April  2022, 16(2): 397-415. doi: 10.3934/ipi.2021055

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 Published  April 2022 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 and Imaging, 2022, 16 (2) : 397-415. doi: 10.3934/ipi.2021055
References:
[1]

H. AmmariG. 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.

[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.

[3]

A. Baker, Transcendental Number Theory, 2$^{nd}$ edition, Cambridge University Press, 1990.

[4]

G. BaoC. 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.

[5]

G. BaoC. Chen and P. Li, Inverse random source scattering for elastic waves, SIAM J. Numer. Anal., 55 (2017), 2616-2643.  doi: 10.1137/16M1088922.

[6]

G. BaoS.-N. ChowP. 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.

[7]

G. BaoJ. 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.

[8]

G. BaoJ. 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.

[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. 

[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.

[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.

[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.

[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.

[14]

A. Devaney, The inverse problem for random sources, J. Math. Phys., 20 (1979), 1687-1691.  doi: 10.1063/1.524277.

[15]

A. Devaney, Inverse source and scattering problems in ultrasonics, IEEE Trans. Sonics Ultrason., 30 (1983), 355-363. 

[16]

M. de HoopL. 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.

[17]

M. ErraouiY. Ouknine and D. Nualart, Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet, Stoch. Dyn., 3 (2003), 121-139.  doi: 10.1142/S0219493703000681.

[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.

[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.

[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.

[21]

V. Isakov, Inverse Source Problems, American Mathematical Society, 1990. doi: 10.1090/surv/034.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[31]

X. Liu and W. Deng, Higher order approximation for stochastic wave equation, arXiv: 2007.02619.

[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.

[33]

E. MarengoM. 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.

[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.

[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.

[36]

D. Nualart, The Malliavin Calculus and Related Topics, Probability and Its Applications, 2$^nd$ edition, Springer-Verlag, Berlin, 2006.

[37]

E. Orsingher, Randomly forced vibrations of a string, Ann. Inst. H. Poincaré Sect. B (N.S.), 18 (1982), 367-394. 

[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.

[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.

[40]

W. A. Strauss, Partial Differential Equations: An Introduction, 2$^{nd}$ edition, John Wiley & Sons, Ltd., Chichester, 2008.

[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.

[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.

[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.

[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.

[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.

show all references

References:
[1]

H. AmmariG. 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.

[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.

[3]

A. Baker, Transcendental Number Theory, 2$^{nd}$ edition, Cambridge University Press, 1990.

[4]

G. BaoC. 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.

[5]

G. BaoC. Chen and P. Li, Inverse random source scattering for elastic waves, SIAM J. Numer. Anal., 55 (2017), 2616-2643.  doi: 10.1137/16M1088922.

[6]

G. BaoS.-N. ChowP. 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.

[7]

G. BaoJ. 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.

[8]

G. BaoJ. 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.

[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. 

[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.

[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.

[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.

[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.

[14]

A. Devaney, The inverse problem for random sources, J. Math. Phys., 20 (1979), 1687-1691.  doi: 10.1063/1.524277.

[15]

A. Devaney, Inverse source and scattering problems in ultrasonics, IEEE Trans. Sonics Ultrason., 30 (1983), 355-363. 

[16]

M. de HoopL. 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.

[17]

M. ErraouiY. Ouknine and D. Nualart, Hyperbolic stochastic partial differential equations with additive fractional Brownian sheet, Stoch. Dyn., 3 (2003), 121-139.  doi: 10.1142/S0219493703000681.

[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.

[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.

[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.

[21]

V. Isakov, Inverse Source Problems, American Mathematical Society, 1990. doi: 10.1090/surv/034.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[31]

X. Liu and W. Deng, Higher order approximation for stochastic wave equation, arXiv: 2007.02619.

[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.

[33]

E. MarengoM. 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.

[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.

[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.

[36]

D. Nualart, The Malliavin Calculus and Related Topics, Probability and Its Applications, 2$^nd$ edition, Springer-Verlag, Berlin, 2006.

[37]

E. Orsingher, Randomly forced vibrations of a string, Ann. Inst. H. Poincaré Sect. B (N.S.), 18 (1982), 367-394. 

[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.

[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.

[40]

W. A. Strauss, Partial Differential Equations: An Introduction, 2$^{nd}$ edition, John Wiley & Sons, Ltd., Chichester, 2008.

[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.

[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.

[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.

[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.

[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.

Figure 1.  The relative errors of the reconstruction for $ f $ (left) and $ g^2 $ (right) with respect to $ N $ ($ H = 0.9, \delta = 0.001 $)
Figure 2.  The reconstruction for $ f $ (left column) and $ g^2 $ (right column) for different $ N = 50, 100 $ ($ H = 0.9, \delta = 0.001 $)
Figure 3.  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 $)
Table 1.  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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