On the regularization of solution of an inverse ultraparabolic equation associated with perturbed final data

In this paper, we study the inverse problem for a class of abstract ultraparabolic equations which is well-known to be ill-posed. We employ some elementary results of semi-group theory to present the formula of solution, then show the instability cause. Since the solution exhibits unstable dependence on the given data functions, we propose a new regularization method to stabilize the solution. then obtain the error estimate. A numerical example shows that the method is efficient and feasible. This work slightly extends to the earlier results in Zouyed et al. \cite{key-9} (2014).

Ultraparabolic equations arise in several areas of science, such as mathematical biology in population dynamics [13] and probability in connection with multiparameter Brownian motion [17], and in the theory of boundary layers [12]. Due to their applications, ultraparabolic equations have gained considerable attention in many mathematical aspects (see, e.g., [2,4,5,9,11,13] and the references therein).
In the mathematical literature, various types of ultraparabolic problems have been solved. There have been some papers dealing with the existence and uniqueness of solutions for ultraparabolic equations, e.g. [13,19,22]. As the pioneer in numerical methods for such equations, Akrivis et al. [4] numerically approximated the solution of a prototype ultraparabolic equation by applying a fixed-step backward Euler scheme and second-order box-type finite difference method. Some extension works for the numerical angle should be mentioned are [21,23] by A. Ashyralyev-S. Yilmaz and Michael D. Marcozzi, respectively. We also remark that, in general, ultraparabolic equations do not possess properties that are closely fundamental to many kinds of parabolic equations including strong maximum principles, a priori estimates, and so on.
In the phase of ultraparabolic ill-posed problems, the authors F. Zouyed and F. Rebbani, very recently, proposed in [9] the modified quasi-boundary value method to regularize the solution of the problem (1) in homogeneous backward case f ≡ 0. In particular, via the instability terms in the form of the solution of (1) (cf. [2, Theorem 1.1]) they established an approximate problem by replacing A α = A I + αA −1 for the operator A and taking the perturbation α into final conditions of the ill-posed problem, and obtained the convergence order α θ , θ ∈ (0, 1). Motivated by that work, this paper is devoted to investigate a new regularization method.
In the past, many approaches have been studied for solving ill-posed problems, especially the backward heat problems. For example, Lattès and Lions [18], Showalter [24] and Boussetila and Rebbani [26] used quasi-reversibility method; in [22] Ames et al. applied the least squares method with Tikhonov-type regularization; Clark and Oppenheimer [15], Denche and Bessila [14] and Trong et al. [29] used quasiboundary value method. Moreover, some other methods should be listed are the mollification method by Hao [32] and the operator-splitting method studied by Kirkup and Wadsworth [27]. To the best of the author's knowledge, although there are many works on several types of parabolic backward problems, the theoretical literature on regularizing the inverse problems for ultraparabolic equations is very scarce. Therefore, proposing a regularization method for the problem (1) is the scope of this paper.
Our work presented in this paper has the following features. Firstly, for ease of the reading, we summarize in Section 2 some well-known facts in semi-group of operator and present the formula of the solution of (1). Secondly, in Section 3 we construct the regularized solution based on our method, then obtain the error estimate. Finally, a numerical example is given in Section 4 to illustrate the efficiency of the result.

