An iterative generalized quasi-boundary value regularization method for the backward problem of time fractional diffusion-wave equation in a cylinder

In this paper, we consider the backward problem for a time fractional diffusion-wave equation in a cylinder. The ill-posedness and a conditional stability of the inverse problem are proved. Based on the generalized quasi-boundary value regularization method, we propose an iterative generalized quasi-boundary value regularization method to deal with the inverse problem, and this iterative method has a higher convergence rate. The convergence rates of the regularized solution under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule are obtained. Numerical examples illustrate the effectiveness and stability of our proposed method.


Introduction
In recent years, the fractional diffusion-wave equation has attracted more and more attention because it has been successfully applied to some important practical fields such as physics, chemistry, biology, and finance [1,2].Due to the memory and genetic properties of the fractional derivative, the fractional diffusion-wave equation has its unique advantages in describing superdiffusion phenomena [3][4][5].
The direct problem for the time fractional diffusion-wave equation is well-posed in the sense of Hadamard, and has been studied with relatively well-developed results [6][7][8].However, the inverse problem for the time fractional diffusion-wave equation is usually ill-posed in the sense of Hadamard.This means that the existence, uniqueness and stability of the solution of the inverse problem cannot be satisfied simultaneously.
The solution to most inverse problems exists, but the solution is not continuously dependent on the measurement data, i.e., a small noise in the measurement data can cause a large perturbation in the solution.In other words, the ill-posedness can lead to unstable result of the direct computation.Therefore, an appropriate regularization should be proposed to obtain a stable solution of the inverse problem.
Currently, some regularization methods are used for the inverse problem of the time fractional diffusion or diffusion-wave equation.Wei and Wang [9] proposed a modified quasi-boundary value method to identify a space-dependent source term of the time fractional diffusion equation.Tuan et al. [10] solved the backward problem for the inhomogeneous time fractional diffusion equation using the Tikhonov regularization method.Tuan et al. [11] proposed a modified regularization method to study the inverse initial value problem of the time fractional diffusion-wave equation.Yang et al. [12] used the truncated regularization method to determine the initial value of the inhomogeneous time fractional diffusion-wave equation.Han et al. [13] studied the backward problem of the time fractional diffusion equation using the fractional Landweber iterative regularization method.Yang et al. [14] proposed an iterated fractional Tikhonov regularization method for solving the backward problem of the spherically symmetric time fractional diffusion equation.Wei and Luo [15] proposed a generalized quasiboundary value regularization method to deal with the inverse source problem in the time fractional diffusion-wave equation.For other regularization methods to solve the inverse problem of the time fractional diffusion or diffusion-wave equation, refer to [16][17][18][19].The available regularization methods can be divided into two categories.One is non-iterative regularization methods, such as Tikhonov regularization method, truncation regularization method, and quasi-boundary value regularization method.The other category is iterative regularization methods, such as iterated Tikhonov regularization method and Landweber iterative regularization method.
In this paper, we combine the generalized quasi-boundary value regularization method and iteration, and thus propose an iterative generalized quasi-boundary value regularization method for solving the backward problem of a time fractional diffusionwave equation in a cylinder.We learned from [15] that the generalized quasi-boundary value regularization method has a higher convergence rate compared to the standard quasi-boundary value regularization method [20] and the modified quasi-boundary value regularization method [9].For the iterative generalized quasi-boundary value regularization method, we can obtain some higher convergence rates than the generalized quasi-boundary value regularization method in [15], which means that the iterative generalized quasi-boundary value regularization method is optimal among the currently available quasi-boundary value regularization methods.
In [21], Fick's classical diffusion model in two-dimensional Euclidean geometry is usually described by where u(r , t) indicates the gas concentration, r = x 2 + y 2 is the radius of the particles, q(r , t) is the diffusion flux expressed by the classical Fick's law with Q being the diffusion coefficient.By the time fractional Fick's law proposed in [22], the model (1.1) can be represented as where D α t u(r , t) is the Caputo fractional derivative of order α(0 < α < 2).Consider the general form of (1.2) as follows this equation was studied by Narahari Achar and Hanneken [23] for a constant concentration at the surface r = R and in the absence of a mass source f (r , t).Motivated by Narahari Achar and Hanneken, Povstenko [24] developed the results in [23] to include the source term f (r , t).Turner et al. [25] investigate the numerical scheme of the equation (1.3).However, in practice, as in the field of fluids and radiation in physics [26,27], the time fractional diffusion-wave equation in a cylinder (1.2) is more applicable compared to (1.3).
where r and z are cylindrical coordinates, f (r , z, t) is source term and D α t u(r , z, t) represents the Caputo fractional derivative of order α(1 < α < 2) defined by Povstenko [28] considered the solutions of the equation (1.4), and Özdemir et al. [29,30] studied the solutions of the equation (1.4) with Riemann-Liouville fractional derivative.To the best of our knowledge, there are few papers on the inverse problem of the equation (1.4).
In this paper, we focus on the equation (1.4) with the initial conditions and the boundary conditions where the boundary condition (1.7) is equivalent to the symmetry boundary condition u r (0, z, t) = 0 [31] and φ(r , z) need to satisfy the compatibility conditions lim If all the data φ(r , z), ψ(r , z) and f (r , z, t) are known, problem (1.4)-(1.9) is a direct problem for the time fractional diffusion-wave equation.The inverse problem here is to recover the initial value φ(r , z) based on the final value Since the measurement is inevitably contaminated by noise, we assume that g δ (r , z) is g(r , z) with measurement noise and satisfies where ) norm and δ > 0 denotes a noise level.The remainder of this paper is composed of five sections.In Section 2, we give some preliminary material.The ill-posedness and a conditional stability of the inverse problem are analyzed in Section 3. In Section 4, we propose an iterative generalized quasi-boundary value regularization method and provide the convergence rates under two regularization parameter choice rules.In Section 5, numerical examples are shown to verify the effectiveness and stability of our method.Finally, a brief conclusion is given in Section 6.

