2018/10/31 by Dong Nguyen, Dong T. P. Nguyen, Dirk Nuyens · 4 citations
Computer Science · Decision Sciences · Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Finite element method #Lattice (music) #Mathematical Approximation and Integration #Mathematical analysis #Mathematics #Multivariate statistics #Polynomial #Probabilistic and Robust Engineering Design #cs.NA #math.NA #msc:65D30 #msc:65D32 #msc:65N30
paper · pdf · doi:10.1007/s00211-021-01212-9
published in Numerische Mathematik 148(3), 633-669 (Springer Science+Business Media) · 31 pages
openalex publication_date 2021/07/01 · arxiv created 2021/07/08 · arxiv updated 2021/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 (MDM), to compute infinite dimensional integrals, with the finite element method (FEM), 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 log-factor which typically appears in error bounds for multivariate quadrature, i.e., cubature, methods; and we take care of the fact that the number of points 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.