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Variance‐based simplex stochastic collocation with model order reduction for high‐dimensional systems
Author(s) -
Giovanis D.G.,
Shields M.D.
Publication year - 2018
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.5992
Subject(s) - sparse grid , collocation (remote sensing) , singular value decomposition , polynomial chaos , mathematics , simplex , uncertainty quantification , orthogonal collocation , mathematical optimization , model order reduction , collocation method , matrix (chemical analysis) , computer science , algorithm , ordinary differential equation , mathematical analysis , monte carlo method , differential equation , projection (relational algebra) , statistics , materials science , geometry , machine learning , composite material
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.