Preliminaries
Throughout this paper, we use L Then, we give some useful definitions and lemmas.

Ill-posedness and conditional stability
In this section, we formulate the inverse problem into an integral equation by the series expression of the weak solution for the direct problem.Furthermore, we consider the ill-posedness and a conditional stability of the inverse problem.
If the source term f (r , z, t), initial conditions, and boundary conditions as (1.4)-(1.9)are known, applying the method of separation of variables and Laplace transform of Mittag-Leffler function, we can obtain the unique solution of the direct problem as follows where φ mn = (φ(r , z), ω mn (r , z)) r , ψ mn = (ψ(r , z), ) form a standard orthogonal basis in L 2 r ( ).The J 0 (x) and J 1 (x) denote the order 0 and order 1 Bessel functions, and {μ n } ∞ n=1 are the sequence of the zeros of J 0 (x).
Proof By Lemma 2.1, we know that there exists C 0 > 0 such that Because λ mn → +∞ when m → +∞ or n → +∞, there are only finite λ mn satisfying λ mn T α ≤ C 0 .
Lemma 3.2 For any λ mn satisfying λ mn ≥ λ 11 > 0 and 1 < α < 2, there exist positive constants C and C which depend on α, T and λ 11 , such that Proof From Lemma 2.2, we have By lemma 3.1 and its proof process, we know that there exists Thus, we have From Lemma 3.1, we can easily get that there exists at most a finite point set I 1 = {(mn) 1 , (mn) 2 , . . ., (mn) j } such that E α,1 (−λ mn T α ) = 0 for mn ∈ I 1 .In particular, note that the point set I 1 may be empty, which can be considered as a special case.
We consider the general case, i.e., I 1 = ∅, then equation (3.2) can be rewritten as 123 Similar to reference [35], we known that the equation (3.3) has infinitely solutions only when h mn = 0 for mn ∈ I 1 and the condition From Lemma 3.2, we note that in (3.4) which indicates that the inverse problem is ill-posed, i.e., a small measurement noise in the data g(r , z) can cause a large perturbation in the solution φ(r , z).Hence, it is necessary to use a regularization method to recover the stability of the solution of the inverse problem.Before that, we give a conditional stability theorem for the inverse problem.

