Weak Unique Continuation Property and a Related Inverse Source Problem for Time-Fractional Diffusion-Advection Equations
In this paper, we first establish a weak unique continuation property for time-fractional diffusion-advection equations. The proof is mainly based on the Laplace transform and the unique continuation properties for elliptic and parabolic equations. The result is weaker than its parabolic counterpart in the sense that we additionally impose the homogeneous boundary condition. As a direct application, we prove the uniqueness for an inverse problem on determining the spatial component in the source term in by interior measurements. Numerically, we reformulate our inverse source problem as an optimization problem, and propose an iteration thresholding algorithm. Finally, several numerical experiments are presented to show the accuracy and efficiency of the algorithm.
|Keywords||fractional diffusion equation, weak unique continuation, inverse source problem, iterative thresholding algorithm|
AMS Subject Classifications 35R11, 26A33, 35R30
Let be an open bounded domain with a sufficiently smooth boundary (e.g., of -class) and . Let and () be positive constants such that . By we denote the Caputo derivative (see, e.g., [26, §2.4.1])
where stands for the Gamma function. For , we define the operator
Here is a symmetric second-order elliptic operator which will be defined at the beginning of Section Weak Unique Continuation Property and a Related Inverse Source Problem for Time-Fractional Diffusion-Advection Equations, and . Without loss of generality, we set . In this paper, we investigate the following initial-boundary value problem for the time-fractional diffusion-advection equation
where denotes the normal derivative associated with the elliptic operator . The conditions on the initial data , the source term , coefficients involved in and the definitions of will be specified later in Section Weak Unique Continuation Property and a Related Inverse Source Problem for Time-Fractional Diffusion-Advection Equations.
In various forms and generalities, the time-fractional parabolic operator in (1.1) has gained increasing popularity among mathematicians within the last few decades, owing to its applicability in describing the anomalous diffusion phenomena in highly heterogeneous media (see [1, 9] and the references therein). The fundamental theory for the single-term (i.e., ) case of (1.1) was established around the early 2010s, represented by the maximum principle proved in Luchko  and the well-posedness, analyticity and asymptotic behavior proved in Sakamoto and Yamamoto . Thereafter, most of the properties were parallelly generalized to the multi-term case (i.e., ) in [23, 12, 13], and especially the maximum principle was recently improved to stronger ones in [21, 19]. Meanwhile, corresponding numerical methods have also been well-developed and we refer e.g. to [11, 10]. In contrast to the usual parabolic equations characterized by the exponential decay in time and Gaussian profile in space, it reveals that the fractional diffusion equations driven by possess properties of slow decay in time and long-tailed profile in space. Nevertheless, we notice that most of the existing literature only treated the symmetric elliptic operator (i.e., in (1.1)), in which the existence of eigensystem provides convenience for the argument.
Other than the above mentioned aspects, the unique continuation property is also one of the remarkable characterizations of parabolic equations, which asserts the vanishment of a solution to a homogeneous problem in an open subset implies its vanishment in the whole domain (see, e.g., ). The unique continuation property is not only important by itself, but also significant in its applications to many related control and inverse problems. However, the publications on its generalization to fractional diffusion equations are rather limited to the best of the authors’ knowledge. For the special half-order fractional diffusion equation (i.e., , and in (1.1)), the unique continuation property was proved in Xu, Cheng and Yamamoto  for and Cheng, Lin and Nakamura  for via Carleman estimates for the operator . For a general fractional order in the interval, Lin and Nakamura  recently obtained a unique continuation property by using a newly established Carleman estimate based on calculus of pseudo-differential operators. We notice that the conclusion in  requires the homogeneous initial condition, which possibly roots in the memory effect of time-fractional diffusion equations.
Regarding the unique continuation property, the first focus of this paper is the investigation of the following problem.
Let be the solution of (1.2), where the source term . Then does in some open subset of implies in under certain conditions?
In Theorem 2.5, we will give an affirmative answer to this problem. Compared with the existing literature, we formulate the problem on the more general time-fractional parabolic operator with non-symmetric elliptic part in space. Meanwhile, we allow non-vanishing initial data at the cost of the homogeneous Dirichlet or Neumann boundary condition.
