Abstract
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. One- and two-dimensional examples are given to verify the efficiency and stability of the proposed method.
1. 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 [4–10] 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 in paper [4]. The authors of papers [11–13] 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 one-dimensional 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 . We consider the following time-fractional diffusion equation:with homogeneous boundary conditionand initial conditionwhere is a bounded domain in and is symmetric uniformly elliptic operator given bythat is, there exists a constant , such that , , and . The coefficients satisfyHere, is the Caputo fractional derivative which is defined bywhere 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 . 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.
2. Weak Form and Weak Solution
The weak form of problem (1)–(3) is finding such thatwhere , denotes the inner product in , and
Definition 1. A function is said to be a weak solution of the direct problem (1)–(3) if ; and the weak form (8) is satisfied.
Lemma 2 (see [4]). If , , there exists a unique weak solution ; to problem (1)–(3), and the expression of the weak solution can be formulated by the following eigenfunction expansion:where is the double-parameter Mittag-Leffler function and is defined by and are the Dirichlet eigenvalues and the orthonormal eigenfunctions of symmetric uniformly elliptic operator , respectively.
The following two propositions will be used in the context.
Proposition 3. is a completely monotonic decreasing function for and satisfies
Proposition 4. Let , , and . Then there exists a constant such that
3. The Regularized Optimization Problem
In this section, we will propose a regularized optimization method together with its implementations for solving the considered backward problem.
3.1. Regularized Optimization Functional
From results of Lemma 2, formula (10) gives a uniquely weak solution for any initial value . Naturally, it defines a forward operator 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 to . Moreover, it is injective and compact.
Proof. From Lemma 2, the solution can be represented byFrom Proposition 3, we know that for ; is a very small positive number. By the orthogonality of and Proposition 4, we obtainwhich implies that . Then by Sobolev embedding theorem, we conclude the compactness of the operator .
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 aswhere and is a regularization parameter balancing the fidelity term and the smoothness of the solution. Due to the -regularization term , 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 .
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 with a regular triangulation of simplicial elements; let be the set of the nodes, and define to be the continuous piecewise linear finite element space defined over ; that is,Then any can be repeated as , where is the value of 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 is partitioned into equal subintervals by using nodal points , with , . Then, the time-fractional derivative at is estimated bywhere , , . Denote by the approximation of and
Now we define the fully discrete finite element method byfor any , where . The space , in which all functions vanish on the boundary , is a subspace of . Clearly, (23) is a linear system about , . Subsequently, there exists a discrete linear operator such that
Theorem 7. Let and be the weak solution of (1)–(3) and the discrete Galerkin finite element solution of (23), respectively. Then there is a constant such that, for ,where is independent of , and .
The proof of Theorem 7 follows the same lines as the proof of Theorem 2.1 in [22]. So, we omit it.
3.3. 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 ofwhere . 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:where , , is the finite element solution of and satisfiesfor any and , whereTherefore, numerical solving of the backward problem is essential to determine the -dimensional real vector .
From the necessary condition for minimizing the approximation function , that is,we obtain a linear algebraic systemLet be the solution of (31) for a given regularization parameter . Then, we obtain the approximation solution of as follows:
4. 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 and , we consider a geometric sequence of regularization parametersThen, we employ the discrepancy principle to choose a regularization parameter after steps withwhere is the finite element solution with respect to and .
5. Numerical Examples
In all one-dimensional examples, , we divide into 100 equal subintervals which means that there are 100 elements and 101 nodes, , . In all two-dimensional examples, , we divide into 1024 equal triangle element, , . In the computational process, the measurement vector is obtained actually at the points of the mesh grid and added by randomly distributed perturbations with relative noise level ; that is, . We take and in all numerical examples. The relative error of the inverse solutions is defined by
Example 1. We take , , and . Let the exact initial value for problem (1)–(3) be . Numerical results for relative noise levels 1% and 5% are shown and listed in Figure 1 and Table 1.
(a)
(b)
Example 2. Let the exact initial value for problem (1)–(3) be, , and . Numerical results with the relative noise levels and are shown in Figure 2 and listed in Table 1.
(a)
(b)
Example 3. Let the exact initial value for problem (1)–(3) be . And we take , , and . Numerical results are listed in Table 1 and shown in Figures 3 and 4 with relative noise levels 1% and 5%, respectively.
(a) Inverse solution
(b) Error surface
(a) Inverse solution
(b) Error surface
Example 4. In this example, the exact initial value for problem (1)–(3) is taken as . And let , , and . Numerical results are listed in Table 1 and shown in Figures 5 and 6 with relative noise levels 1% and 5%, respectively.
(a) Inverse solution
(b) Error surface
(a) Inverse solution
(b) Error surface
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by National Natural Science Foundation of China (11161002), Young Scientists Training Project of Jiangxi Province (no. 20122BCB23024), Natural Science Foundation of Jiangxi Province of China (no. 20142BAB201008), Ground Project of Science and Technology of Jiangxi Universities (no. KJLD14051), and National High-Tech R&D Program of China (2012AA061504).