Iterative regularization method and convergence rate
In this section, we propose an iterative generalized quasi-boundary value regularization method to solve the backward problem of time fractional diffusion-wave equation in a cylinder.The convergence rates of the regularized solution under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule are given.
We define a linear operator A : λ mn v mn ω mn (r , z), and where γ ≥ 0 is a real number and v mn = (v(r , z), ω mn (r , z)) r .
Let the function u δ,k μ (r , z, t) be the solution of the following iterative regularization problem for k = 1, 2, 3, . . ., and the initial value φ δ,0 μ (r , z) = 0, where μ > 0 is a regularization parameter and k is the number of iterations.If we fix k = 1, this iterative regularization method is transformed into the generalized quasi-boundary value regularization method in [15].Then, based on the iterative regularization form, we have where g δ mn = (g δ (r , z), ω mn (r , z)) r .Furthermore, we can deduce that where From (4.1), we can find that ) k may tend to infinity when E α,1 (−λ mn T α ) < 0, which causes trouble in obtaining the error estimate between φ δ,k μ (r , z) and φ(r , z).Thus, we modify the regularized solution φ δ,k μ (r , z) as For simplicity, let then we obtain the regularized solution as

A priori regularization parameter choice rule
We first prove some useful lemmas, and to ensure that the following lemmas and theorems are strictly valid, assume that λ mn ≥ 1.This holds easily because λ mn = ( μ n R ) 2 + ( mπ L ) 2 , so it is sufficient to take L ≤ π to satisfy λ mn ≥ 1. .
Proof By Lemma 2.3, we have Introduce a new variable x = λ mn , let For γ ≥ 0, the function A(x) is continuous.Since A(x) ≥ 0, lim x→0 A(x) = 0 and lim x→+∞ A(x) = 0, the maximum point satisfies A (x * ) = 0. Thus, we obtain and Proof By λ mn ≥ 1 and Lemma 3.2, we have Let and Theorem 4.1 Let the exact solution φ(r , z) be given by (3.4) and the regularized solution φ δ,k μ (r , z) be given by (4.2).If the noise data g δ (r , z) satisfies (1.11) and the exact solution φ(r , z) satisfies the a priori bound condition (3.5), the following convergence rates are satisfied.
, we have a convergence rate p+2 , we have a convergence rate Proof Using the triangle inequality, we obtain From (1.11), (4.2) and Lemma 4.1, we have Choose μ by then we have

A posteriori regularization parameter choice rule
We consider an a posteriori regularization parameter choice rule which is in accordance with the Morozov discrepancy principle.
First, we define a linear operator B : L 2 r ( ) → L 2 r ( ) and an orthogonal operator P : L 2 r ( ) → span{ω mn (r , z), mn ∈ I 1 } ⊥ as follows where v mn = (v(r , z), ω mn (r , z)) r .Then, the Morozov difference principle here is to find the regularization parameter μ so that where τ > 1 is a constant and According to the following lemma, we know that there exists a unique solution for (4.12) if Ph δ (r , z) > τδ > 0.
If h δ (r , z) > 0, then the following results are obtained

is a strictly increasing function over (0, +∞).
Proof From (4.2) and (4.13), we have This implies that From (4.14), the results are obtained.
we have a convergence rate where Proof From (4.12) and Lemma 4.4, we have mn φ mn ω * mn (r , z) Therefore, we have (4.20) From (3.4) and (4.2), we have and From (4.21) to (4.23), we get Case 2: 0 < p < 2k(1 + γ ) − 2, we have Then and From (4.25) to (4.27), we get Combining (4.24) and (4.28), we have (4.29) Combining (4.20) and (4.29), we obtain ) are the convergence rates of the generalized quasi-boundary value regularization method under the a priori regularization parameter choice rule and the a posteriori regularization parameter choice rule, respectively.This is obvious because both functions k+kγ 1+k+kγ and k+kγ −1 k+kγ are strictly increasing functions with respect to k ∈ N.

Numerical implementation
In this section, we present some examples to illustrate the effectiveness and stability of the proposed method.To avoid the "inverse crime", the finite difference methods [36] are used to calculate the direct problem.The finite difference schemes are sketched as follows.
First, we denote the discrete points in the time interval [0, T ] as t n = nτ (n = 0, 1, • • • , N ) with the time step size τ = T N , the grid points in the space interval [0, R] as where h r = R M and h z = L M represents the space step size.The value of each grid point is u n i, j = u(r i , z j , t n ).Second, based on the finite difference scheme, we discretize the equation where b and (5.1) can be simplified as Then we have the following matrix equation where A and B are matrices 1) is identity matrix.
Final, the final value u(r , z, T ) = g(r , z) is obtained.
After obtaining the final value g(r , z), we generate noisy data g δ (r , z) by adding a random disturbance, i.e., and the corresponding noise level is calculated by δ = g δ − g .
To show the accuracy of the regularization method, we calculate the relative error between the exact solution and the regularized solution by e r = φ δ,k μ (r , z) − φ(r , z) φ(r , z) .
In numerical experiments, we always take R = π, L = π , T = 1, N = 10, M = 40.The exact solution φ(r , z) is given by (3.4) and the regularized solution φ δ,k μ (r , z) is given by (4.2).The regularization parameter μ under the a priori choice rule is given by Theorem 4.1, and the regularization parameter μ under the a posteriori choice rule is chosen by (4.12) with τ = 1.1.Moreover, to demonstrate the validity of our proposed iterative generalized quasi-boundary value regularization method (IGM), we compare it with the generalized quasi-boundary value regularization method (GM).Since our method adds iteration to GM, it can be regarded as an improvement of GM.And we have already mentioned in the theoretical part of this paper that the IGM is consistent with the GM when k = 1.Here we choose k = 20 in IGM, and γ = 2.

