Variance‐based simplex stochastic collocation with model order reduction for high‐dimensional systems

• Published in: International journal for numerical methods in engineering Vol. 117; no. 11; pp. 1079 - 1116
• Main Authors: Giovanis, D.G., Shields, M.D.
• Format: Journal Article
• Language: English
• Published: Bognor Regis Wiley Subscription Services, Inc 16.03.2019
• Subjects: Adaptive systems; Anisotropy; Collocation methods; higher‐order singular value decomposition; Mathematical models; Matrix algebra; Matrix methods; Methods; Model reduction; multi‐element collocation; polynomial chaos; Polynomials; reduced‐order model; simplex stochastic collocation; Singular value decomposition; Stochastic systems; Tensors; Variance
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.5992

Summary In this work, an adaptive simplex stochastic collocation method is introduced in which sample refinement is informed by variability in the solution of the system. The proposed method is based on the concept of multi‐element stochastic collocation methods and is capable of dealing with very high‐dimensional models whose solutions are expressed as a vector, a matrix, or a tensor. The method leverages random samples to create a multi‐element polynomial chaos surrogate model that incorporates local anisotropy in the refinement, informed by the variance of the estimated solution. This feature makes it beneficial for strongly nonlinear and/or discontinuous problems with correlated non‐Gaussian uncertainties. To solve large systems, a reduced‐order model (ROM) of the high‐dimensional response is identified using singular value decomposition (higher‐order SVD for matrix/tensor solutions) and polynomial chaos is used to interpolate the ROM. The method is applied to several stochastic systems of varying type of response (scalar/vector/matrix) and it shows considerable improvement in performance compared to existing simplex stochastic collocation methods and adaptive sparse grid collocation methods.