A non-intrusive model order reduction approach for parameterized time-domain Maxwell's equations

Abstract

International audienceWe present a non-intrusive model order reduction (NIMOR) method with an offline-online decoupling for the solution of parameterized time-domain Maxwell's equations. During the offline stage, the training parameters are chosen by using Smolyak sparse grid method with an approximation level L (L ≥ 1) over a target parameterized space. This method can deal with the so-called curse of dimensionality in high dimensional space. For each selected parameter, the snapshot vectors are first produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. In order to minimize the overall computational cost in the offline stage and to improve the accuracy of the NIMOR method, a radial basis function (RBF) interpolation method is then used to construct more snapshot vectors at the sparse grid with approximation level L + 1, which includes the sparse grids from approximation level L. A nested proper orthogonal decomposition (POD) method is employed to extract time-and parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the reduced coefficient matrices of the high-fidelity solutions onto the reduced-order subspace spaned by the POD basis functions are extracted. Moreover, a Gaussian process regression (GPR) method is proposed to approximate the dominating time-and parameter-modes of the reduced coefficient matrices. During the online stage, the reduced-order solutions for new time and parameter values can be rapidly recovered via outputs from the regression models without using the DGTD method. The performance of this NIMOR method is illustrated numerically by considering two classical test cases: the scattering of a plane wave by a 2-D dielectric disk and the scattering of a plane wave by a multi-layer heterogeneous medium. The prediction capabilities of the NIMOR method are evaluated by varying the relative permittivity. Numerical results indicate that the NIMOR method is a promising approach for simulating accurately and in fast way parameterized timedomain electromagnetic problems