Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation

Chan Liu; Jin Wen; Zhidong Zhang

doi:10.3934/ipi.2020053

Article Contents

2020, Volume 14, Issue 6: 1001-1024. Doi: 10.3934/ipi.2020053

This issue Previous Article Two-dimensional seismic data reconstruction using patch tensor completion Next Article Direct and inverse time-harmonic elastic scattering from point-like and extended obstacles

Reconstruction of the time-dependent source term in a stochastic fractional diffusion equation

1.
School of Mathematical Sciences, Fudan University, Shanghai 200433, China
2.
Department of Mathematics, Northwest Normal University, Lanzhou 730070, Gansu, China
3.
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai 519082, Guangdong, China

^* Corresponding author
^* Corresponding author

Received: November 2019

Revised: June 2020

Early access: August 2020

Published: December 2020

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Abstract

In this work, an inverse problem in the fractional diffusion equation with random source is considered. The measurements we use are the statistical moments of the realizations of single point observation $ u(x_0,t,\omega). $ We build a representation of the solution $ u $ in the integral sense, then prove some theoretical results like uniqueness and stability. After that, we establish a numerical algorithm to solve the unknowns, where a mollification method is used.

Keywords:

Inverse problem,
stochastic fractional diffusion equation,
random source,
Volterra integral equation,
mollification.

Mathematics Subject Classification: 35R11, 35R30, 65C30, 65M32.

Citation:

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Figure 1. Experiment $ (e1) $, $ \hat E $ (top) and $ g_1, \hat g_1 $ (bottom) for different amount of samples

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Figure 2. Experiment $ (e1), $ $ \hat E,J_\epsilon * \hat E $ (top) and $ g_1, \hat g_1 $ (bottom) for different $ \epsilon $, $ 10^3 $ realizations

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Figure 3. Experiment $ (e1) $, $ \hat V $ (top) and $ |g_2|,|\hat g_2| $ (bottom) for different amount of samples

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Figure 4. Experiment $ (e1) $, $ \hat V, J_\epsilon *\hat V $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \epsilon $, $ 10^3 $ realizations

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Figure 11. Experiment $ (e1) $, $ er(g_1) $ and $ er(|g_2|) $ under different $ \epsilon $

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Figure 5. Experiment $ (e1) $, $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \epsilon $

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Figure 6. Experiment $ (e1) $, $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $

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Figure 7. Experiment $ (e2) $, $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $

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Figure 8. Experiment $ (e3) $, $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $

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Figure 9. Experiment $ (e4) $, $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $

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Figure 10. Experiment $ (e5) $, $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $

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Figure 12. Experiment $ (e1) $, $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $

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Figure 13. Experiment $ (e2) $, $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $

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Figure 14. Experiment $ (e3) $, $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $

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Figure 15. Experiment $ (e4) $, $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $

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Figure 16. Experiment $ (e5) $, $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $

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Table 1. Errors without mollification

	$ 10^3 $ samples		$ 10^4 $ samples		$ 10^5 $ samples
	$ er(g_1) $	$ er(\|g_2\|) $	$ er(g_1) $	$ er(\|g_2\|) $	$ er(g_1) $	$ er(\|g_2\|) $
$ (e1) $	0.299794	0.056932	0.091171	0.018218	0.028705	0.009218
$ (e2) $	0.449422	0.084018	0.142783	0.029317	0.046238	0.012576
$ (e3) $	0.447672	0.083309	0.231421	0.043802	0.081429	0.020119
$ (e4) $	1.570322	0.288016	0.503009	0.104327	0.151976	0.055286
$ (e5) $	1.542568	0.310195	0.509957	0.115601	0.171542	0.057249

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Table 2. Errors for different $ \epsilon $, $ 10^3 $ realizations used

	Without mollification		$ \epsilon=0.005 $		$ \epsilon=0.05 $
	$ er(g_1) $	$ er(\|g_2\|) $	$ er(g_1) $	$ er(\|g_2\|) $	$ er(g_1) $	$ er(\|g_2\|) $
$ (e1) $	0.299794	0.056932	0.124498	0.030254	0.050135	0.055231
$ (e2) $	0.449422	0.084018	0.203604	0.039741	0.097088	0.034309
$ (e3) $	0.447672	0.083309	0.216069	0.041439	0.096336	0.033850
$ (e4) $	1.570322	0.288016	0.827782	0.172930	0.266984	0.262352
$ (e5) $	1.542568	0.310195	0.624018	0.195870	0.165958	0.281415

