

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.

Citation: Xiaoli Feng, Meixia Zhao, Peijun Li, Xu Wang. An inverse source problem for the stochastic wave equation. Inverse Problems & Imaging, doi: 10.3934/ipi.2021055
##### References:
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Zhou, An inverse random source problem for the Helmholtz equation, Math. Comp., 83 (2014), 215-233.  doi: 10.1090/S0025-5718-2013-02730-5.  Google Scholar [7] G. Bao, J. Lin and F. Triki, A multi-frequency inverse source problem, J. Differential Equations, 249 (2010), 3443-3465.  doi: 10.1016/j.jde.2010.08.013.  Google Scholar [8] G. Bao, J. Lin and F. Triki, An inverse source problem with multiple frequency data, C. R. Math., 349 (2011), 855-859.  doi: 10.1016/j.crma.2011.07.009.  Google Scholar [9] A. L. Bukhgeim and M. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 17 (1981), 269-272.   Google Scholar [10] P. Caithamer, The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise, Stoch. Dyn., 5 (2005), 45-64.  doi: 10.1142/S0219493705001286.  Google Scholar [11] J. R. Cannon and P. Duchateau, An inverse problem for an unknown source term in a wave equation, SIAM J. Appl. 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Isakov, Inverse Source Problems, American Mathematical Society, 1990. doi: 10.1090/surv/034.  Google Scholar [22] H. Leszczyński and M. Wrzosek, Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion, Math. Biosci. Eng., 14 (2017), 237-248.  doi: 10.3934/mbe.2017015.  Google Scholar [23] M. Li, C. Chen and P. Li, Inverse random source scattering for the Helmholtz equation in inhomogeneous media, Inverse Problems, 34 (2018), 015003. doi: 10.1088/1361-6420/aa99d2.  Google Scholar [24] P. Li, An inverse random source scattering problem in inhomogeneous media, Inverse Problems, 27 (2011), 035004. doi: 10.1088/0266-5611/27/3/035004.  Google Scholar [25] P. Li and X. Wang, Inverse random source scattering for the Helmholtz equation with attenuation, SIAM J. Appl. Math., 81 (2021), 485-506.  doi: 10.1137/19M1309456.  Google Scholar [26] P. Li and X. Wang, An inverse random source problem for Maxwell's equations, Multiscale Model. Simul., 19 (2021), 25-45.  doi: 10.1137/20M1331342.  Google Scholar [27] P. Li and W. Wang, An inverse random source problem for the one-dimensional Helmholtz equation with attenuation, Inverse Problems, 37 (2021), 015009. doi: 10.1088/1361-6420/abcd43.  Google Scholar [28] 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.  Google Scholar [29] S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping and potential coefficients of inverse hyperbolic problem, Nonlinear Anal. Real World Appl., 12 (2011), 1562-1590.  doi: 10.1016/j.nonrwa.2010.10.014.  Google Scholar [30] S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with nonhomogeneous Newmann B.C. through an additional Dirichlet boundary trace, SIAM J. Math. Anal., 43 (2011), 1631-1666.  doi: 10.1137/100808988.  Google Scholar [31] X. Liu and W. Deng, Higher order approximation for stochastic wave equation, arXiv: 2007.02619. Google Scholar [32] E. Marengo and A. Devaney, The inverse source problem of electromagnetics: Linear inversion formulation and minimum energy solution, IEEE Trans. Antennas Propag., 47 (1999), 410-412.  doi: 10.1109/8.761085.  Google Scholar [33] E. Marengo, M. Khodja and A. Boucherif, Inverse source problem in nonhomogeneous background media: II. Vector formulation and antenna substrate performance characterization, SIAM J. Appl. Math., 69 (2008), 81-110.  doi: 10.1137/070689875.  Google Scholar [34] L. H. Nguyen, An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method, Inverse Problems, 35 (2019), 35007. doi: 10.1088/1361-6420/aafe8f.  