RESOLUTION OF THE CAUCHY PROBLEM AND UNCERTAINTY QUANTIFICATION VIA THE STEKLOV-POINCARÉ APPROACH
Résumé
When carrying out non-destructive testing of mechanical parts, the identification of boundary conditionsfrom measurements on other boundaries is problematic. This problem is known as data completionproblem, or Cauchy problem, which is present in a wide range of applications.However, the Cauchy problem for elliptic PDEs is known to be mathematically ill-posed in the sense ofHadamard. Its theoretical properties have direct consequences on the feasibility of the resolution in anindustrial context.A range of methods have been developed in the past decade to adress the ill-posedness of the problem.One method has been selected in this study, namely the Steklov-Poincar ́e’s method [1], that relies on theinversion of a linear system that implies the Steklov-Poincar ́e operator by a Krylov solver.In [2], it was shown that it is possible to compute at very low cost the Ritz values and vectors, thatapproximate the eigenvalues and vectors of the Steklov-Poincar ́e’s operator that are arisen by the righthand side. These Ritz elements are then used to filter the solution.In the present work, we will show that it is possible to use this Ritz decomposition as a reduced ordermodel of the inverse problem, that will make the uncertainty quantification much less costly in termsof computing time. In particular, in the framework of Gaussian uncertainties, the computation of theuncertainty on the posterior distribution requires to invert the forward operator, which is trivial if thisoperator is replaced by its reduced form, which is diagonal in the Ritz basis. This procedure has been benumerically illustrated on 2D and 3D test cases