Example 5.1 Take the initial values
and the source term Then the exact solution of the direct problem is given by Example 5.2 Take the initial values  and the source term f (r , z, t) = 2t 2−α r sin(z).
Figures 1 and 2 give the comparisons and the absolute error between the exact solution φ(r , z) and the regularized solutions of GM and IGM under the a priori and a posteriori regularization parameter choice rules with = 0.001 for Example 5.1.Table 1 shows the relative errors of GM and IGM under the a priori and a posteriori regularization parameter choice rules with different noise levels of Example 5.1.Figure 3 gives the exact solution φ(r , z) and its a priori and a posteriori regularized solutions of IGM for Example 5.1 under = 0.001 in the case of α = 1.2, 1.5, 1.8.Table 2 shows the relative errors of IGM under = 0.001 in the case of α = 1.2, 1.5, 1.8 for Example 5.1.
Figures 4 and 5 give the comparisons and the absolute error between the exact solution φ(r , z) and the regularized solutions of GM and IGM under the a priori and a posteriori regularization parameter choice rules with = 0.001 for Example 5.2.Table 3 shows the relative errors of GM and IGM under the a priori and a posteriori regularization parameter choice rules with different noise levels of Example 5.2.
Figure 6 gives the exact solution φ(r , z) and its a priori and a posteriori regularized solutions of IGM for Example 5.2 under = 0.001 in the case of α = 1.2, 1.5, 1.8.Table 4 shows the relative errors of IGM under = 0.001 in the case of α = 1.2, 1.5, 1.8 for Example 5.2.
From Figs. 1 and 2 and Figs. 4 and 5 and Tables 1 and 3, we can find that IGM is even comparable to GM.It can also be concluded that the smaller , the more effective the regularization method is.In addition, it seems that the results obtained from the a priori regularization parameter choice rule and the a posteriori regularization parameter choice rule are very close to each other.This is because the a priori regularization parameter choice rule is related to the a priori bound condition.If a better a priori bound condition is chosen, the result is closer to the a posteriori regularization parameter choice rule.However, if a smaller or larger a priori bound condition is chosen, the result is worse.From Fig. 3 and Fig. 6 and Table 2 and Table 4, we find that the larger the α, the better the regularization approximation of the IGM.In summary, numerical examples verify the effectiveness and stability of our iterative regularization method.

Conclusion
In this paper, we propose an iterative generalized quasi-boundary value regularization method for solving the backward problem of time fractional diffusion-wave equation in a cylinder from final value.The ill-posedness and a conditional stability of the inverse problem are proved.Based on an a priori assumption for the exact solution, we obtain the convergence rates of the regularized solutions under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule.Numerical examples illustrate the effectiveness and stability of our proposed method.

Remark 4 . 1
For the iterative generalized quasi-boundary value regularization method, there is no saturation in the convergence rates.If the smoothness index p of the unknown initital value φ(r , z) is large enough, we can choose a slightly larger k such that the convergence rate O(δ k+kγ 1+k+kγ ) is better than O(δ 1+γ 2+γ ) under the a priori regularization parameter choice rule and the convergence rate O(δ k+kγ −1 k+kγ ) is better than O(δ γ 1+γ ) under the a posteriori regularization parameter choice rule.

Fig.
Fig. The of regularized solutions of GM and IGM for Example 5.2 with α = 1.5

Table 1
The relative errors for different of Example 5.1 with α = 1.5

Table 2
The relative errors for different α of Example 5.1 with = 0.001 for IGM

Table 3
The relative errors for different of Example 5.2 with α = 1.5

Table 4
The relative errors for different α of Example 5.2 with = 0.001 for IGM