Google Scholar [35] P. Niu, T. Helin and Z. Zhang, An inverse random source problem in a stochastic fractional diffusion equation, Inverse Problems, 36 (2020), 045002. doi: 10.1088/1361-6420/ab532c.  Google Scholar [36] D. Nualart, The Malliavin Calculus and Related Topics, Probability and Its Applications, 2$^nd$ edition, Springer-Verlag, Berlin, 2006.  Google Scholar [37] E. Orsingher, Randomly forced vibrations of a string, Ann. Inst. H. Poincaré Sect. B (N.S.), 18 (1982), 367-394.   Google Scholar [38] L. Quer-Sardanyons and S. Tindel, The 1-d stochastic wave equation driven by a fractional Brownian sheet, Stochastic Process. Appl., 117 (2007), 1448-1472.  doi: 10.1016/j.spa.2007.01.009.  Google Scholar [39] P. Sattari Shajari and A. Shidfar, Application of weighted homotopy analysis method to solve an inverse source problem for wave equation, Inverse Probl. Sci. Eng., 27 (2019), 61-88.  doi: 10.1080/17415977.2018.1442447.  Google Scholar [40] W. A. Strauss, Partial Differential Equations: An Introduction, 2$^{nd}$ edition, John Wiley & Sons, Ltd., Chichester, 2008.  Google Scholar [41] D. Tang and Y. Wang, The stochastic wave equations driven by fractional and colored noised, Acta Math. Sin. (Engl. Ser.), 26 (2010), 1055-1070.  doi: 10.1007/s10114-010-8469-9.  Google Scholar [42] J. B. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Math., Springer, Berlin, 1180 (1986), 265–439. doi: 10.1007/BFb0074920.  Google Scholar [43] M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Problems, 11 (1995), 481-496.  doi: 10.1088/0266-5611/11/2/013.  Google Scholar [44] M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pures Appl., 78 (1999), 65-98.  doi: 10.1016/S0021-7824(99)80010-5.  Google Scholar [45] M. Yamamoto and X. Zhang, Global uniqueness and stability for a class of multidimensional inverse hyperbolic problems with two unknowns, Appl. Anal. Optim., 48 (2003), 211-228.  doi: 10.1007/s00245-003-0775-5.  Google Scholar

show all references

##### References:
 [1] H. Ammari, G. Bao and J. Fleming, An inverse source problem for Maxwell's equations in magnetoencephalography, SIAM J. Appl. Math., 62 (2002), 1369-1382.  doi: 10.1137/S0036139900373927.  Google Scholar [2] M. A. Anastasio, J. Zhang, D. Modgil and P. J. La Rivière, Application of inverse source concepts to photoacoustic tomography, Inverse Problems, 23 (2007), S21–S35. doi: 10.1088/0266-5611/23/6/S03.  Google Scholar [3] A. Baker, Transcendental Number Theory, 2$^{nd}$ edition, Cambridge University Press, 1990.  Google Scholar [4] G. Bao, C. Chen and P. Li, Inverse random source scattering problems in several dimensions, SIAM/ASA J. Uncertain. Quantif., 4 (2016), 1263-1287.  doi: 10.1137/16M1067470.  Google Scholar [5] 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.  Google Scholar [6] G. Bao, S.-N. Chow, P. Li and H. Zhou, An inverse random source problem for the Helmholtz equation, Math. Comp., 83 (2014), 215-233.  doi: 10.1090/S0025-5718-2013-02730-5.  Google Scholar [7] G. Bao, J. Lin and F. Triki, A multi-frequency inverse source problem, J. Differential Equations, 249 (2010), 3443-3465.  doi: 10.1016/j.jde.2010.08.013.  Google Scholar [8] G. Bao, J. Lin and F. Triki, An inverse source problem with multiple frequency data, C. R. Math., 349 (2011), 855-859.  doi: 10.1016/j.crma.2011.07.009.  Google Scholar [9] A. L. Bukhgeim and M. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 17 (1981), 269-272.   Google Scholar [10] P. Caithamer, The stochastic wave equation driven by fractional Brownian noise and temporally correlated smooth noise, Stoch. Dyn., 5 (2005), 45-64.  