Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method

In this paper, we propose a modified quasi-boundary value method to solve an inverse source problem for a time fractional diffusion equation. Under some boundedness assumption, the corresponding convergence rate estimates are derived by using an a priori and an a posteriori regularization parameter choice rules, respectively. Based on the superposition principle, we propose a direct inversion algorithm in a parallel manner.


(Communicated by Gang Bao)
Abstract. In this paper, we propose a modified quasi-boundary value method to solve an inverse source problem for a time fractional diffusion equation. Under some boundedness assumption, the corresponding convergence rate estimates are derived by using an a priori and an a posteriori regularization parameter choice rules, respectively. Based on the superposition principle, we propose a direct inversion algorithm in a parallel manner.
1. Introduction. In this paper, we consider an inverse source problem for a time fractional diffusion equation with time independent source term as follows: u(x, t) = 0, x ∈ ∂Ω, t ≥ 0, with the final data
−L is a symmetric uniformly elliptic operator defined on D(−L) = H 1 0 (Ω) ∩ H 2 (Ω) given by i.e., there exists a constant v > 0, such that v The coefficients satisfy: In practical measurement, noise is inevitable, we can only obtain the terminal observation data gδ(x) with noise polluted, whereδ is the noise level such that We notice that the inverse problems for the time fractional diffusion equation have attracted more and more researchers' attention. In [3], Cheng et al. gave a uniqueness result of determining the order of fractional derivative and diffusion coefficient in a fractional diffusion equation. In [25,19,15], the method of the eigenfunction expansion, the integral equation method and the separation of variables method were adopted respectively to recover the space-dependent or time-dependent source term for time fractional diffusion equation. In [9,21,17,24], some backward problems were investigated. For inverse potential problems, we refer to [8,10,22].
The quasi-boundary value method, also called the non-local boundary value problem method in [7], has been extensively used to solve the inverse problems of different equations, such as time fractional diffusion equation [20,18,23], parabolic equation [7,2,4], elliptic and hyper-parabolic equations [6,5,14]. The main idea is to approximate the ill-posed problem by a well-posed problem. In this study, we propose a modified quasi-boundary value method to solve an inverse source problem for a time fractional diffusion equation. The main idea is by constructing a coupled inverse source problem as a regularised problem to approximate the originally inverse source problem. We derive two kinds of convergence rate estimates by using an a priori and an a posteriori regularization parameter choice rule, respectively.
The paper is organized as follows. In Section 2, we give some preliminary results and conditional stability conclusion of the inverse source problem. In Section 3, we propose a modified quasi-boundary value method and derive the convergence estimates under an a priori assumption for the exact solution and an a posteriori regularization parameter choice rule, respectively. In Section 4, we give a direct inversion algorithm for the modified quasi-boundary value method. In final, several numerical examples are given to test the performance of the inversion algorithm.
2. Preliminaries and conditional stability. Throughout this paper, we use the following definition.
Definition 2.1. The two-parameters Mittag-Leffler function is where α and β are arbitrary constants, Γ(·) is the gamma function.
For convenient, we collect some properties of Mittag-Leffler function as follows [11].
(c) For any λ k such that λ k ≥ λ 1 > 0, there have two positive constants c − , c + only depending on α, T, λ 1 such that We prove the following two lemmas which will be used for the proof of convergence rate estimates. Lemma 2.3. For any constants µ > 0, β 0 > 0, p > 0, s ≥ λ 1 > 0, we have where C 1 , C 2 are two positive constants which are only dependent on λ 1 , p and β 0 .
Proof. If 0 < p ≤ 2, we have lim If p > 2, then for s ≥ λ 1 > 0, we have The proof is finished.
where C 3 , C 4 are two positive constants which are only dependent on λ 1 , p, β 1 and β 2 .

ZHOUSHENG RUAN, SEN ZHANG AND SICAN XIONG
Proof. The same argument as used in lemma 2.3 can be used to prove the lemma.
Here we omit the details.
Since −L is the symmetric uniformly elliptic operator, we can assume λ 1 ≤ λ 2 ≤ · · · ≤ λ n ≤ · · · , lim n→∞ λ n = +∞, and {ϕ n (x)} ∞ n=1 becomes an orthonormal basis of space L 2 (Ω). Define where < ·, · > is the scalar product in L 2 (Ω). It is not difficult to derive that the space D((−L) p 2 ) is a Hilbert space with the following norm For any given source function f (x) ∈ L 2 (Ω), by solving Eq. (1) we can formally define a forward linear operator K : . From the lemma 2.1 in [12] or theorem 2.1 in [3], we know g(x) ∈ H 2 (Ω). Then compactness of the map K follows by Sobolev compact embedding theorem 6.61 in [1]. The forward operator K is a smoothing operator and hence the associated inverse problem is illposed. From Theorem 3.1 in [20], we know that there exists a unique solution u(x, t) and f (x) for the inverse source problem (1)-(2) and it has the following conditional stability.
then we have where C 5 is a positive constant only dependent on α, T, p, λ 1 .