Preliminaries
The operator −∆ is a positive self-adjoint unbounded linear operator on L 2 (0, π). Therefore, it can be applied to some elementary results in [2,6,7,9]. Particularly, the formula of the solution of the problem (1) can be obtained by L. Lorenzi et al. [2] and the authors in [6,7] gave a detailed description on fundamental properties of the generalized operator. In this section, we thus recall those results in which we want to apply to our main results in this paper. We list them and skip their proofs for conciseness.
In fact, we shall study in this section the generalized formula of the solution by the following operator equation in terms of semi-group theory.
where A is a positive self-adjoint unbounded linear operator on the Hilbert space H.
We denote by {E λ , λ > 0} the spectral resolution of the identify associated to A.
for all u ∈ D (A). In this connection, u ∈ D (A) iff the integral (3) exists, i.e., For this family of operators {S (t)} t≥0 we have: 1. S (t) ≤ 1 for all t ≥ 0; 2. the function t → S (t) , t > 0 is analytic; 3. for every real r ≥ 0 and t > 0, the operator S (t) ∈ L (H, D (A r )); 4. for every integer k ≥ 0 and t > 0, Remark 1 In the sequel, let us denote and make some conditions on the given functions as follows: In the following theorems, we show the formula of solution of the problem (2) by employing Theorem 1.1 in [2] with a 1 (t) = a 2 (s) = 1 and following the steps in [9].
admits a unique solution u presented by the following formula. For any (t, s) ∈ D 1 , and for any (t, s) ∈ D 2 , Moreover, the solution u belongs to the space

Theorem 3 Under the conditions (A1)-(A4), if the problem
admits a solution u, then this solution can be presented by It follows that Thus, we obtain by the maps ζ = T − η in the integrals. We can see by the initial conditions of (5) that By virtual of semi-group properties, we get Substituting (7) into (6), we thus have Theorem 4 Under the conditions (A1), (A2) and (A4), if the problem (2) with ϕ (T ) = ψ (T ) admits a solution u, then this solution can be given by Proof Now we put τ = T − t and ξ = T − s, then write Using Theorem 3, the solution v (τ, ξ) can be presented by It follows that Hence, we obtain which completes the proof. Now we return to the consideration of problem (1). All of our results in this paper apply to more general problems, for which the boundary conditions are generalized in Robin-type, for example, α 3 u (π, t, s) + α 4 u x (π, t, s) = 0, or we can consider, in general, the operator equations with the self-adjoint operator A having a discrete spectrum on an abstract Hilbert space H and satisfying the condition that −A generates a compact contraction semi-group on H, like the problem (2) considered above. However, for the sake of simplicity, we confine our attention to the problem (1) in which the homogeneous Dirichlet boundary conditions at the endpoints of [0, π] are given. In this problem, we have H = L 2 (0, π) and D (A) = H 1 0 (0, π) ∩ H 2 (0, π), so there exists an orthonormal basis of L 2 (0, π), {φ n } n∈N satisfying (see e.g. [33, p. 181 The Laplace operator thus has a discrete spectrum σ (A) = {λ n } n≥1 with λ n = n 2 and gives the orthonormal eigenbasis φ n = 2 π sin (nx) for n ∈ N, n ≥ 1. Then, thanks to those theorems above, the solution has the form where We can see that the instability is caused by all of the exponential functions. In fact, let us see the case (t, s) ∈ D 1 in (8). Since the discrete spectrum increases monotonically as n tends to infinity, the rapid escalation of e (T −t)n 2 and e (T −η)n 2 is mainly the instability cause. Even though these exact given functions (ψ n , f n ) may tend to zero very fast, performing classical calculation is impossible. It is because that the given data may be diffused by a variety of reasons such as round-off errors, measurement errors. A small perturbation in the data can arbitrarily generate a large error in the solution. A regularization method is thus required.

Theoretical results
In this section, assuming that the problem has an exact solution u satisfying various corresponding assumptions, we construct the regularized solution depending continuously on the data such that converges to the exact solution u in some sense. Moreover, the accuracy of regularized solution is estimated.
The solution of (1) can be given by We shall replace all instability terms by the better ones, particularly, ε + e −pn 2 t−T p and ε + e −pn 2 s−T p where p ≥ 1 is a real number. Then, the regularized solution corresponding to the exact data is for any (t, s) ∈ D 1 , and for any (t, s) ∈ D 2 . We also denote the regularized solution corresponding to the perturbed data by for any (t, s) ∈ D 1 , and for any (t, s) ∈ D 2 . Now we shall show two elementary inequalities in the following lemmas.
Proof It is obvious that ε + e −n 2 p