doi: 10.1142/S0219493705001286.  Google Scholar [11] J. R. Cannon and P. Duchateau, An inverse problem for an unknown source term in a wave equation, SIAM J. Appl. Math., 43 (1983), 553-564.  doi: 10.1137/0143036.  Google Scholar [12] B. Chen, Y. Guo, F. Ma and Y. Sun, Numerical schemes to reconstruct three-dimensional time-dependent point sources of acoustic waves, Inverse Problems, 36 (2020), 075009. doi: 10.1088/1361-6420/ab8f85.  Google Scholar [13] R. C. Dalang and N. E. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab., 26 (1998), 187-212.  doi: 10.1214/aop/1022855416.  Google Scholar [14] A. Devaney, The inverse problem for random sources, J. Math. Phys., 20 (1979), 1687-1691.  doi: 10.1063/1.524277.  Google Scholar [15] A. Devaney, Inverse source and scattering problems in ultrasonics, IEEE Trans. Sonics Ultrason., 30 (1983), 355-363.   Google Scholar [16] M. de Hoop, L. Oksanen and J. Tittelfitz, Uniqueness for a seismic inverse source problem modeling a subsonic rupture, Commun. Partial. Differ. Equ., 41 (2016), 1895-1917.  doi: 10.1080/03605302.2016.1240183.  Google Scholar [17] M. Erraoui, Y. Ouknine and D. Nualart, Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet, Stoch. Dyn., 3 (2003), 121-139.  doi: 10.1142/S0219493703000681.  Google Scholar [18] X. Feng, P. Li and X. Wang, An inverse random source problem for the time fractional diffusion equation driven by a fractional Brownian motion, Inverse Problems, 36 (2020), 045008. doi: 10.1088/1361-6420/ab6503.  Google Scholar [19] A. Hasanov and B. Mukanova, Fourier collocation algorithm for identification of a spacewise dependent source in wave equation from Neumann-type measured data, Appl. Numer. Math., 111 (2017), 49-63.  doi: 10.1016/j.apnum.2016.09.006.  Google Scholar [20] O. Y. Imanuvilov and M. Yamamoto, Global Lipschitz stability in an inverse hyperbolic problem by interior observations, Inverse Problems, 17 (2001), 717-728.  doi: 10.1088/0266-5611/17/4/310.  Google Scholar [21] V. Isakov, Inverse Source Problems, American Mathematical Society, 1990. doi: 10.1090/surv/034.  Google Scholar [22] H. Leszczyński and M. Wrzosek, Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion, Math. Biosci. Eng., 14 (2017), 237-248.  doi: 10.3934/mbe.2017015.  Google Scholar [23] M. Li, C. Chen and P. Li, Inverse random source scattering for the Helmholtz equation in inhomogeneous media, Inverse Problems, 34 (2018), 015003. doi: 10.1088/1361-6420/aa99d2.  Google Scholar [24] P. Li, An inverse random source scattering problem in inhomogeneous media, Inverse Problems, 27 (2011), 035004. doi: 10.1088/0266-5611/27/3/035004.  Google Scholar [25] P. Li and X. Wang, Inverse random source scattering for the Helmholtz equation with attenuation, SIAM J. Appl. Math., 81 (2021), 485-506.  doi: 10.1137/19M1309456.  Google Scholar [26] P. Li and X. Wang, An inverse random source problem for Maxwell's equations, Multiscale Model. Simul., 19 (2021), 25-45.  doi: 10.1137/20M1331342.  Google Scholar [27] P. Li and W. Wang, An inverse random source problem for the one-dimensional Helmholtz equation with attenuation, Inverse Problems, 37 (2021), 015009. doi: 10.1088/1361-6420/abcd43.  Google Scholar [28] 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.  Google Scholar [29] S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping and potential coefficients of inverse hyperbolic problem, Nonlinear Anal. Real World Appl., 12 (2011), 1562-1590.  doi: 10.1016/j.nonrwa.2010.10.014.  Google Scholar [30] S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping coefficient of an inverse hyperbolic problem with nonhomogeneous Newmann B.C. through an additional Dirichlet boundary trace, SIAM J. Math. Anal., 43 (2011), 1631-1666.  