Least squares solvers for ill-posed PDEs that are conditionally stable

Abstract

ABSTRACT. This paper is concerned with the design and analysis of least squares solvers for ill-posed PDEs that are conditionally stable. The norms and the regularization term used in the least squares functional are determined by the ingredients of the conditional stability assumption. We are then able to establish a general error bound that, in view of the conditional stability assumption, is qualitatively the best possible, without assuming consistent data. The price for these advantages is to handle dual norms which reduces to verifying suitable inf-sup stability. This, in turn, is done by constructing appropriate Fortin projectors for all sample scenarios. The theoretical findings are illustrated by numerical experiments.

Optimal Approximation of Unique Continuation

“…In this paper we have shown that the convergence order of the approximation error for unique continuation problems, obtained by combining the approximation orders of the data fitting and the pde-residual with the conditional stability, can not be improved without increasing the sensitivity to perturbations. This shows that the asymptotic accuracy of the methods for unique continuation discussed in [10,11,[14][15][16][17]21] is optimal, in the sense that it is impossible to design a method with better convergence properties. The only remaining possibilities to enhance the accuracy of approximation methods is either to resort to adaptivity, or to introduce some additional a priori assumption to make the continuous problem more stable, such as finite dimensionality of target quantities (see [19]).…”

“…For ill-posed PDEs that are conditionally stable, error estimates in terms of the modulus of continuity in the conditional stability, the consistency error and the best approximation error have also been obtained in [21]. Based on least squares with the norms and the regularisation term dictated by the conditional stability estimate, this variation of quasi-reversibility relies on working with discrete dual norms and constructing Fortin projectors.…”