MDFEM: Multivariate decomposition finite element method for elliptic PDEs with uniform random diffusion coefficients using higher-order QMC and FEM

Abstract

We introduce the multivariate decomposition finite element method (MDFEM) for solving elliptic PDEs with uniform random diffusion coefficients. We show that the MDFEM can be used to reduce the computational complexity of estimating the expected value of a linear functional of the solution of the PDE. The proposed algorithm combines the multivariate decomposition method, to compute infinite-dimensional integrals, with the finite element method, to solve different instances of the PDE. The strategy of the MDFEM is to decompose the infinite-dimensional problem into multiple finite-dimensional ones which lends itself to easier parallelization than to solve a single large dimensional problem. Our first result adjusts the analysis of the multivariate decomposition method to incorporate the \((\ln (n))^d\)-factor which typically appears in error bounds for d-dimensional n-point cubature formulae and we take care of the fact that n needs to come, e.g., in powers of 2 for higher order approximations. For the further analysis we specialize the cubature methods to be two types of quasi-Monte Carlo (QMC) rules, being digitally shifted polynomial lattice rules and interlaced polynomial lattice rules. The second and main contribution then presents a bound on the error of the MDFEM and shows higher-order convergence w.r.t. the total computational cost in case of the interlaced polynomial lattice rules in combination with a higher-order finite element method. We show that the cost to achieve an error \(\epsilon \) is of order \(\epsilon ^{-a_{\mathrm {MDFEM}}}\) with \(a_{\mathrm {MDFEM}} = 1/\lambda + d'/\tau \) if the QMC cubature errors can be bounded by \(n^{-\lambda }\) and the FE approximations converge like \(h^\tau \) with cost \(h^{d'}\), where \(\lambda = \tau (1-p^*) / (p^* (1+d'/\tau ))\) and \(p^*\) is a parameter representing the “sparsity” of the random field expansion. A comparison with a dimension truncation algorithm shows that the MDFEM will perform better than the truncation algorithm if \(p^*\) is sufficiently small, i.e., the representation of the random field is sufficiently sparse.