Lemma 6
For all x > 0, 0 < α < 1 we have Proof The proof of this lemma is based on the fact that x α < (x + 1) α . Therefore, we have which leads to In the sequel, we only prove the case (t, s) ∈ D 1 in our main result because of the similarity. The results are about the regularized solution depending continuously on the corresponding data and the convergence of that solution to the exact solution. Now we shall use two elementary lemmas above to support the proof of the main results.
Proof Let u ε 1 and u ε 2 be two solutions of (10)- (11) corresponding to the data ϕ 1 , ψ 1 and ϕ 2 , ψ 2 , respectively. By using Parseval relation, for (t, s) ∈ D 1 we have Similarly, for any (t, s) ∈ D 2 , we get Theorem 8 Under the conditions (A1), (A2) and (A4), if the problem (1) with ϕ (T ) = ψ (T ) admits a unique solution u satisfying and where u n (t, s) =ˆπ 0 u (x, t, s) sin (nx) dx, let (ϕ ε , ψ ε ) be perturbed functions satisfying the conditions (A1)-(A2), respectively, and let v ε be the regularized solution, given by (12)- (13), corresponding to the perturbed data (ϕ ε , ψ ε ), then for (t, s) Proof For any (t, s) ∈ D 1 , we have Using triangle inequality, in order to get the error estimate, we have to estimate v ε (·, t, s) − u ε (·, t, s) and u ε (·, t, s) − u (·, t, s) . Indeed, we get v ε (·, t, s) − u ε (·, t, s) Next, u ε (·, t, s) − u (·, t, s) can be estimated as follows. We put and it is of order ε for (t, s) = (T, T ). Then, if p is very large for any fixed T > 0, the order ε p−T p may approach ε. This creates the globally stability behavior of the error in numerical sense. On the other hand, the natural acceptance of (14)- (15) can be obtained at (t, s) = (0, 0). Namely, by letting p = T the conditions become Moreover, the error is of order O ε this error is faster than the order ln ε −1 −q , q > 0 as ε → 0 which is studied in many works, such as [6,14,15,29]. Combining the strong points above, the reader can infer that our method is feasible.

A numerical example
In order to see how well the method works, we consider as an example the problem (1) by choosing Now let us take perturbation on data functions as follows. For m ∈ N, we define Thus, the solution corresponding to the perturbed data functions is  It is easy to see that (ϕ m , ψ m ) converges to (ϕ, ψ) over the norm L 2 (0, π) as m → ∞. To observe the ill-posedness, we can compute, for example, u ex x, Therefore, we get as m → ∞. This divergence is also showed in Figure 1 with m = 2 and m = 3.  (19)- (20) with m = 10 10 .
In Fig. 3, we have drawn the exact solution x → u ex (x, 0, 0) ≡ sin x and the approximate solution x → v m (x, 0, 0) where m are 5 × 10 9 , 7 × 10 9 and 10 10 , respectively, in order to see the convergence at (t, s) = (0, 0) as m becomes very large, namely, the bound ε in theoretical result tends to zero. As in Figs. 2-3, we can conclude that the regularized solution converges to the exact one as the error becomes smaller and smaller. Moreover, convergence is, particularly, observed from the absolute (abs.) errors in Table 1. Hence, our numerical results are all reasonable for the theoretical result.

Conclusion
In this work, a regularization method has been successfully applied to the inverse ultraparabolic problem. This method is to replace the instability terms appearing in the formula of the solution which is employed by semi-group theory. Therefore, such a way forms the so-called regularized solution which strongly converges to the exact solution in L 2 -norm. We also obtain the error estimate which is of order ε p−T p , p > T . By a numerical example, application of the method is flexible and calculation of successive approximations is direct and straightforward. This work is more general than [9], a recent work of Zouyed et al., in both error estimate and the considered problem.