This work addresses uncertainty quantification of electromagnetic devices determined by the eddy current problem. The multilevel Monte Carlo (MLMC) method is used for the treatment of uncertain parameters while the devices are discretized in space by the finite element method. Both methods yield numerical approximations such that the total errors is split into stochastic and spatial contributions. We propose a particular implementation where the spatial error is controlled based on a Richardson-extrapolation-based error indicator. The stochastic error in turn is efficiently reduced in the MLMC approach by distributing the samples on multiple grids. The method is applied to a toy problem with closed-form solution and a permanent magnet synchronous machine with uncertainties. The uncertainties under consideration are related to the material properties in the stator and the magnets in the rotor. The examples show that the error indicator works reliably, the meshes used for the different levels do not have to be nested and, most importantly, MLMC reduces the computational cost by at least one order of magnitude compared to standard Monte Carlo.

中文翻译：

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低频电磁器件高维不确定度量化的多级蒙特卡罗方法

这项工作解决了由涡流问题确定的电磁设备的不确定性量化问题。多级蒙特卡罗（MLMC）方法用于处理不确定参数，而设备通过有限元方法在空间中离散。这两种方法都会产生数值近似值，从而将总误差分为随机和空间贡献。我们提出了一种特定的实现方式，其中空间误差是基于基于理查森外推的误差指标来控制的。通过将样本分布在多个网格上，MLMC 方法又有效地减少了随机误差。该方法应用于具有封闭形式解的玩具问题和具有不确定性的永磁同步电机。所考虑的不确定性与定子中的材料特性和转子中的磁体有关。示例表明，误差指示器工作可靠，用于不同级别的网格不必嵌套，最重要的是，与标准蒙特卡罗相比，MLMC 将计算成本降低了至少一个数量级。