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

Abstract

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

Simultaneous identification of timewise terms and free boundaries for the heat equation

“…Moreover, within lsqnonlin , we apply the interior-reflective Newton approach based Trust Region Reflective algorithm (Coleman and Li, 1996). Each iteration involves a large linear system of equations whose solution, based on a preconditioned conjugate gradient method, allows a regular and sufficiently smooth decrease of the objective functional (40) or (41), (Baranger et al , 2014). We have compiled this routine with the following specifications: Number of variables M = N = 40. Maximum number of iterations =10 2 × (number of variables). Maximum number of objective function evaluations = 10 3 × (number of variables). Termination tolerance on the function value, (TolFun) = 10 –20 . Solution tolerance, (xTolx) = 10 –20 . …”

Data Recovery from Cauchy Measurements in Transient Heat Transfer

“…The particularity of the related mathematical problem is the ill-posedness. Missing data to reconstruct suffer from high instabilities generated by unavoidable perturbations affecting the available data because of the finite accuracy of measuring instruments (see [2,3,14,17,18,22,31,32]). As a consequence, computing methods crudely employed for numerical handling of these inverse problems are most often doomed to fail unless they are used with relevant regularization techniques combined to suitable automatic selection rules of the regularization parameter(s).…”

Missing boundary data reconstruction for the heat equation

“…The considered inverse problem encountered in many practical applications such as the identification of multiple leak zones in saturated unsteady flow, see previous work 13 and obstacle identification problems. In the framework of the parabolic equation, many approaches have been proposed in the literature for solving this ill‐posed inverse problem with varying degrees of success, we can cite the boundary integral equation approach, see Chapko and Johansson 14 and Onyango et al, 15 iterative regularization methods, we quote previous works 13,16 and Johansson, 17 quasi‐reversibility process, see Dardé, 18 the minimization of a cost functional, see Rischette et al, 19 and the decomposition approach see Lesnic and Elliott 20 …”