Convergence analysis for finite element approximation to an inverse Cauchy problem

Abstract

In this paper, we propose an approximate optimal control formulation of the Cauchy problem, for an elliptic equation, equivalent to the original one under some regularity assumptions on the data. We prove the existence of a solution and show the convergence of a subsequence of approximate solutions, obtained via finite element approximation, to a solution of the continuous problem. Finally, we give some numerical results showing the efficiency of our approach and confirming the convergence result.

Misfit functional for recovering data in 2D ElectroCardioGraphy problems

“…Notice that this least-squares formulation has been studied in Chakib et al [7] resorting to the optimal control tools.…”

A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source

“…Nevertheless, the literature devoted to the Cauchy problem for linear homogeneous elliptic equations is very rich, see e.g. [4,5,7,9,12,13,16,21,23,29,33,35] and the references therein. Recently, a linear inhomogeneous version of Helmholtz equation (i.e.…”

A non-typical Lie-group integrator to solve nonlinear inverse Cauchy problem in an arbitrary doubly-connected domain

“…Particularly in the noisy case, a small error in the given data can terribly detract the accuracy of numerical solution. In the past few years, the inverse Cauchy problems of elliptic equations have been studied extensively, and owing to its highly ill-posed nature [2], the inverse Cauchy problems associated with the elliptic type partial differential equations (PDEs) have been studied by using different numerical methods [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. There are two types approaches to the solution of inverse Cauchy problems.…”