Addressing the curse of dimensionality in SSFEM using the dependence of eigenvalues in KL expansion on domain size

Abstract

This work is aimed at reducing the dimensionality in the spectral stochastic finite element method (SSFEM)–thus the computational cost–through a domain decomposition (DD) method. This reduction hinges on some new mathematical results on domain size dependence of the Karhunen–Loève (KL) expansion. It has been reported in the literature that for few covariance kernels a lower domain size leads to a faster convergence of the KL eigenvalues. This observation leads to an early truncation of the KL expansion, and this reduction in stochastic dimensionality brings down the total computational cost. In this work first we mathematically show the generalization of this faster convergence, that is, for any arbitrary covariance kernel. This is achieved via developing a bound on eigenvalues as a function of the domain size. Then we prove that for a chosen number of terms in the KL expansion with any kernel for a one-dimensional process, the approximation error in the trace norm reduces with the domain size. Based on this domain size dependence, we propose an algorithm for solving a stochastic elliptic equation in a DD framework. The computational cost gain is demonstrated by a numerical study and is observed that the serial implementation of the proposed algorithm is about an order of magnitude faster compared to the existing method. The cost saving increases with the stochastic dimensionality in the global domain. The structure of this algorithm provides scope for parallelization, which would help in efficiently solving large scale problems. The sharpness of the proposed eigenvalue bounds is also tested for Gaussian and exponential kernels. The generalization opens avenues for developing further DD based SSFEM solvers.