On the other hand, parallelly with the intensive attention paid to forward problems for time-fractional diffusion equations, there are also rapidly growing publications on the related inverse problems with various combinations of unknown functions and observation data. Here we do not intend to give a full list of bibliographies, but just mention [5, 16, 24, 33, 15, 14] and the references therein for readers’ curiosity. Nevertheless, it turns out that the majority of them concentrate on coefficient inverse problems. In contrast, the study on inverse source problems is far from satisfactory and mainly restricts to several special cases due to the lack of specified techniques. In the one-dimensional case, Zhang and Xu  proved the uniqueness for determining a time-independent source term by the partial boundary data, and a conditional stability for the recovery of the spatial component in the source term was proved for the half-order case in Yamamoto and Zhang . With the final overdetermining data, Sakamoto and Yamamoto  showed the generic well-posedness for reconstructing the spatial component. Similarly to the situation of the forward problems reviewed above, it reveals that almost all papers treating the related inverse problems also rely heavily on the symmetry of the involved elliptic operator, regardless of the practical importance of the non-symmetric case.
Keeping the above points in mind, we are also interested in studying the following inverse source problem, which is the second focus of this paper.
Let be the solution of (1.2), where the initial data and the source term takes the form of separated variables, namely . Provided that the temporal component is known, can we uniquely determine the spatial component by the partial interior observation of in some open subset of under certain conditions?
Theorem 2.6 answers this problem affirmatively. Obviously, the above problem is closely related to Problem 1.1 in the sense that both are concerned with the partial interior information of the solution. Practically, the formulation of Problem 1.2 is applicable in the determination of the space distribution modeling the contaminant source, where the anomalous diffusion phenomena is described by (1.2) and the time evolution of the contaminant is known in advance. As far as the authors know, the above problem has not yet been considered in form of the generalized time-fractional parabolic operator .
By restricting the open subset in Problems 1.1–1.2 as a cylindrical subdomain, first we will give an affirmative answer to Problem 1.1 in two cases, that is, either the multi-term fractional diffusion equation without an advection term or the single-term one with an advection term. The statement concluded in Theorem 2.5 will be called as the weak unique continuation property because we impose the homogeneous Dirichlet or Neumann boundary condition, which is absent in the usual parabolic prototype. As a direct application, the uniqueness for Problem 1.2 can be immediately proved with the aid of a fractional version of Duhamel’s principle. For the numerical reconstruction, we reformulate Problem 1.2 as an optimization problem with Tikhonov regularization. After the derivation of the corresponding variational equation, we can characterize the minimizer by employing the associated backward fractional diffusion equation, which results in an efficient iterative method.
The remainder of this paper is organized as follows. Preparing all necessities about the weak solution of (1.2), in Section Weak Unique Continuation Property and a Related Inverse Source Problem for Time-Fractional Diffusion-Advection Equations we state the main results answering Problems 1.1 and 1.2 in Theorems 2.5 and 2.6, respectively. Then Section Weak Unique Continuation Property and a Related Inverse Source Problem for Time-Fractional Diffusion-Advection Equations is devoted to the proofs of the above theorems. In Section Weak Unique Continuation Property and a Related Inverse Source Problem for Time-Fractional Diffusion-Advection Equations, we propose the iterative thresholding algorithm for the numerical treatment of our inverse source problem, followed by several numerical examples illustrating the performance of the proposed method in Section Weak Unique Continuation Property and a Related Inverse Source Problem for Time-Fractional Diffusion-Advection Equations. As technical details, we provide the proofs for the well-posedness of the weak solutions of (1.2) in Appendix Weak Unique Continuation Property and a Related Inverse Source Problem for Time-Fractional Diffusion-Advection Equations.
sectionPreliminaries and Main Results
In this section, we first set up notations and terminologies, and review some of standard facts on the fractional calculus. Let be a usual -space with the inner product and , , etc. denote the usual Sobolev spaces. Especially, for we define the fractional Sobolev space in time (see Adams ). The elliptic operator is defined for or as
where and denotes the outward unit normal vector to at . Here we assume (), and there exists a constant such that
When the zeroth order coefficient in , we introduce the eigensystem of such that , () and forms a complete orthonormal basis of . Considering the possibility of , we define . Then the corresponding eigenvalues are all strictly positive, and the fractional power is defined for (e.g., ) as
and is a Hilbert space with the norm
On the other hand, the first order coefficient in the operator is assumed to be in .
By we denote the Riemann-Liouville integral operator, which is defined by
Then the Caputo derivative can be rephrased as . Parallelly, we define the backward Riemann-Liouville integral operator by
and the backward Riemann-Liouville fractional derivative by .
