True Error Control for the Localized Reduced Basis Method for Parabolic Problems

TL;DR

This paper develops a comprehensive a posteriori error estimation framework for localized reduced basis methods applied to parabolic problems, enabling true error certification and improved model reduction accuracy.

Contribution

It extends existing error estimation techniques to localized RB methods for parabolic equations, providing offline/online decomposable error bounds including discretization and model reduction errors.

Findings

Effective error bounds demonstrated through numerical experiments

Framework generalizes localized RB with true error certification

Applicable to scalar parabolic evolution equations

Abstract

We present an abstract framework for a posteriori error estimation for approximations of scalar parabolic evolution equations, based on elliptic reconstruction techniques [10, 9, 3, 5]. In addition to its original application (to derive error estimates on the discretization error), we extend the scope of this framework to derive offline/online decomposable a posteriori estimates on the model reduction error in the context of Reduced Basis (RB) methods. In addition, we present offline/online decomposable a posteriori error estimates on the full approximation error (including discretization as well as model reduction error) in the context of the localized RB method [14]. Hence, this work generalizes the localized RB method with true error certification to parabolic problems. Numerical experiments are given to demonstrate the applicability of the approach.