Abstract. We consider the following data completion problem for the Laplace equation in the cylindrical domain:smooth bounded open set and a > 0), limited by the faces Γ 0 = {0}×O and Γ a = {a}×O. The Neumann and Dirichlet boundary conditions are given on Γ 0 while no condition is given on Γ a . The completion data problem consists in recovering a boundary condition on Γ a . This problem has been known since Hadamard [12] to be ill-posed. The problem is set as an optimal control problem with a regularized cost function. To obtain directly an approximation of the missing data on Γ a we use the method of factorization of elliptic boundary value problems. This method allows 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).

In this paper, we investigate the Cauchy problem for the modified Helmholtz equation. We consider the data completion problem in a bounded cylindrical domain on which the Neumann and the Dirichlet conditions are given in a part of the boundary. Since this problem is ill-posed, we reformulate it as an optimal control problem with an appropriate cost function. The method of factorization of boundary value problems is used to immediately obtain an approximation of the missing boundary data. In order to regularize this problem, we firstly scrutinize two classical regularizations for the cost function. Then we propose a new numerical regularization named “adaptive Runge–Kutta regularization”, which does not require any penalization term. Finally, we compare them numerically.

In this paper, we establish an identifiability result for the coefficient identification problem in a fractional diffusion equation in a bounded domain from the observation of the Cauchy data on particular subsets of the boundary.