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References

[1]	R. A. Adams and J. J. Fournier, Sobolev Spaces, 140, Elsevier, 2003.
[2]	A. Babaei and S. Banihashemi, Reconstructing unknown nonlinear boundary conditions in a time-fractional inverse reaction-diffusion-convection problem, Numer. Methods Partial Differential Equations, 35 (2019), 976-992. doi: 10.1002/num.22334.
[3]	R. L. Bagley and P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983), 201-210. doi: 10.1122/1.549724.
[4]	D. Baleanu and A. M. Lopes, Handbook of Fractional Calculus with Applications, De Gruyter, 2019.
[5]	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.
[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.
[7]	G. Bao and X. Xu, An inverse random source problem in quantifying the elastic modulus of nanomaterials, Inverse Problems, 29 (2013), 015006. doi: 10.1088/0266-5611/29/1/015006.
[8]	E. Barkai, R. Metzler and J. Klafter, From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E, 61 (2000), 132-138. doi: 10.1103/PhysRevE.61.132.
[9]	B. Berkowitz, A. Cortis, M. Dentz and H. Scher, Modeling non-Fickian transport in geological formations as a continuous time random walk, Rev. Geophys., 44. doi: 10.1029/2005RG000178.
[10]	H. Brunner, Volterra Integral Equations, Cambridge Monographs on Applied and Computational Mathematics, 30, Cambridge University Press, Cambridge, 2017 doi: 10.1017/9781316162491.
[11]	X. Cao, Y.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Probl. Imaging, 13 (2019), 197-210. doi: 10.3934/ipi.2019011.
[12]	X. Cao and H. Liu, Determining a fractional Helmholtz equation with unknown source and scattering potential, Commun. Math. Sci., 17 (2019), 1861-1876. doi: 10.4310/CMS.2019.v17.n7.a5.
[13]	M. Cekić, Y.-H. Lin and A. Rüland, The Calderón problem for the fractional Schrödinger equation with drift, Calc. Var. Partial Differential Equations, 59 (2020), Paper No. 91. doi: 10.1007/s00526-020-01740-6.
[14]	X. Cheng, L. Yuan and K. Liang, Inverse source problem for a distributed-order time fractional diffusion equation, J. Inverse Ill-Posed Probl., 28 (2020), 17-32. doi: 10.1515/jiip-2019-0006.
[15]	B. De Finetti, Theory of Probability: A Critical Introductory Treatment, 6, John Wiley & Sons, 2017. doi: 10.1002/9781119286387.
[16]	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.
[17]	T. Ghosh, A. Rúland, M. Salo and G. Uhlmann, Uniqueness and reconstruction for the fractional Calderón problem with a single measurement, J. Funct. Anal., 279 (2020), 108505. doi: 10.1016/j.jfa.2020.108505.
[18]	T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, Anal. PDE, 13 (2020), 455-475. doi: 10.2140/apde.2020.13.455.
[19]	R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto, Fractional calculus and continuous-time finance. Ⅲ. The diffusion limit, in Mathematical Finance (Konstanz, 2000), Trends Math., Birkh¨auser, Basel, 2001, 171-180.
[20]	B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation Ⅱ. General potentials and stability, SIAM J. Math. Anal., 52 (2020), 402-436. doi: 10.1137/19M1251576.
[21]	D. Hou, M. T. Hasan and C. Xu, Müntz spectral methods for the time-fractional diffusion equation, Comput. Methods Appl. Math., 18 (2018), 43-62. doi: 10.1515/cmam-2017-0027.
[22]	X. Huang, Z. Li and M. Yamamoto, Carleman estimates for the time-fractional advection-diffusion equations and applications, Inverse Problems, 35 (2019), 045003. doi: 10.1088/1361-6420/ab0138.
[23]	B. Jin, R. Lazarov and Z. Zhou, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. Numer. Anal., 36 (2016), 197-221. doi: 10.1093/imanum/dru063.
[24]	B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems, 31 (2015), 035003. doi: 10.1088/0266-5611/31/3/035003.
[25]	B. Kaltenbacher and W. Rundell, On an inverse potential problem for a fractional reaction-diffusion equation, Inverse Problems, 35 (2019), 065004. doi: 10.1088/1361-6420/ab109e.
[26]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B. V., Amsterdam, 2006.
[27]	J. Klafter and R. Silbey, Derivation of the continuous-time random-walk equation, Phys. Rev. Lett., 44 (1980), 55-58. doi: 10.1103/PhysRevLett.44.55.
[28]	R. C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, Trans. ASME J. Appl. Mech., 51 (1984), 299-307. doi: 10.1115/1.3167616.
[29]	R.-Y. Lai and Y.-H. Lin, Global uniqueness for the fractional semilinear Schrödinger equation, Proc. Amer. Math. Soc., 147 (2019), 1189-1199. doi: 10.1090/proc/14319.
[30]	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.
[31]	P. Li, An inverse random source scattering problem in inhomogeneous media, Inverse Problems, 27 (2011), 035004. doi: 10.1088/0266-5611/27/3/035004.
[32]	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.
