Function approximation with redundancy: stability and applications
Daan Huybrechs, KU Leuven
Aula Beltrami, Dipartimento di Matematica – Martedì 14 Gennaio 2020 h.15:00
Abstract. It is common to discretize continuous problems in science and engineering using a basis, a non-redundant set of functions that is complete in a relevant function space. We show that great flexibility is gained by allowing redundancy in the discretization. Using redundancy one can easily deal with domains of complicated shape, one can introduce known singularities of the solution into the approximation space, or use other analytical knowledge that does not match well with a known basis. We supply several examples for partial differential equations and integral equations. Redundancy does lead to ill-conditioning of the discretization: we show that with a suitable strategy, based on least squares approximations, this ill-conditioning is largely benign and computations can be performed in a stable, and often even very efficient, manner.