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

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  • 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.

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

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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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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