A Directly Numerical Algorithm for a Backward Time-Fractional Diffusion Equation Based on the Finite Element Method

We study a backward problem for a time-fractional diffusion equation, which is formulated into a regularized optimization problem. After solving a sequence of well-posed direct problems by the finite element method, a directly numerical algorithm is proposed for solving the regularized optimization problem. In order to obtain a reasonable regularization solution, we utilize the discrepancy principle with decreasing geometric sequence to choose regularization parameters. Oneand two-dimensional examples are given to verify the efficiency and stability of the proposed method.


Introduction
Nowadays, there is increasing attention on fractional diffusion equations which can be used to describe anomalous diffusion phenomena instead of classical diffusion process.These new fractional-order models are more efficient than the integer-order models, because the fractional-order derivatives and integrals enable the description of the memory and hereditary properties of different substance [1].By an argument similar to the derivation of the classical diffusion equation from Brownian motion, one can derive a fractional diffusion equation from continuous-time random walk.For example, in paper [2] the authors illustrated a fractional diffusion with respect to a non-Markovian diffusion process, while the authors discussed continuous-time random walks on fractals in paper [3].
We notice that mathematical and numerical analysis of the direct problems of the time-fractional diffusion equations has aroused wide concern in recent years; see [4][5][6][7][8][9][10] and references therein.At the same time, the inverse problems for the time-fractional diffusion equations have attracted more and more attention, not only for theoretical analysis but also for popular applications.The authors concluded that there exists a unique weak solution for the backward time-fractional diffusion equation problem under the overdetermined condition (, ) ∈  2 (Ω) ∩  1 0 (Ω) in paper [4].The authors of papers [11][12][13] considered the backward problem of the time-fractional diffusion equation and proposed, respectively, a quasi-reversibility method, an optimization method, and a data regularization method for reconstructing the initial value.Inverse source problems for time-fractional diffusion equations were studied by using the method of the eigenfunction expansion [14], the integral equation method [15], and the separation of variables method [16], respectively, for recovering the space-dependent or time-dependent source term.In [17], the authors recovered the temperature function from one measured temperature at one interior point of a one-dimensional semi-infinite fractional diffusion equation based on Dirichlet kernel mollification techniques.The authors studied an inverse problem of identifying a spatially varying potential term in a onedimensional time-fractional diffusion equation from the flux measurements taken at a single fixed time corresponding to a given set of input sources in [18].Recently, for determining the space-dependent source in a parabolic equation, the authors [19] proposed a regularized optimization method together with the linear model function method [19,20] for choosing regularization parameters.Inspired by this noniterative optimization method, we develop it to solve the backward problem for a time-fractional diffusion equation in this paper.
Let  be a constant such that 0 <  < 1.We consider the following time-fractional diffusion equation: with homogeneous boundary condition and initial condition where Ω is a bounded domain in   ( ≥ 1) and  is symmetric uniformly elliptic operator given by that is, there exists a constant V > 0, such that V ∑    2  ≤ ∑  ,=1  ,     ,  ∈ Ω, and  ∈ R  .The coefficients satisfy Here,   (, )/  is the Caputo fractional derivative which is defined by where Γ(1 − ) is the Gamma function.
If the function () and the coefficients in (1) are all known, problem (1)-( 3) is the so-called direct problem that can be solved stably by the finite element method, the finite difference, the spectrum method, and so forth.Here, we focus on the backward problem; that is, we try to determine the initial value () by the additional data () which is the measurement of the exact value (, ) and satisfies for some known error level  > 0. As we all know, the backward problem is ill-posed, which means that the solution does not depend continuously on the given data and any small perturbation in the given data may cause large change to the solution.For overcoming the ill-posedness we will adopt Tikhonov regularization in our treatment.The rest of the paper is organized as follows.In Section 2, we reformulate the direct problem in a weak and variational sense.Then we formulate the inverse problem into a regularized optimization problem in Section 3. In Section 4, we give implementations of the regularized optimization method.Finally, numerical results are given to illustrate the efficiency and stability of the proposed method.
The following two propositions will be used in the context.

