In this paper, we study the final value problem
where we aim at recovering the initial distribution given the final distribution . In (1), and denotes the Caputo fractional derivative [kilbas2006theory] defined as
where is nothing but the Gamma function : .
Time-fractional diffusion equations usually model sub-diffusion processes such as slow and anomalous diffusion processes which failed to be described by classical diffusion models [balakrishnan1985anomalous, chechkin2005fractional, metzler2000random, podlubnv1999fractional]. Due to the high diversity of such phenomena which are not properly modeled by classical diffusion, time-fractional diffusion problems have gained much attention in last decades. Beyond applications in diffusion processes, time-fractional equation (1) has also been applied to image de-blurring [wang2013total] where the time-fractional derivative allows to capture the memory effect in image blurring.
It is well-known that the ill-posedness of equation (1) comes from the irreversibility of time of the diffusion equation, which is caused by the very smoothing property of the forward diffusion. As a result, very small perturbation of the final distribution may cause arbitrary large error in the initial distribution . Hence, a regularization method is crucial in order to recover stable approximate of the initial distribution . In this regard, many regularization methods have been applied to final value time-fractional diffusion equation. Let us mention the mollification method [van2020mollification, yang2014mollification], Fourier regularization [xiong2012inverse, yang2015fourier], the method of quasi-reversibility [liu2010backward], Tikhonov method [wang2015optimal], total variation regularization [wang2013total], boundary condition regularization [yang2013solving], non-local boundary value method [hao2019stability], truncation method [wang2012data]. Yet, the set of regularization methods applied to backward time-fractional diffusion equation still presents some sparsity, especially compared to the set regularization methods for backward classical diffusion problems.
In this paper, we describe how in the context of regularization of the final value time-fractional diffusion equation (1), the Fourier regularization [yang2015fourier] and mollification [van2020mollification] are nothing but examples of approximate-inverse [louis1990mollifier] regularization. Next, we investigate a regularization technique which yields a better trade-off between stability and accuracy compared to the Fourier regularization [yang2015fourier] and the mollification technique of Van Duc N. et al [van2020mollification]. We consider noisy setting where is approximated by a noisy data satisfying
and we derive order-optimal convergence rates between our approximate solution and the exact solution under classical Sobolev smoothness condition
We also provide error estimates under the more realistic setting where both the data and the forward diffusion operator are only approximately known. The motivation here being that, in practice, the Mittag-Leffler function which plays a major role in the resolution of equation (1) can only be approximated in practice.
The outline of this article is as follows:
In Section 2, we discuss existence of solution of equation (1) and reformulate the equation into an operator equation of the form in which is a bounded linear operator on . We present key estimates necessary for the regularization analysis and illustrate the ill-posedness of recovering from . Next, we introduce the framework of regularization which includes Fourier regularization, mollification, and approximate inverse. At last, we introduce our regularization approach.
Section 3 deals with error estimates and order-optimality of our regularization technique under the smoothness condition (2). In this section, we derive error estimates between the approximate solution and the exact solution in Sobolev spaces with . We also give error estimates for the approximation of early distribution with . We end the section by presenting analogous error estimates for the case where both the data and the forward diffusion operator are only approximately known.
Section 4 is devoted to parameter selection rules which is a critical step in the application of a regularization method. Here we propose a Morozov-like a-posteriori parameter choice rule leading to order-optimal convergence rates under smoothness condition (2). We also present analogous error estimates as that obtained in Section 3 corresponding to the a-posteriori rule prescribed.
Finally, we study four numerical examples in Section 5 to illustrate the effectiveness of the regularization approach coupled with the parameter choice rule described in Section 4. Moreover, in this Section, we also carry out a numerical convergence rates analysis in order to confirm the theoretical convergence rates given in Section 3.
In the sequel, or always refers to the -norm of the function on , denotes the Sobolev norm of on and denotes operator norm of a bounded linear mapping. Throughout the paper, or (resp.
) denotes the Fourier (resp. inverse Fourier) transform of the functiondefined as
Let us start by the following result about the existence and uniqueness of solution of equation (1).
Proposition 1 is merely generalization of [yang2015fourier, Lemma 2.2] where only the case is considered. The idea of the proof is merely to check that the formal solution defined by (5) is the weak solution.
By applying the Laplace transform with respect to variable in (3), one gets
where is the Mittag Leffler function [podlubnv1999fractional] defined as
From (4), we can deduce the following relation between the solution and the data
in the frequency domain:
Moreover, we can also derive the following relation for early distribution with
From equations (5) and (6), we can see that the Mittag Leffler function plays an important role in time-fractional diffusion equation (1). Hence, let us recall some key estimates about the function that will be repeatedly used in the sequel.
Let , there exists constants and depending only on and such that
Assume , then for every and ,
In (8), denotes the maximum while denotes the minimum, that is, and .
From (10), we see that very small perturbations in high frequencies in the data leads to arbitrary large errors in the solution . Therefore, one needs a regularization method to recover stable estimates of .
