Discrete non standard formulation of PDE inverse problems

In this paper, we are interested in the computation of the unknown initial state for the simulation and prediction of PDE systems where the solution measures are partially known over a time interval. Such a problem is usually solved by an ill-posed optimal control problem. Based on an appropriate collocation approximation, we obtain a discrete inverse problem. To solve this problem, a non-standard discrete approach is used. This allows to obtain a transformation of the original problem into a well-posed ones based on the zero controllability of a discrete system. The desired control is then calculated as well as the discrete approximation of the initial state values


Introduction
Let Ω be a bounded domain of R n (n ≥ 1) with a Lipschitzian boundary Γ.
Consider ω ⊂⊂ Ω and A an elliptic operator. We assume that we are given of the following equations system: where f and g are given data respectively in L 2 (0, T ; L 2 (Ω)) and in L 2 (0, T ; L 2 (ω)) .
Given an observed data y obs ∈ L 2 (0, T ; L 2 (ω)) we are concerned with the problem of finding y(T ) where y satisfies system equations (1) - (2) and the equation This is a problem of the reconstruction of the state of a system whose solution is partially known. Such inverse problems arise in many scientific fields ranging from medicine to geology etc. They are often solved by variational data assimilation methods which are known to be ill-posed problems [5,6,11]. However these classical methods require regularization techniques which are very difficult to implement [2,3,4]. In this article, another approach is developed in order to better take into account the aspects concerning the size of the observed data without having the additional information necessary for regularization techniques. The most direct non-standard formulation of the inverse problem was given by Puel and al. [1,9,10]. This, first, consists in transforming the considered inverse problem into a zero controllability problem. Then approximation schemes of exact controllability problems are applied for a numerical resolution. We note that it is well known that numerical approximation schemes which are stable for solving initial boundary value problems develop instabilities when applied to controllability problems. Several methods have been used to deal with these instabilities. One can refer to [8,12].
We propose here rather a discrete inverse formulation that we solve by a non-standard discrete approach.

Discrete inverse formulation
For a discrete formulation of the problem (1)-(3) we consider a space collocation method associated with an Euler scheme according to the temporal variable. To this end, assume that we are given of N collocation points x i in Ω. Let's pose We assume that the interval [0, T ] is divided into m uniform subintervals of step size δt = T /m and we set t k = kδt. Let y k i be an approximation of y at the point (t k , x i ). Using any colocation method, one can approximate where the a ij are the components of a N by N matrix and the Λ k i are coefficients depending on the datum g. Then by applying the progressive Euler scheme to equation (1), we formulate the inverse problem (1) -(3) in the following discrete form: Find y m i which approximates y(T ) at the point x i such that where y obs,k Let us now consider a sequence z k i which will be chosen so as to easily calculate the unknowns y m i . Multiplying the equation (5) by z k+1 i and summing according to the indices i and k, we obtain which can also be written as follows Let us introduce a sequence of control values v k i and a discrete χ function on I ω which satisfies χ I ω (i) = 1 if i ∈ I ω and 0 otherwise. We assume that the sequence z k i satisfies the following backward discrete equations system for arbitrary given real µ i . Given the sequence z k i which satisfies (9) -(10) then the equation (8) becomes Now, for fixed l, we set µ i = δ il and we denote by z k,l i and v k,l i corresponding state -control that satisfies the discrete retrograde equations systems (9) -(10). From equation (11) we then obtain Since terms y 0 i are unknown, the terms y m i are completely determined if for each l, we can find v k,l i such that the corresponding solution z k,l i of the system (9) -(10) satisfies z 0,l i = 0 for all i.

Calculation of the null control
Consider the following bi adjoint equations system for given α i . Let multiply (13) by z k+1,l i . Then we obtain Thanks to (9) Using final condition (10) we obtain Let fix l ′ and set α i = δ il ′ . If we denote by s k,l ′ i the corresponding solution of the biadjoint system (13)-(14) then from the equation (18) we obtain for each l ′ The condition which ensures z 0,l l ′ = 0 for each l ′ reads This system clearly admits an infinity of solutions since there are more unknowns than equations. The choice of controls v k,l i which reduce the terms z 0,l l ′ to zero may be subject to additional conditions. For exact zero controllability of continuous problems, for example, the null control is required to satisfy the minimal L 2 -norm [7]. Here, it suffices to impose that a norm of the control be less than or equal to an arbitrarily chosen value that is sufficiently small. Consequently, thanks to the relation (12), we easily deduce the following result. This proposition gives the direct calculation of y m l approximation of the values of the final state y(T ) at all the points x l of the domain Ω. We note that, given the expression (21), this solution is quite stable with respect to the observed data.

Conclusion
In this paper we have developed a discrete procedure for the reconstruction of a state of a boundary value problem from which we have partial data. The approach adopted is not classic. Starting from the original problem we gave a discrete formulation of the inverse problem and then we adapted a nonstandard approach already used in continuous problems. We note that the solution calculated in the approach developed in this paper is easily implemented and perfectly suited for large-scale problems that can be encountered, for example, in oceanography.