A finite element model for the data completion problem: analysis and assessment

Abstract

We investigate the computational advantages of the Steklov-Poincaré variational formulation, based on the uniqueness Holmgren theorem, for the badly ill-posed data completion problem. We study the discrete solution with a finite element approximation. The uniqueness issue is dealt with and we check that it is related to a discrete Holmgren result which requires a particular assumption on the mesh. Then, the important point is to show how the finite element variational problem may be recast into a least-squares problem. Lavrentiev's method turns out to be a Tikhonov regularization of an ‘underlying’ equation that will never be explicited. When employed in conjunction with the Morozov discrepancy principle to select the regularization parameter, the overall mathematical studies realized specifically for the Tikhonov method extends as well to the Lavrentiev method to conclude to a convergent regularization strategy. Similarly, the conjugate gradient method (CGM) may be considered as applied to a normal equation of that same ‘hidden underlying’ problem. We conduct a brief discussion about this method to explain why it yields a convergent strategy. We close with several numerical simulations for different geometries to assess the Lavrentiev finite element or the PCG finite element solution of the data completion problem.