First we state the well-posedness result for the homogeneous case of the initial-boundary value problem (1.2).
Assume , and let be a given constant. Then there exists a unique solution to the problem (1.2). Moreover, the solution is analytic and can be analytically extended to . Further, there exists a constant such that
Now we turn to the inhomogeneous problem, i.e., and . Since [7, Theorem 1.1] asserts the regularity of the solution, we see that the initial value becomes delicate in the case of because the time-regularity does not make sense pointwisely anymore. Following the same line as that in , we shall redefine the weak solution to (1.2).
Definition 2.3 (Weak solution)
Let . We say that is a weak solution to the initial-boundary value problem (1.2) with if
Here denotes the inverse operator of the Riemann-Liouville integral operator .
Within this framework, we can prove the following well-posedness result.
Lemma 2.4 (Well-posedness of Definition 2.3)
Let and . Then the initial-boundary value problem (1.2) admits a unique weak solution . Moreover, there exists a constant such that
The proof of the above lemma will be given in Appendix Weak Unique Continuation Property and a Related Inverse Source Problem for Time-Fractional Diffusion-Advection Equations.
By Lemma 2.1 and the unique continuation for elliptic and parabolic equations, we can establish the weak type unique continuation for the fractional parabolic equation.
Let , and be the solution to (1.2). Let be an arbitrarily chosen open subdomain. Then
in either of the following two cases.
Case 1 , i.e., is a single-term time-fractional parabolic operator.
Case 2 and in , i.e., the first order coefficient in vanishes and the zeroth order one is non-negative.
Sakamoto and Yamamoto  proved Theorem 2.5 for the symmetric single-term time-fractional diffusion equation by use of the eigenfunction expansion and the unique continuation property for elliptic equations. However, their method cannot work for the non-symmetric counterpart because their argument relies heavily on the symmetry of the elliptic operator.
As an immediate application of the above property, we can give a uniqueness result for Problem 1.2.
Proof of Theorem 2.5.
According to our assumptions and Lemma 2.1, the solution to the initial-boundary value problem (1.2) can be analytically extended from to . For simplicity, we still denote the extension by . Then we arrive at the following initial-boundary value problem
and the condition in implies
immediately. We divide the proof into the two cases described in Theorem 2.5.
Case 1 . For simplicity, we write . We perform the Laplace transform (denoted by ) in (3.1) and use the formula
to derive the transformed algebraic equation
with a parameter , where is a sufficiently large constant. Multiplying both sides of the above equation by and setting , we then obtain the following boundary value problem for an elliptic equation
On the other hand, let us consider an initial-boundary value problem for a parabolic equation
Again, applying the Laplace transform yields
where the parameter and is a sufficiently large constant. After the change of variable , we find
In comparison with (3.3), it follows from the uniqueness result for boundary value problems of elliptic type that
Since (3.2) gives in , we conclude from the above identities that
Consequently, the uniqueness of the inverse Laplace transform indicates in . According to the unique continuation property for parabolic equations (see, e.g., ), we conclude in and thus in . Now that the initial value vanishes, it is readily seen that in from the uniqueness of the solution to (1.2), which completes the proof of the first part of Theorem 2.5.
where and is a sufficiently large constant. Then it follows from (3.2) that
Setting , we see that varies over some domain as varies over . Therefore, we obtain
Moreover, it is readily seen that (3.4) holds for . Then for any , we can take a sufficiently small circle centered at which does not include distinct eigenvalues, and integrating (3.4) on this circle yields
Since satisfies the elliptic equation in , the unique continuation for elliptic equations implies in for all . By the orthogonality of in , we conclude for all and thus in , which indicates in again by the uniqueness of the solution to (1.2). This completes the proof of Theorem 2.5. ∎
Now let us turn to the proof of the uniqueness of the inverse source problem. The argument is mainly based on the weak unique continuation and the following Duhamel’s principle for time-fractional parabolic equations.
Lemma 3.1 (Duhamel’s principle)
Let and , where and . Then the weak solution to the initial-boundary value problem (1.2) allows the representation
where solves the homogeneous problem
and is the unique solution to the fractional integral equation
The above conclusion is almost identical to [21, Lemma 4.1] for the single-term case and [19, Lemma 4.2] for the multi-term case, except for the existence of non-symmetric part. Since the same argument still works in our setting, we omit the proof here.