3.
A modified quasi-boundary value method and convergence rate estimates. In this section, a modified quasi-boundary value method is proposed to solve problem (1)- (2). The main idea of the modified quasi-boundary value method is replacing inverse source problem (1)-(2) with a coupled inverse source problem (8)-(9) which is well posed, then using the solutions of the coupled problem to construct approximate solutions of (1)- (2).
whereα ∈ (0, 1) is an arbitrary constant, µ is a regularization parameter, T 1 is an arbitrary positive constant. Generally, we take T 1 < T .
Proof. By the method of separation of variables, we can derive the solution to (9) in the form of series as follows where In the same way, we known that the solution u µ (x, t) to (8a)-(8c) can be expressed as follows From (12), we know if g(x) = 0, we have f µ (x) = 0. Further, by (10) and (11), we known u µ (x, t) = 0, v µ (x, t) = 0. This ends the proof.
Remark 1. If T 1 → 0, the coupled regularization problem restores to the inverse source problem (1) -(2) itself. If T 1 → ∞, the above regularization problem is identical to the standard quasi-boundary value method for the inverse source problem (1) - (2). Therefore, in some sense, the regularized approach proposed in this paper can be understood as a generalization of the standard quasi-boundary value method.
3.1. Convergence rate estimate under an a priori regularization parameter choice strategy. We denote (uδ µ (x, t), vδ µ (x, t), fδ µ (x)) as the solutions to the following regularized problem In the same way, we can obtain whereC 1 ,C 2 are two positive constants independent ofδ, but may be dependent on α,α, T, T 1 , λ 1 , c + , c − , p, E.
Proof. By direct computation and lemma 2.3, we can get where C 6 , C 7 are two positive constants only dependent on α,α, T, T 1 , λ 1 , p.
By the triangle inequality, we get then theorem 3.2 holds.

3.2.
Convergence rate estimate under an a posterior regularization parameter choice strategy. As we know, the boundedness E is not easy to obtain when considering a priori regularization parameter selection strategy. The unknown of constant E hampers the selection of the regularization parameter by the a priori rule. We now derive the convergence rate estimate for an a posterior regularisation parameter choice strategy, based on the following Morozov's discrepancy principle: Set (µ) = µ(µI + K) −1 (Kfδ µ − gδ) , then from expressions (15) and (16), we can get From expression (19), we can get the following conclusion.
4. Inversion algorithm. We take the finite element technique to solve the inverse source problem. Triangulating the domain Ω with a regular triangulation of simplicial elements. Let T h be a quasi-uniform triangulation of Ω, {p i } N1 i=0 be the set of the nodes. By the interpolation of finite element, the source term f (x) can be approximated in the finite element form of where f i := f (p i ), ψ i is the pyramid function, i.e.,

Experiments setting.
We take the computational domain Ω = (0, 1) and Ω = (0, 1) × (0, 1) for one dimensional and two dimensional example, respectively. The finite element space V h (for final time measurement) is continuous P 1 over triangulations T h . In one dimensional case, T h is chosen as a uniform partition with h = 1 100 . In two dimensional case, we let T h with 256 triangles. The final time is chosen as T = 1. We take the discrete Galerkin finite element method to compute the final time measurement and the temporal discretization parameter τ is 1 100 and 1 80 in one and two dimensional case, respectively. The fraction time derivative ∂ α u(x,t) ∂t α at t k is approximated as where ω l = l 1−α − (l − 1) 1−α , l = 1, · · · , L. To obtain the (noisy) additional data gδ, we first give the true solution f (x) and solve the direct problem (1), then add pointwise noise by gδ( is the finite element solution of direct problem (1) at the point (x, t) = (p i , T ). δ is the relative noise level and ξ is a uniform random variable in [−1, 1]. The corresponding absolute noise level isδ = δ u L h (·) . Meanwhile, the regularization parameter µ is very crucial for the inversion algorithm 1. Here we apply a continuation strategy for the regularization parameter µ (cf. [16,13]), i.e., given a decreasing sequence {µ k }, µ k = µ 0 r k (0 < r < 1), then we solve the regularization problem (26) with µ = µ k . Once the discrepancy principle (18) is satisfied, we stop the algorithm and let f µ,h (x) be f (x).

Numerical tests.
To observe the performance of the inversion algorithm, we define the error function as and the L 2 relative error Define the convergence rate(C r ) as follows: Example 1. Let α = 0.7, and the exact source function to be sin(πx).
Error curves of numerical inversion for example 1 are showed in figure 1 with relative noise level 0.05%, 0.1%, 0.2% (left) and 0.4%, 0.8%, 1.6%(right), respectively. Table 1 and Table 2 include the relative errors and convergence orders of the inversion solution for Example 1 and Example 2, respectively, with different relative error levels for the additional data gδ(x). The numerical error is decreasing as the level of relative noise becomes smaller and the convergence order is about 0.5, which is consistent to our convergence estimate.