A Certified Natural-Norm Successive Constraint Method for Parametric Inf-Sup Lower Bounds

TL;DR

This paper introduces a certified version of the Natural-Norm Successive Constraint Method (cNNSCM) that provides guaranteed bounds for inf-sup constants, improving the reliability of reduced basis error analysis in parametric problems.

Contribution

The paper develops cNNSCM, which adds a certification step to NNSCM, providing guaranteed bounds and allowing control over the gap between bounds.

Findings

cNNSCM effectively provides certified lower and upper bounds.

Numerical experiments validate the accuracy and efficiency of cNNSCM.

The method improves reliability in reduced basis error estimation.

Abstract

We present a certified version of the Natural-Norm Successive Constraint Method (cNNSCM) for fast and accurate Inf-Sup lower bound evaluation of parametric operators. Successive Constraint Methods (SCM) are essential tools for the construction of a lower bound for the inf-sup stability constants which are required in {\it a posteriori} error analysis of reduced basis approximations. They utilize a Linear Program (LP) relaxation scheme incorporating continuity and stability constraints. The natural-norm approach {\em linearizes} inf-sup constant as a function of the parameter. The Natural-Norm Successive Constraint Method (NNSCM) combines these two aspects. It uses a greedy algorithm to select SCM control points which adaptively construct an optimal decomposition of the parameter domain, and then apply the SCM on each domain. Unfortunately, the NNSCM produces no guarantee for the quality of the lower bound. The new cNNSCM provides an upper bound in addition to the lower bound and let the user control the gap, thus the quality of the lower bound. The efficacy and accuracy of the new method is validated by numerical experiments.