Proof of Theorem 2.6.
Let satisfy the initial-boundary value problem (1.2) with and , where and . Then takes the form of (3.5) according to Lemma 3.1. Performing the linear combination of the Riemann-Liouville integral operators to (3.5), we deduce
where we applied Fubini’s theorem and used the relation (3.7). Then the vanishment of in immediately yields
Differentiating the above equality with respect to , we obtain
Owing to the assumption that , we estimate
sectionIterative Thresholding Algorithm
Based on the theoretical uniqueness result explained in the previous sections, this section mainly aims at developing an effective numerical method for Problem 1.2, that is, the numerical reconstruction of the spatial component of the source term in a time-fractional parabolic equation.
As a representative, in the sequel we consider the initial-boundary value problem for a single-term time-fractional diffusion equation with the homogeneous Neumann boundary condition
for any test function with in , where stands for the backward Riemann-Liouville derivative. This is easily understood in view of integration by parts and the following lemma.
For and , there holds
Henceforth, we specify as the true solution to Problem 1.2 and investigate the numerical reconstruction by the noise contaminated observation data in satisfying , where stands for the noise level. For avoiding ambiguity, we interpret out of so that it is well-defined in .
By a classical Tikhonov regularization technique, the reconstruction of the source term can be reformulated as the minimization of the following output least squares functional
where is the regularization parameter. As the majority of efficient iterative methods do, we need the information about the Fréchet derivative of the objective functional . For an arbitrarily fixed direction , it follows from direct calculations that
Here denotes the Fréchet derivative of in the direction , and the linearity of (4.1) immediately yields
In order to reduce the computational costs for computing the Fréchet derivatives, we follow the argument used in  to introduce the adjoint system of (4.1), that is, the following system for a backward time-fractional diffusion equation
Parallelly to Definition 2.3, we give the definition of the weak solution to the backward fractional diffusion equation with Riemann-Liouville derivatives.
Let . We say that is a weak solution to (4.5) if
Correspondingly, we can also show the well-posedness of the solution defined above as that in Lemma 2.4.
Lemma 4.3 (Well-posedness for Definition 4.2)
Let . Then the problem (4.5) admits a unique weak solution such that . Moreover, there exists a constant such that
In a similar manner of the proof of [8, Proposition 4.1], one can also show Lemma 4.3 by using the eigenfunction expansion. For conciseness, we omit the proof here. On the other hand, from Lemma 4.3 and integration by parts, it turns out that the solution to problem (4.5) satisfies
for any test function with in .
Here denotes the characterization function of , and we write the solution of (4.7) as . Then for any , it follows from Lemma 4.1 and Remark 2.2 that and can be taken as mutual test functions in definitions (4.2) and (4.6). In such a manner, we can further treat the first term in (4.4) as
This suggests a characterization of the solution to the minimization problem (4.3).
Adding () to both sides of (4.8) and rearranging in view of the iteration, we are led to the iterative thresholding algorithm
At this stage, we are well prepared to propose the iterative thresholding algorithm for the reconstruction.
By [6, Theorem 3.1], we see that the sequence generated by the iteration (4.9) converges strongly to the solution of the minimization problem (4.3). Meanwhile, we can also see from (4.9) that at each iteration step, we only need to solve the forward problem (4.1) once for and the backward problem (4.7) once for subsequently. As a result, the numerical implementation of Algorithm 4.5 is easy and computationally cheap. Moreover, although (4.7) involves the backward Riemann-Liouville derivative, we know that the solution coincides with the following problem with a backward Caputo derivative
In this section, we will apply the iterative thresholding algorithm established in the previous section to the numerical treatment of Problem 1.2 in one and two spatial dimensions, that is, the identification of the spatial component in the source term of the initial-boundary value problem (4.1).
To begin with, we assign the general settings of the reconstructions as follows. Without loss of generality, in (4.1) we set
With the true solution , we produce the noisy observation data by adding uniform random noises to the true data, i.e.,
Here denotes the uniformly distributed random number in and is the noise level. Throughout this section, we will fix the known temporal component in the source term, the regularization parameter and the initial guess as
respectively. We shall demonstrate the reconstruction method by abundant test examples in one and two spatial dimensions. Other than the illustrative figures, we mainly evaluate the numerical performance by the relative -norm error