A Multilevel Monte Carlo Method for High-Dimensional Uncertainty Quantification of Low-Frequency Electromagnetic Devices

This paper 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 error 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 to 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.