Estimate (11) illustrates the fact that the backward time-fractional diffusion equation is less ill-posed (mildly ill-posed) on the contrary to the classical backward diffusion equation which is exponentially ill-posed. This is actually due to the asymptotic slow decay of the Mittag Leffler function compared to .
where is the linear forward diffusion operator operator which maps the initial distribution to the final distribution , that is,
In the sequel, given a data , we aim at recovering . Let be a smooth real-valued function in satisfying . It is well-known that the family of functions defined by
where is nothing but the convolution of the functions and defined as For , let be the mollifier operator defined by
From (14), we see that the family of operators is an approximation of unity in , that is,
Let be the solution of the equation
which yields that
From (18), noticing that and using the fact that the function is increasing on with , we can see that (20) implies that for every and , the mapping which maps the data to is bounded with . Moreover, from (16) and (19), we deduce that for every , converges to in as goes to .
From Proposition 2, we can see that, choosing a kernel which satisfies condition (20) allows to defines a regularization method for equation (1). Now, let us show that the family of regularization methods defined in this way actually coincides with approximate-inverse introduced by Louis and Maass [louis1990mollifier].
From the first equation in (18), we can derive that
Moreover, one can easily check that is nothing but the solution of the adjoint equation
Hence, we deduce that the solution of equation (17) actually corresponds to the approximate-inverse [louis1990mollifier] regularized solution of equation (12), the pair being what is usually called reconstruction kernel - mollifier.
This setting of regularization we just described encompasses many regularization methods that appears distinctively in the literature of the regularization of the final value time-fractional diffusion equation. Each regularization method being a particular choice of the mollifier kernel .
For the Fourier regularization [yang2015fourier] (where ), we have
denotes the characteristic function of the setequal to on and elsewhere. By comparing (23) and (18), we readily get that
In this case, condition (20) merely reads
Using (8), we can see that this condition is satisfies with where .
For the mollification method of N. Van Duc et al. [van2020mollification], the mollifier operator is denoted where is the convolution by the so called Dirichlet kernel defined as
Hence we can deduce that, this merely corresponds to
Given that the Fourier transform of the kernel is given by
we see that condition (20) merely reads
which is fulfilled with where .
By the way, from (24) and (25), we can see that the Fourier regularization and the mollification approach of N. Van Duc et al. actually coincides, the latter approach being a generalization of the former to dimensional case. From (26), we can conclude that both regularization approaches are nothing but truncation methods. That is, the regularization is done by merely throwing away high frequency components of the data, which are responsible of the ill-posedness, and conserving unchanged the remaining frequency components. In order word, these two methods can be regarded as spectral cut-off methods. It is important to notice that though high frequency components are responsible of ill-posedness, nevertheless, the still carry non-negligible information on the sought solution. Therefore, it is desirable to apply a regularization which do not suppress high frequency components but which applies much regularization to those components compared to low frequency components. Let us point out that mere truncation of high frequency components usually entails Gibbs phenomena and oscillation of the approximate solution which should be avoided as far as possible. This is actually possible by choosing a kernel whose Fourier transform is supported on the whole domain .
Now on, let us consider a mollifier kernel defined by
where and are two free positive parameters. From (27), we can see that and satisfies .
Let and be two positive numbers, consider the function . Then there exists a constant depending only on such that
whence (20) with .
By defining the mollifier kernel as in (27), we can see that the regularization technique induces a more suitable treatment of frequency components. Indeed, with our choice of mollifier kernel, the amount of regularization smoothly depends on the magnitude of the frequency components: The higher the frequency, the stronger the regularization applied, and similarly, the lower the frequency, the lower the regularization applied. This is actually desirable for a regularization method given that as the frequency gets higher, the noise in the frequency components gets much more amplified, and as the frequency gets lower, the noise in the frequency components gets less and less amplified.
From Proposition 2, we can see that the family of function defined by (27) allows to define a family of regularization methods, each regularization method being determined by the choice of the free parameter . For instance, the choice means considering a Cauchy convolution kernel while means taking a Gaussian convolution kernel.
Let us end this section by the following Lemma which gives rates of convergence of the mollifier operator corresponding to kernel defined by (27) on Sobolev spaces with .
Proof. Let . If , using Parseval identity, we have
3 Error estimates
Henceforth, denotes the mollifier kernel defined by (27) and denotes a noisy data satisfying the noise level condition
where is the exact final distribution. Let us introduce the regularized solution corresponding to the noisy data as the solution of equation
Equivalently, we can define in the frequency domain by
It is well known that without assuming a smoothness condition on the exact solution (or on the exact data ), it is impossible to exhibit a rate of convergence of regularized solution towards the exact solution [schock1985approximate]. Henceforth, we consider the following classical Sobolev smoothness condition:
Before presenting the main results of this section, let us state some lemmas which will be useful in the sequel.
Let , and be a solution of equation on . If , then for every , and
The proof follows readily by applying Parseval identity and estimate (8) to equation which links and in the frequency domain.
The next lemma illustrates the fact that the Sobolev smoothness condition (34) is nothing but a Hölder source condition.
Proof. For , formally define in the frequency domain by
From (13), we can verify that the above definition of from is merely reformulation of the equation in the frequency domain. Next, we can check that is well defined and belongs to . Finally, estimate (36) is deduced from (8).
The next Lemma which generalizes [van2020mollification, Lemma 3] will be useful in the sequel for establishing Sobolev norm error estimates.
Let and be a solution of equation on . If then
where . Moreover,
Proof. Let and be a solution of equation with . For , using Hölder inequality, we have