Combined energy method and regularization to solve the Cauchy problem for the heat equation

Abstract

This 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.