Solving Cauchy problems by minimizing an energy-like functional

Abstract

An energy-like error functional is introduced in the context of the ill-posed problem of boundary data recovering, which is well known as a Cauchy problem. Links with existing methods for data completion are detailed. Here the problem is converted into an optimization problem; the computation of the gradients of the energy-like functional is given for both the continuous and the discrete problems. Numerical experiments highlight the efficiency of the proposed method as well as its robustness in the model context of Laplace's equation, but also for anisotropic conductivity problems.

An iterative method for the Cauchy problem in linear elasticity with fading regularization effect

“…The FEM computing code used is Cast3m [7]. The boundary mesh used has 360 non-uniformly distributed nodes, the density being higher on the Γ u part (120 uniformly distributed nodes) corresponding to 1 8 of Γ. The boundary part Γ d corresponds to 8 9 of Γ and is distributed on both sides of both contact areas ( Figure 13 Figure 15) is less accurate and gives with a relative precision the maximal value of the contact pressure distribution.…”

“…In order to solve Cauchy problems for elliptic equations, many regularization methods have been introduced. The references [1,5,8,9,11,12,14,20,21,23,25,26] propose different methods of solving the Cauchy problem for the Laplace equation. References [2,3,13,22,23,27,28,29,30,31,32,33,34,35,36,37,38,40,41] deal with the Cauchy problem in linear elasticity.…”

“…References [2,3,13,22,23,27,28,29,30,31,32,33,34,35,36,37,38,40,41] deal with the Cauchy problem in linear elasticity. These methods can be classified as Tikhonov type methods [15,22,27,29,33,34,35,38,39,40,41], quasi-reversibility type methods [5,21,25], iterative methods [1,2,3,8,9,11,12,13,14,15,20,23,26,28,32,36,37],... Quasi reversibility methods and Tikhonov regularization methods present the advantage of leading to well posed problems after mod...…”

“…Some iterative methods are based on the use of a sequence of well-posed problems and others on the minimization of an energy-like functional. Numerical algorithms are implemented using different numerical methods, such as the finite element method (FEM) [1,2,3,5,8,9,11,13,27,38], the boundary element method (BEM) [12,14,20,22,26,23,28,29,30,31,32,33,36,37,40,41], the finite difference method [21] or meshless methods [34,35]. Some papers present comparisons between different numerical methods [10,31,37].…”

A numerical solution of a Cauchy problem for an elliptic equation by Krylov subspaces

“…There are many engineering applications of ill-posed Cauchy problems, see [25,26,36] and the references therein. A standard approach for solving Cauchy problems of this type is to apply an iterative procedure, where a certain energy functional is minimized; a recent example is given in [1]. Very general (non-cylindrical) problems can be handled, but if the procedure from [1] were to be applied to our problem, then at each iteration four well-posed elliptic equations would have to be solved over the whole three-dimensional domain.…”