To be or not to be intrusive? The solution of parametric and stochastic equations - the "plain vanilla" Galerkin case

TL;DR

This paper explores how to approximate solutions to parametric and stochastic equations using Galerkin methods in a non-intrusive manner, enabling the use of existing solvers without modifying them.

Contribution

It demonstrates that simple Galerkin formulations can be computed non-intrusively, challenging the common view that they are inherently intrusive.

Findings

Non-intrusive Galerkin methods are feasible for simple formulations.

Coupled systems can be approximated without modifying original solvers.

The approach simplifies the application of Galerkin methods in practice.

Abstract

In parametric equations - stochastic equations are a special case - one may want to approximate the solution such that it is easy to evaluate its dependence of the parameters. Interpolation in the parameters is an obvious possibility, in this context often labeled as a collocation method. In the frequent situation where one has a "solver" for the equation for a given parameter value - this may be a software component or a program - it is evident that this can independently solve for the parameter values to be interpolated. Such uncoupled methods which allow the use of the original solver are classed as "non-intrusive". By extension, all other methods which produce some kind of coupled system are often - in our view prematurely - classed as "intrusive". We show for simple Galerkin formulations of the parametric problem - which generally produce coupled systems - how one may compute the approximation in a non-intusive way.