Gaussian process emulators for the stochastic finite element method

Abstract

SUMMARYThis paper explores a method to reduce the computational cost of stochastic finite element codes. The method, known as Gaussian process emulation, consists of building a statistical approximation to the output of such codes based on few training runs. The incorporation of emulation is explored for two aspects of the stochastic finite element problem. First, it is applied to approximating realizations of random fields discretized via the Karhunen-Loève expansion. Numerical results of emulating realizations of Gaussian and lognormal homogeneous two-dimensional random fields are presented. Second, it is coupled with the polynomial chaos expansion and the partitioned Cholesky decomposition in order to compute the response of the typical sparse linear system that arises due to the discretization of the partial differential equations that govern the response of a stochastic finite element problem. The advantages and challenges of adopting the proposed coupling are discussed.

Multivariate function approximations using the D-MORPH algorithm

“…Using the D-MORPH algorithm for estimating the unknown coefficients allows the proposed method to formulate the approximate polynomial based on a considerably lower number of data points compared with other methods, such as Kriging polynomial approximation [43,44,1,6,36] and polynomial chaos (PC)-based polynomial approximation [32][33][34][35][36][37]. Thus, the computational effort is lower with the proposed method.…”

“…Furthermore, unlike http://dx.doi.org/10.1016/j.apm.2015.03.008 0307-904X/Ó 2015 Elsevier Inc. All rights reserved. popular techniques, such as PCE [32][33][34][35][36][37], HDMR [45][46][47][48][49][8][9][10][50][51][52]36], and the kriging method [43,44,1,6,36], our proposed approach is capable of solving systems with more unknown parameters than sample points. We also propose a new weight matrix for use with D-MORPH.…”

“…This method is quick but the accuracy is questionable. Other methods for constructing approximate functions include the projection pursuit algorithm [23][24][25][26][27], multilayer perceptron [28][29][30][31], polynomial chaos expansion (PCE) [32][33][34][35][36][37], radial basis functions [38][39][40][41][42], Kriging method [43,44,1,6,36], high-dimensional model representation (HDMR) [45][46][47][48][49][8][9][10][50][51][52]36], and Bayesian emulators [53]. In this study, we introduce a new method for high-dimensional polynomial approximation based on diffeomorphic modulation under observable response preserving homotopy (D-MORPH) regression [54,55].…”

Explicit solution for linear systems with parametric stiffness and application to uncertain structures

“…The explicit relationships proposed herein could also be resorted to in the static analysis of interval structural problems [16][17][18][19][20][21][22], fuzzy problems [23], optimization and anti-optimization [24,25], stochastic finite elements methods [26][27][28][29][30], stability analysis of uncertain structures [31][32][33], re-analysis [34][35][36][37][38].…”

A Multi-scale Finite Element Approach for the Random Mechanical Response of Honeycomb-cored Structures

“…After conditioning on the training data and updating, the mean of the resulting posterior distribution provides a fast approximation to the code's output at any untried input, whereas it returns the known value of the code at each of the initial runs. Gaussian process emulators have already been implemented in several scientific fields, such as test crash modelling 19 and structural dynamics 20,21 .…”