This paper is concerned with solving the Cauchy problem for an elliptic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, numerical convergence analysis of the energy-like minimization method is carried out and leads to adapted stopping criteria for the minimization process depending on the noise rate. Numerical examples involving smooth and singular data are presented.

International audienceThis paper is concerned with solving Cauchy problem for parabolic equation by minimizing an energy-like error functional and by taking into account noisy Cauchy data. After giving some fundamental results, numerical convergence analysis of the energy-like minimization method is carried out and leads to an adapted stopping criteria depending on noise rate for the minimization process. Numerical experiments are performed and confirm theoretical convergence order and the good behavior of the minimization process

International audienceThis paper deals with an energy method coupled with total variation regularization and an adequate stopping criterion in order to solve a Cauchy problem for the heat equation when using noisy data. First, the Cauchy problem is written as a data completion one, then it is split into two well-posed thermal problems. Therefore, a pseudo-energy functional measuring the gap between solutions of these two problems is introduced and minimized. The problem is then converted into one of constrained optimization; the computation of the gradient of this functional is given for the full time-space discrete problems by means of the adjoint method. In order to deal with noisy data, two regularization techniques were used, the first one which fits into the optimization procedure is an adequate stopping criterion depending on the noise level and it avoids numerical instability. The second one is a total variation regularization method which can be carried out a priori and/or a posteriori of the optimization procedure. Numerical experiments are performed on the noisy Cauchy data and/or the identified data. Numerical experiments highlight the efficiency and weakness of the coupled methods

International audienceThis paper focuses on the regularization of noisy Cauchy data and unknown boundary conditions approximated by the energy-like minimization method. After providing considerations necessary for the numerical analysis leading to an adequate stopping criterion for the minimization process, we describe the denoising procedure proposed. Numerical experiments involving singular data are presented.Copyright © 2013 John Wiley & Sons, Ltd