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：轴向对称反应堆中等离子体的平衡构型，在数学上通过与其著名的Grad-Shafranov方程相关的自由边界问题来描述。模型参数中的不确定性引入了量化预测变异性的需求。这通常通过在一组计算网格上对参数值集合计算大量模型解，然后获取解的统计特性估计值来实现。在本研究中，我们探讨了使用多层蒙特卡洛方法所能获得的成本节省——该方法通过在一系列比简单蒙特卡洛模拟通常采用的网格更粗糙的空间网格上执行大部分计算来降低成本。我们使用一系列均匀细化网格和基于离散误差估计器引导的自适应细化网格来检验这一方法。数值实验表明，多层方法大幅降低了模拟成本，成本降低通常达60倍或更多，最大可能达到200倍。自适应网格划分能够更精确地计算几何量，例如与模型相关的X点。