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.

翻译：在轴对称反应堆中，等离子体的平衡构型在数学上由与著名的Grad-Shafranov方程相关的自由边界问题描述。模型参数中存在的不确定性要求量化预测结果的变异性。这通常通过在一组参数值的计算网格上求解大量模型解，继而获得解统计特性的估计来实现。本研究探索了通过多层蒙特卡罗方法所能实现的节省——该方法通过在比简单蒙特卡罗模拟通常采用的空间网格更粗糙的一系列网格上执行大部分计算来降低成本。我们使用一组均匀细化网格和由离散误差估计器引导的自适应细化网格分别检验了该方法。数值实验表明，多层方法显著降低了模拟成本，成本降低幅度通常达到60倍以上，最高可达200倍。自适应网格划分能够更精确地计算几何量，例如模型中与X点相关的几何特征。