The Regularized Optimization Problem
In this section, we will propose a regularized optimization method together with its implementations for solving the considered backward problem.
Clearly, the forward operator  is a linear map and has the following property.
Lemma 5.The operator  is a well-defined bounded linear operator from  2 (Ω) to  2 (Ω).Moreover, it is injective and compact.
Results of Lemma 5 show that the backward problem is ill-posed due to the compactness of operator .Thus, regularization is necessary for recovering the initial value ().To this end, we consider a Tikhonov functional as where () ∈  2 (Ω) and  is a regularization parameter balancing the fidelity term and the smoothness of the solution.Due to the  2 -regularization term (/2)‖‖ 2  2 (Ω) , the cost functional () is strongly convex.Subsequently, the unique existence of the minimizer can be obtained by standard arguments.Theorem 6.There exists a unique minimizer  ⋆ to () for any given  > 0. Now, we formulate the backward problem into the following minimization problem: 3.2.Finite Element Method Approximation.Obviously, problem ( 18) is a function space minimization problem.Here, we use the finite element method to approximate it.Similar to that done in [19,21], we first triangulate the domain Ω with a regular triangulation  ℎ of simplicial elements; let be the set of the nodes, and define  ℎ to be the continuous piecewise linear finite element space defined over  ℎ ; that is, Then any  ℎ ∈  ℎ can be repeated as =0     , where   is the value of  ℎ () at point   , and   is the pyramid function; that is, Next, we need to consider the discretization of the bounded linear operator .We will adopt the discrete Galerkin method to solve the direct problem (1)-( 3).The time interval [0, ] is partitioned into  2 equal subintervals by using nodal points 0 =  0 <  1 < ⋅ ⋅ ⋅ <   2 −1 <   2 = , with   = ,  = / 2 .Then, the time-fractional derivative   (, )/  at   is estimated by where   =  1− − ( − 1) 1− ,  = 1, 2, . . ., ,  = 1, 2, . . .,  2 .Denote by   ℎ ∈  ℎ the approximation of (⋅,   ) and Now we define the fully discrete finite element method by (24) Theorem 7. Let  and   ℎ be the weak solution of ( 1)-( 3) and the discrete Galerkin finite element solution of (23), respectively.Then there is a constant  > 0 such that, for 0 <  < 1,      (⋅,   ) −   ℎ      2 (Ω) ≤  ( 2− + ℎ) ,  = 1, 2, . . .,  2 , (25 where  is independent of ℎ, , and .
The proof of Theorem 7 follows the same lines as the proof of Theorem 2.1 in [22].So, we omit it.

Implementations of the Regularized Optimization Method.
Applying the interpolation of finite element, the initial value function () can be written approximately in the finite element form of where   := (  ).Due to the linearity of the homogeneous governing equation and the homogeneous boundary condition, we easily see that problem ( 1)-( 3) satisfies the principle of superposition.Here, we also use this principle of superposition to formulate the continuous problem ( 18) into the following discrete problem: min where   ℎ, ,  = 1, 2, . . .,  2 , is the finite element solution of (, ) and satisfies for any   ∈ ∘  ℎ and  = 1, 2, . . .,  2 , where Therefore, numerical solving of the backward problem is essential to determine the ( 1 + 1)-dimensional real vector Φ = [ 0 , . . .,   1 ]  .From the necessary condition for minimizing the approximation function ( ℎ ), that is, we obtain a linear algebraic system Let Φ * = [ * 0 ,  * 1 , . . .,  *  1 ]  be the solution of (31) for a given regularization parameter .Then, we obtain the approximation solution of  as follows:

Method for Choosing Regularization Parameters
As we all know, the backward problem for determining the initial value is an ill-posed problem; that is, the round-off errors and the measurement noises may be highly amplified due to the choice of an unreasonable regularization parameter, therefore making the regularization solution completely useless [19,20].Because of the important role of regularization parameters, a good strategy for selecting regularization parameters should be taken in the computational process.For a fixed 0 <  < 1 and  0 > 0, we consider a geometric sequence of regularization parameters Then, we employ the discrepancy principle to choose a regularization parameter   * after  * steps with where   ℎ (, ) is the finite element solution with respect to  ℎ and   .

Figure 1 :
Figure 1: Comparison between exact solution and inverse solution for Example 1.

Figure 2 :
Figure 2: Comparison between exact solution and inverse solution for Example 2.