December  2020, 14(6): 1001-1024. doi: 10.3934/ipi.2020053

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

Received  November 2019 Revised  June 2020 Published  August 2020

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.

Citation: 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
References:
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show all references

References:
[1]

R. A. Adams and J. J. Fournier, Sobolev Spaces, 140, Elsevier, 2003.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[4]

D. Baleanu and A. M. Lopes, Handbook of Fractional Calculus with Applications, De Gruyter, 2019.  Google Scholar

[5]

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

E. BarkaiR. 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.  Google Scholar

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[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.  Google Scholar

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[14]

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[15]

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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

[17]

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[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.  Google Scholar

[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.  Google Scholar

[21]

D. HouM. 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.  Google Scholar

[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.  Google Scholar

[23]

B. JinR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

Z. LiY. 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.  Google Scholar

[35]

J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. Ⅰ, Springer-Verlag, New York-Heidelberg, 1972.  Google Scholar

[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.  Google Scholar

[37]

Z. LiuF. 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.  Google Scholar

[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.  Google Scholar

[39]

C. LvM. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[42]

D. MurioC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[45]

B. Øksendal, Stochastic Differential Equations, 6$^th$ edition, Universitext, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-14394-6.  Google Scholar

[46] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.   Google Scholar
[47]

S. QasemiD. 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.  Google Scholar

[48]

Z. RuanS. 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.  Google Scholar

[49]

A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003. doi: 10.1088/1361-6420/aaac5a.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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Figure 1.  Experiment $ (e1) $, $ \hat E $ (top) and $ g_1, \hat g_1 $ (bottom) for different amount of samples
Figure 2.  Experiment $ (e1), $ $ \hat E,J_\epsilon * \hat E $ (top) and $ g_1, \hat g_1 $ (bottom) for different $ \epsilon $, $ 10^3 $ realizations
Figure 3.  Experiment $ (e1) $, $ \hat V $ (top) and $ |g_2|,|\hat g_2| $ (bottom) for different amount of samples
Figure 4.  Experiment $ (e1) $, $ \hat V, J_\epsilon *\hat V $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \epsilon $, $ 10^3 $ realizations
Figure 11.  Experiment $ (e1) $, $ er(g_1) $ and $ er(|g_2|) $ under different $ \epsilon $
Figure 5.  Experiment $ (e1) $, $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \epsilon $
Figure 6.  Experiment $ (e1) $, $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $
Figure 7.  Experiment $ (e2) $, $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $
Figure 8.  Experiment $ (e3) $, $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $
Figure 9.  Experiment $ (e4) $, $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $
Figure 10.  Experiment $ (e5) $, $ g_1, \hat g_1 $ (top) and $ |g_2|, |\hat g_2| $ (bottom) for different $ \sigma $
Figure 12.  Experiment $ (e1) $, $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $
Figure 13.  Experiment $ (e2) $, $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $
Figure 14.  Experiment $ (e3) $, $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $
Figure 15.  Experiment $ (e4) $, $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $
Figure 16.  Experiment $ (e5) $, $ er(g_1) $ (left) and $ er(|g_2|) $ (right) for different $ \sigma $
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
$ 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
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
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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