Multilevel Monte Carlo methods for the Grad-Shafranov free boundary problem

TL;DR

This paper demonstrates that multilevel Monte Carlo methods significantly reduce computational costs in solving the Grad-Shafranov free boundary problem under uncertainty, especially when combined with adaptive grid refinement.

Contribution

It introduces multilevel Monte Carlo techniques for the Grad-Shafranov equation and shows their effectiveness in reducing computational costs and improving accuracy with adaptive grids.

Findings

Cost reductions of 60-200 times compared to standard methods

Adaptive grids improve accuracy of geometric quantities

Multilevel methods are effective for uncertainty quantification in plasma equilibrium

Abstract

The equilibrium configuration of a plasma in an axially symmetric reactor is described mathematically by a free boundary problem associated with the celebrated Grad--Shafranov equation. The presence of uncertainty in the model parameters introduces the need to quantify the variability in the predictions. This is often done by computing a large number of model solutions on a computational grid for an ensemble of parameter values and then obtaining estimates for the statistical properties of solutions. In this study, we explore the savings that can be obtained using multilevel Monte Carlo methods, which reduce costs by performing the bulk of the computations on a sequence of spatial grids that are coarser than the one that would typically be used for a simple Monte Carlo simulation. We examine this approach using both a set of uniformly refined grids and a set of adaptively refined grids guided by a discrete error estimator. Numerical experiments show that multilevel methods dramatically reduce the cost of simulation, with cost reductions typically on the order of 60 or more and possibly as large as 200. Adaptive gridding results in more accurate computation of geometric quantities such as x-points associated with the model.