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Boundary data completion: the method of boundary value problem factorization

Amel Ben Abda 1 Jacques Henry 2, 3 Fadhel Jday 1 
3 ANUBIS - Tools of automatic control for scientific computing, Models and Methods in Biomathematics
Université Bordeaux Segalen - Bordeaux 2, Université Sciences et Technologies - Bordeaux 1, Inria Bordeaux - Sud-Ouest, CNRS - Centre National de la Recherche Scientifique : UMR
Abstract : We consider the following data completion problem for the Laplace equation in the cylindrical domain: = ]0, a[×O,O ⊂ Rn−1 (O is a smooth bounded open set anda > 0), limited by the faces 0 = {0}×O and a = {a}×O. The Neumann and Dirichlet boundary conditions are given on 0 while no condition is given on a. The completion data problem consists in recovering a boundary condition on a. This problem has been known to be ill-posed since Hadamard [12]. The problem is set as an optimal control problem with a regularized cost function. To obtain directly an approximation of the missing data on a we use the method of factorization of elliptic boundary value problems. This method allows us to factorize a boundary value problem in the product of two parabolic problems. Here it is applied to the optimality system (i.e. jointly on the state and adjoint state equations).
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Submitted on : Monday, August 29, 2011 - 1:11:00 PM
Last modification on : Friday, February 4, 2022 - 3:11:42 AM
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Amel Ben Abda, Jacques Henry, Fadhel Jday. Boundary data completion: the method of boundary value problem factorization. Inverse Problems, IOP Publishing, 2011, 27 (5), ⟨10.1088/0266-5611/27/5/055014⟩. ⟨inria-00617511⟩



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