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: October 31, 2019

Revised: May 31, 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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