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Abstract:

The stochastic collocation and Galerkin method are two state-of-the-art tools for solving stochastic partial differential equations (PDE). The former method is based on sampling the random variables present in a stochastic PDE into a set of multidimensional collocation points. The latter converts a stochastic PDE into a coupled set of deterministic PDEs after applying a spectral discretization in the stochastic dimension. While the stochastic collocation method can straightforwardly be extended to nonlinear stochastic PDEs, it is not trivial to apply the stochastic Galerkin procedure to a nonlinear PDE and approximations are required. In this talk, we shall apply both stochastic solvers to a particular nonlinear stochastic PDE and compare their accuracy and computational cost. We shall consider the stochastic simulation of a ferromagnetic cylinder rotating at high speed. This type of model can be found as part of solid-rotor induction machines in various machining tools. A precise design requires to take ferromagnetic saturation effects into account and needs to deal with uncertainty on the nonlinear magnetic material properties. We shall determine to what extent uncertainty on the material properties influences the machine properties. Both the stochastic collocation as well as the stochastic Galerkin method yield high-order accurate stochastic solutions. The stochastic collocation procedure applies either a tensor product grid or a sparse grid of collocation points. The stochastic Galerkin method requires in general more computational time than the stochastic collocation method to reach the same level of accuracy.