doi: 10.1137/100808988.  Google Scholar [31] X. Liu and W. Deng, Higher order approximation for stochastic wave equation, arXiv: 2007.02619. Google Scholar [32] E. Marengo and A. Devaney, The inverse source problem of electromagnetics: Linear inversion formulation and minimum energy solution, IEEE Trans. Antennas Propag., 47 (1999), 410-412.  doi: 10.1109/8.761085.  Google Scholar [33] E. Marengo, M. Khodja and A. Boucherif, Inverse source problem in nonhomogeneous background media: II. Vector formulation and antenna substrate performance characterization, SIAM J. Appl. Math., 69 (2008), 81-110.  doi: 10.1137/070689875.  Google Scholar [34] L. H. Nguyen, An inverse space-dependent source problem for hyperbolic equations and the Lipschitz-like convergence of the quasi-reversibility method, Inverse Problems, 35 (2019), 35007. doi: 10.1088/1361-6420/aafe8f.  Google Scholar [35] P. Niu, T. Helin and Z. Zhang, An inverse random source problem in a stochastic fractional diffusion equation, Inverse Problems, 36 (2020), 045002. doi: 10.1088/1361-6420/ab532c.  Google Scholar [36] D. Nualart, The Malliavin Calculus and Related Topics, Probability and Its Applications, 2$^nd$ edition, Springer-Verlag, Berlin, 2006.  Google Scholar [37] E. Orsingher, Randomly forced vibrations of a string, Ann. Inst. H. Poincaré Sect. B (N.S.), 18 (1982), 367-394.   Google Scholar [38] L. Quer-Sardanyons and S. Tindel, The 1-d stochastic wave equation driven by a fractional Brownian sheet, Stochastic Process. Appl., 117 (2007), 1448-1472.  doi: 10.1016/j.spa.2007.01.009.  Google Scholar [39] P. Sattari Shajari and A. Shidfar, Application of weighted homotopy analysis method to solve an inverse source problem for wave equation, Inverse Probl. Sci. Eng., 27 (2019), 61-88.  doi: 10.1080/17415977.2018.1442447.  Google Scholar [40] W. A. Strauss, Partial Differential Equations: An Introduction, 2$^{nd}$ edition, John Wiley & Sons, Ltd., Chichester, 2008.  Google Scholar [41] D. Tang and Y. Wang, The stochastic wave equations driven by fractional and colored noised, Acta Math. Sin. (Engl. Ser.), 26 (2010), 1055-1070.  doi: 10.1007/s10114-010-8469-9.  Google Scholar [42] J. B. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Math., Springer, Berlin, 1180 (1986), 265–439. doi: 10.1007/BFb0074920.  Google Scholar [43] M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method, Inverse Problems, 11 (1995), 481-496.  doi: 10.1088/0266-5611/11/2/013.  Google Scholar [44] M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pures Appl., 78 (1999), 65-98.  doi: 10.1016/S0021-7824(99)80010-5.  Google Scholar [45] M. Yamamoto and X. Zhang, Global uniqueness and stability for a class of multidimensional inverse hyperbolic problems with two unknowns, Appl. Anal. Optim., 48 (2003), 211-228.  doi: 10.1007/s00245-003-0775-5.  Google Scholar
The relative errors of the reconstruction for $f$ (left) and $g^2$ (right) with respect to $N$ ($H = 0.9, \delta = 0.001$)
The reconstruction for $f$ (left column) and $g^2$ (right column) for different $N = 50, 100$ ($H = 0.9, \delta = 0.001$)
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 $\delta = 0.1, 0.001$ ($N = 9$)
The relative errors of the reconstruction for $f$ and $g^2$ with respect to $\delta$ ($N = 9, H = 0.9$)
 $\delta$ $0.001$ $0.005$ $0.01$ $0.05$ $0.1$ $f$ $0.0260$ $0.0261$ $0.0261$ $0.0286$ $0.0837$ $g^2$ $0.0141$ $0.0147$ $0.0237$ $0.0729$ $0.0683$
 $\delta$ $0.001$ $0.005$ $0.01$ $0.05$ $0.1$ $f$ $0.0260$ $0.0261$ $0.0261$ $0.0286$ $0.0837$ $g^2$ $0.0141$ $0.0147$ $0.0237$ $0.0729$ $0.0683$
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