[33]	Z. Li, X. Cheng and G. Li, An inverse problem in time-fractional diffusion equations with nonlinear boundary condition, J. Math. Phys., 60 (2019), 091502. doi: 10.1063/1.5047074.
[34]	Z. Li, Y. Luchko and M. Yamamoto, Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem, Comput. Math. Appl., 73 (2017), 1041-1052. doi: 10.1016/j.camwa.2016.06.030.
[35]	J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. Ⅰ, Springer-Verlag, New York-Heidelberg, 1972.
[36]	Y. Liu and Z. Zhang, Reconstruction of the temporal component in the source term of a (time-fractional) diffusion equation, J. Phys. A, 50 (2017), 305203. doi: 10.1088/1751-8121/aa763a.
[37]	Z. Liu, F. Liu and F. Zeng, An alternating direction implicit spectral method for solving two dimensional multi-term time fractional mixed diffusion and diffusion-wave equations, Appl. Numer. Math., 136 (2019), 139-151. doi: 10.1016/j.apnum.2018.10.005.
[38]	Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351 (2009), 218-223. doi: 10.1016/j.jmaa.2008.10.018.
[39]	C. Lv, M. Azaiez and C. Xu, Spectral deferred correction methods for fractional differential equations, Numer. Math. Theory Methods Appl., 11 (2018), 729-751. doi: 10.4208/nmtma.2018.s03.
[40]	C. Lv and C. Xu, Error analysis of a high order method for time-fractional diffusion equations, SIAM J. Sci. Comput., 38 (2016), A2699–A2724. doi: 10.1137/15M102664X.
[41]	M. Magdziarz, A. Weron, K. Burnecki and J. Klafter, Fractional Brownian motion versus the continuous-time random walk: A simple test for subdiffusive dynamics, Phys. Rev. Lett., 103 (2009), 180602. doi: 10.1103/PhysRevLett.103.180602.
[42]	D. Murio, C. E. Mejía and S. Zhan, Discrete mollification and automatic numerical differentiation, Comput. Math. Appl., 35 (1998), 1-16. doi: 10.1016/S0898-1221(98)00001-7.
[43]	R. Nigmatullin, The realization of the generalized transfer equation in a medium with fractal geometry, Physica Status Solidi (B), 133 (1986), 425-430. doi: 10.1002/pssb.2221330150.
[44]	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.
[45]	B. Øksendal, Stochastic Differential Equations, 6$^th$ edition, Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.
[46]	I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.
[47]	S. Qasemi, D. Rostamy and N. Abdollahi, The time-fractional diffusion inverse problem subject to an extra measurement by a local discontinuous Galerkin method, BIT, 59 (2019), 183-212. doi: 10.1007/s10543-018-0731-z.
[48]	Z. Ruan, S. Zhang and S. Xiong, Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method, Evol. Equ. Control Theory, 7 (2018), 669-682. doi: 10.3934/eect.2018032.
[49]	A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003. doi: 10.1088/1361-6420/aaac5a.
[50]	W. Rundell and Z. Zhang, Fractional diffusion: Recovering the distributed fractional derivative from overposed data, Inverse Problems, 33 (2017), 035008. doi: 10.1088/1361-6420/aa573e.
[51]	W. Rundell and Z. Zhang, Recovering an unknown source in a fractional diffusion problem, J. Comput. Phys., 368 (2018), 299-314. doi: 10.1016/j.jcp.2018.04.046.
[52]	K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447. doi: 10.1016/j.jmaa.2011.04.058.
[53]	S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon, 1993.
[54]	S. Shen, F. Liu and V. V. Anh, The analytical solution and numerical solutions for a two-dimensional multi-term time fractional diffusion and diffusion-wave equation, J. Comput. Appl. Math., 345 (2019), 515-534. doi: 10.1016/j.cam.2018.05.020.
[55]	T. N. Thach, T. N. Huy, P. T. M. Tam, M. N. Minh and N. H. Can, Identification of an inverse source problem for time-fractional diffusion equation with random noise, Math. Methods Appl. Sci., 42 (2019), 204-218. doi: 10.1002/mma.5334.
[56]	F. G. Tricomi, Integral Equations, Dover Publications, Inc., New York, 1985.
[57]	N. H. Tuan, V. C. H. Luu and S. Tatar, An inverse problem for an inhomogeneous time-fractional diffusion equation: A regularization method and error estimate, Comput. Appl. Math., 38 (2019), Art. 32. doi: 10.1007/s40314-019-0776-x.
[58]	X. B. Yan and T. Wei, Inverse space-dependent source problem for a time-fractional diffusion equation by an adjoint problem approach, J. Inverse Ill-Posed Probl., 27 (2019), 1-16. doi: 10.1515/jiip-2017-0091.
[59]	F. Zeng, C. Li, F. Liu and I. Turner, The use of finite difference/element approaches for solving the time-fractional subdiffusion equation, SIAM J. Sci. Comput., 35 (2013), A2976–A3000. doi: 10.1137/130910865.
[60]	Z. Zhang, An undetermined time-dependent coefficient in a fractional diffusion equation, Inverse Probl. Imaging, 11 (2017), 875-900. doi: 10.3934/ipi.2017041.
[61]	Z. Zhang and Z. Zhou, Recovering the potential term in a fractional diffusion equation, IMA J. Appl. Math., 82 (2017), 579-600. doi: 10.1093/imamat/hxx004.