Randomized residual-based error estimators for parametrized equations

TL;DR

This paper introduces a randomized a posteriori error estimator for reduced order models of parametrized PDEs, which is efficient, probabilistically reliable, and does not require stability constants, demonstrated on a Helmholtz problem.

Contribution

It presents a novel randomized error estimator that is efficient, does not need stability constants, and extends to multiple queries with probabilistic guarantees.

Findings

Estimator effectivity close to unity with high probability

No need for stability constant calculations

Effective reduced dual spaces for Helmholtz problem

Abstract

We propose a randomized a posteriori error estimator for reduced order approximations of parametrized (partial) differential equations. The error estimator has several important properties: the effectivity is close to unity with prescribed lower and upper bounds at specified high probability; the estimator does not require the calculation of stability (coercivity, or inf-sup) constants; the online cost to evaluate the a posteriori error estimator is commensurate with the cost to find the reduced order approximation; the probabilistic bounds extend to many queries with only modest increase in cost. To build this estimator, we first estimate the norm of the error with a Monte-Carlo estimator using Gaussian random vectors whose covariance is chosen according to the desired error measure, e.g. user-defined norms or quantity of interest. Then, we introduce a dual problem with random right-hand side the solution of which allows us to rewrite the error estimator in terms of the residual of the original equation. In order to have a fast-to-evaluate estimator, model order reduction methods can be used to approximate the random dual solutions. Here, we propose a greedy algorithm that is guided by a scalar quantity of interest depending on the error estimator. Numerical experiments on a multi-parametric Helmholtz problem demonstrate that this strategy yields rather low-dimensional reduced dual spaces.