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Backward problem for a time-space fractional diffusion equation

  • * Corresponding author: Jigen Peng

    * Corresponding author: Jigen Peng 
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  • In this paper, a backward problem for a time-space fractional diffusion process has been considered. For this problem, we propose to construct the initial function by minimizing data residual error in Fourier space domain with variable total variation (TV) regularizing term which can protect the edges as TV regularizing term and reduce staircasing effect. The well-posedness of this optimization problem is obtained under a very general setting. Actually, we rewrite the time-space fractional diffusion equation as an abstract fractional differential equation and deduce our results by using fractional operator semigroup theory, hence, our theoretical results can be applied to other backward problems for the differential equations with more general fractional operator. Then a modified Bregman iterative algorithm has been proposed to approximate the minimizer. The new features of this algorithm is that the regularizing term altered in each step and we need not to solve the complex Euler-Lagrange equation of variable TV regularizing term (just need to solve a simple Euler-Lagrange equation). The convergence of this algorithm and the strategy of choosing parameters are also obtained. Numerical implementations are provided to support our theoretical analysis to show the flexibility of our minimization model.

    Mathematics Subject Classification: Primary: 35R30, 35R11; Secondary: 65L09.


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  • Figure 1.  Left: Initial data; Right: The solution of the fractional diffusion equation (2) at time $T = 1$ with $\alpha = 0.6$, $\beta = 0.9$.

    Figure 2.  Left: Boundaries of the initial data; Right: Boundaries of the solution of the fractional diffusion equation (2) at time $T = 1$ with $\alpha = 0.6$, $\beta = 0.9$.

    Figure 3.  Left: Original function; Middle: Recovered function by variable TV model with $\delta = 0.0005$; Right: Recovered function by variable TV model with $\delta = 0.005$ for Example 1.

    Figure 4.  Initial function for Example 2.

    Figure 5.  Left: Recovered function by variable TV model with $\delta = 0.0005$ for Example 2; Right: Recovered function by the variable TV model with $\delta = 0.005$ for Example 2.

    Figure 6.  The curve of the relative error of the recovered data for different values of parameter $\alpha$

    Table 1.  The values of RelErr of three methods for Example 1

    RelErr TV model Tikhonov model Variable TV model
    $\sigma = 0.0005$ $3.8283\%$ $0.3857\%$ $0.3696\%$
    $\sigma = 0.005$ $8.8646\%$ $0.6559\%$ $0.6597\%$
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    Table 2.  The values of RelErr of three methods for Example 2

    RelErr TV model Tikhonov model Variable TV model
    $\sigma = 0.0005$ $13.0053\%$ $13.7772\%$ $13.0666\%$
    $\sigma = 0.005$ $22.7222\%$ $25.2101\%$ $22.7810\%$
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    Table 3.  The values of RelErr with different parameters λ of Variable TV model for Example 2

    $\lambda = 10^{11}$ $\lambda = \frac{1}{4}\times 10^{11}$ $\lambda = \frac{1}{16}\times 10^{11}$
    $\sigma = 0.0005$ $\text{M} = 9$ $\text{M} = 34$ $\text{M} = 150$
    $\text{RelErr} = 13.0792\%$ $\text{RelErr} = 13.0342\%$ $\text{RelErr} = 13.0275\%$
     | Show Table
    DownLoad: CSV
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