Interpolatory Rational Model Order Reduction of Parametric Problems Lacking Uniform Inf-Sup Stability

Davide Pradovera

doi:10.1137/19m1269695

Abstract

We present a technique for the approximation of a class of Hilbert space-valued maps which arise within the framework of Model Order Reduction for parametric partial differential equations, whose solution map has a meromorphic structure. Our MOR stategy consists in constructing an explicit rational approximation based on few snapshots of the solution, in an interpolatory fashion. Under some restrictions on the structure of the original problem, we describe a priori convergence results for our technique, hereafter called minimal rational interpolation, which show its ability to identify the main features (e.g. resonance locations) of the target solution map. We also investigate some procedures to obtain a posteriori error indicators, which may be employed to adapt the degree and the sampling points of the minimal rational interpolant. Finally, some numerical experiments are carried out to confirm the theoretical results and the effectiveness of our technique.

Highlights

Mathematical models based on PDEs are employed to analyze numerically a wide array of physical, financial, and engineering-related phenomena
Provided the set K ⊂ K of values of μ, for which a unique u(μ) solving (6) exists, is non-empty, we focus on the solution map associated to (6): u:K →V (7)
The form of the solution map is, in general, not (8): for instance, it cannot be guaranteed that the poles are simple, i.e. a λ-dependent exponent may appear in the denominator of (8)

Summary

Introduction

Mathematical models based on PDEs are employed to analyze numerically a wide array of physical, financial, and engineering-related phenomena. If one applies carelessly the RB method (in both its main versions, POD and greedy [35, 36]) to (1), one may observe the appearance of spurious (“non-physical”) resonances in the surrogate model Discussions concerning this effect can be found in works related to MOR methods for dynamical systems (e.g. Krylov subspace methods [20, 22]), where problems of similar form are quite common, due to the need for frequency-domain computations. In the wake of these methods for dynamical systems, and somehow trying to profit from their main advantages, univariate LS Pade approximants have been introduced and studied, both in a standard [9, 11] and a fast [10] version, in the context of a single parameter Such techniques are based on multiple solves of the FOM at a single parameter value, in the same spirit as Krylov subspace methods, but yield an explicit rational approximant like VF.

Description of the method

Parametric problem framework

Convergence theory

Preliminaries for convergence theory

Convergence of minimal rational interpolant poles

Pole convergence with respect to denominator degree

Convergence of minimal rational interpolants

Error convergence with respect to denominator degree

Generalizations to non-orthogonal residues

A posteriori error indicators

Extensions to multiple parameters

Numerical examples