Non-Intrusive Reduced-Order Modeling of Parameterized Electromagnetic Scattering Problems using Cubic Spline Interpolation

Abstract

This paper presents a non-intrusive model order reduction (MOR) for the solution of parameterized electromagnetic scattering problems, which needs to prepare a database offline of full-order solution samples (snapshots) at some different parameter locations. The snapshot vectors are produced by a high order discontinuous Galerkin time-domain (DGTD) solver formulated on an unstructured simplicial mesh. Because the second dimension of snapshots matrix is large, a two-step or nested proper orthogonal decomposition (POD) method is employed to extract timeand parameter-independent POD basis functions. By using the singular value decomposition (SVD) method, the principal components of the projection coefficient matrices (also referred to as the reduced coefficient matrices) of full-order solutions onto the RB subspace are extracted. A cubic spline interpolation-based (CSI) approach is proposed to approximate the dominating time-and parameter-modes of the reduced coefficient matrices without resorting to Galerkin projection. The generation of snapshot vectors, the construction of POD basis functions and the approximation of reduced coefficient matrices based on the CSI method are completed during the offline stage. The RB solutions for new time and parameter values can be rapidly recovered via outputs from the interpolation models in the online stage. In particular, the offline and online stages of the proposed RB method, termed as the POD-CSI method, are completely decoupled, which ensures the computational validity of the method. Moreover, a surrogate error model is constructed as an efficient error estimator for the POD-CSI method. Numerical experiments for the scattering of plane wave by a 2-D dielectric cylinder and a multi-layer heterogeneous medium nicely illustrate the performance of POD-CSI method.

Data Assimilation Predictive GAN (DA-PredGAN) Applied to a Spatio-Temporal Compartmental Model in Epidemiology

“…In the third step, the high-dimensional system is projected onto the low-dimensional space for a projection-based ROM [ 6 ], whereas for a non-intrusive ROM (NIROM), the snapshots are projected onto the low-dimensional space and the dynamics of the system in this space for unseen parameters are represented by interpolation. This can be performed by classical interpolation methods such as cubic splines [ 36 ] or radial basis functions (RBF) [ 4 ]. The RBF approach was extended by [ 65 ] who used a Smolyak grid to sample the parameter space; by [ 34 ] who interpolated values of model parameters and time levels using one parametrisation; and by [ 2 ] who used adaptive sampling in time.…”

Co-Design Secure Control Based on Image Attack Detection and Data Compensation for Networked Visual Control Systems

“…If the images are judged as "invalid" and discarded by the above double-level attack detector, data compensation must be designed to guarantee the stability of NVCSs. Due to good characteristics such as continuity of derivatives and interpolation and low computational burden, etc., cubic spline interpolation algorithm [46] is adopted, as shown in Section I. C of the supplementary materials.…”

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

“…Based on this idea, one could construct a reduced-order model (ROM) by using efficient model order reduction (MOR) methods dramatically reducing the number of DoFs. The overall goal of the MOR method is to reduce the computational cost by several orders of magnitude while maintaining an acceptable level of accuracy [44,22].…”

“…Different from the IMOR method, the NIMOR method only uses the approximation mappings obtained by some data-driven methods [6], such as interpolation, regression, and artificial neural networks (ANN) methods, to calculate the reducedorder coefficients for new time/parameter values. There have been a lot of works [22,13,46,53,49,1,2,7,10,19,42,11] on the NIMOR method in the very recent years. Audouze et al in [2] presented a NIMOR method for nonlinear parametric time dependent PDEs using POD and radial basis function (RBF) approximation.…”

“…However, the drawback of NIMOR method is the high cost of the offline stage, since a large number of high-fidelity simulations are needed to generate a sufficiently large data set to train an accurate regression model [9]. Therefore, we consider a NIMOR method based on the Smolyak sparse grid, RBF and GPR methods for parameterized time-domain Maxwell's equations in this study, which is different from our previous work [22], in which the training parameter samples are generated via uniform sampling and the cubic spline interpolation-based approach is proposed to approximate the dominating time-and parameter-modes of the reduced coefficient matrices. In current work, a small size snapshot set is firstly generated from a high fidelity DGTD solver at the Smolyak sparse grid with approximation level L (L ≥ 1), and the RBF method is then used to produce more snapshot vectors at the sparse grid with approximation level L + 1 to minimize the overall computational offline cost and improve the accuracy of the NIMOR method.…”