A modified quasi-boundary value method for an abstract ill-posed biparabolic problem

Abstract In this paper, we are concerned with the problem of approximating a solution of an ill-posed biparabolic problem in the abstract setting. In order to overcome the instability of the original problem, we propose a modified quasi-boundary value method to construct approximate stable solutions for the original ill-posed boundary value problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution. Moreover, numerical tests are presented to illustrate the accuracy and efficiency of this method.


Ill-posedness of the problem and a conditional stability result
We point out here some results established in [2].
Let use consider the following well-posed problem.
In this case, the high-frequency θ n (t, λ n ) is equal to e (T−t)λ n and the problem is severely ill-posed. -In the case of biparabolic model, we have σ n = r n θ n , where r n = + tλ n + Tλ n , is the relaxation coe cient resulting from the hyperbolic character of the biparabolic model. and where From this remark, we observe that the degree of ill-posedness in the biparabolic model is relaxed compared to the classical parabolic case.

. Conditional stability estimate
We would like to have estimates of the form for some function Ψ (.) such that Ψ (s) → as s → .
Since the problem of determining u(t) from the knowledge of {u(T) = g, u ′ ( ) = } is ill-posed, an estimate such as the above will not be possible unless we restrict the solution u(t) to certain source set M ⊂ H.
In our model, we will see that we can employ the method of logarithmic convexity to identify this source set: On the basis {φ n } we introduce the Hilbert scale (H s ) s∈R (resp. (E s ) s∈R ) induced by A as follows We give here a result of conditional stability. The demonstration is given in the paper [2].
Theorem 2.5. The problem 1 is conditionally well-posed on the set Moreover, if u(t) ∈ M ρ , then we have the following Hölder continuity where γ = +Tλ λ T−t T .

Regularization and error estimates
In this work, we propose a modi ed quasi-boundary value method (MQBVM) to solve the inverse problem 1, i.e., replacing the nal condition u(T) = g with the functional time nonlocal condition, to form an approximate regularized problem where r > is a real parameter and α > is the regularization parameter.  αλ r n + ( + Tλ n ) e −Tλ n e −tλ n g n φ n , g n = ⟨g, φ n ⟩ (18)

Moreover, the following inequality holds
where and C = max (C , C ) , C = (rT) r , C = r.
Proof. We compute andH Our goal here is to prove that Indeed, we have We denote s = Tλ n and the function The function attains its maximum at λ * , Hence, we get sup If λ ≥ λ * , we can writeH Putting C = (rT) r , C = r, C = max (C , C ).
• If < r < and < α < , we observe that Then, for α su ciently small, we have • If r ≥ and < α < , we have lim Then, for α su ciently small, we have From (26) and (27), we obtain the desired estimate: Proof. We have u( ) = ∞ n= + Tλ n e Tλ n ⟨g, φ n ⟩φ n , From this equality we can write We have We observe thatF (λ n ) = αλ r n αλ r n +( +Tλ n )e −Tλn ≤ , then we can write The other quantity can be estimated as follows If we choose the parameter α such that αλ r N e Tλ N u ( ) ≤ ε √ , we obtain Which shows that u α ( ) → u ( ) , as α → .
We conclude this paper by constructing a family of regularizing operators for the problem 1. (1) if for each solution u(t), ≤ t ≤ T of (1) with nal element g, and for any η > , there exists α(η) > , such that
De ne R α (t) = (I + tA) αA r + ( + TA) e −TA − e −tA . It is clear that R α (t) ∈ L(H) (see (19)). In the following we will show that R α (t) is a family of regularizing operators for the problem 1. Proof. We have where We observe that Now, by Theorem 3.3 we have uniformly in t. Combining (41) and (42) we obtain This shows that R α (t) is a family of regularizing operators for the problem 1.

Numerical results
In this section we give a two-dimensional numerical test to show the feasibility and e ciency of the proposed method. Numerical experiments where carried out using MATLAB. We consider the following inverse problem where f (x) = u(x, ) is the unknown initial condition and u(x, ) = g(x) is the nal condition. It is well known that the operator is positive, self-adjoint with compact resolvent (A is diagonalizable). The eigenpairs (λ n , φ n ) of A are In this case, the formula (7) takes the form In the following, we consider an example which has an exact expression of solutions (u(x, t), f (x)). is the exact solution of the problem (43). Consequently, the data function is g(x) = u(x, ) = π e sin(x). By using the central di erence with step length h = π N+ to approximate the rst derivative u x and the second derivative u xx , we can get the following semi-discret problem (ordinary di erential equation): where A h is the discretisation matrix stemming from the operator A = − d dx : is a symmetric, positive de nite matrix. We assume that it is ne enough so that the discretization errors are small compared to the uncertainty δ of the data; this means that A h is a good approximation of the di erential operator A = − d dx , whose unboundedness is re ected in a large norm of A h . The eigenpairs (µ k , e k ) of A h are given by Adding a random distributed perturbation (obtained by the Matlab command randn) to each data function, we obtain the vector g δ : g δ = g + εrandn(size(g)), where ε indicates the noise level of the measurement data and the function "randn(.)" generates arrays of random numbers whose elements are normally distributed with mean , variance σ = , and standard deviation σ = . "randn(size(g))" returns an array of random entries that is the same size as g. The bound on the measurement error δ can be measured in the sense of Root Mean Square Error (RMSE) according to The discret approximation of (18) takes the form where I N is the identity matrix. In our numerical computations we always take N = and consider only the cases when ε = . , . . The regularization parameter (α, r) is chosen in the following way: for any xed r ∈ { , , , }, we try to nd a satisfactory error by varying the second parameter α = ε s with step length s = . . We note α one of the best choice which gives this result. Now, for α = α xed, we try to nd an acceptable error by varying the rst parameter r = , , , in order to obtain the best possible convergence rate. It is important to note that this choice is of heuristic nature and the multiparameter discrepancy principle is quite scarce in the literature.
The relative error RE(f ) is given by

Conclusion and discussion
Numerical results are shown in Figures 1-8 and Tables 1-2. In this paper, we have proposed an improved two-parameter regularization method (MQBVM) to solve an illposed biparabolic problem. The convergence and stability estimates have been obtained under a priori bound assumptions for the exact solution. Finally, some numerical tests show that our proposed regularization method is e ective and stable.    According to the numerical tests, we observe the following regularizing e ect: • In the case r = , ε = . and α = .
(resp. r = , ε = . and α = . ), the approximate solution is far from the exact solution. But for the case r = , , , we observe that the solution becomes precise and very near to the exact solution (in particular for r = , ). This shows that our approach has a nice regularizing e ect and gives a better approximation with comparison to the classical QBV-method.