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

Abstract

We introduce the multivariate decomposition finite element method for elliptic PDEs with lognormal diffusion coefficient $a=\exp (Z)$ where $Z$ is a Gaussian random field defined by an infinite series expansion $Z(\mathbf{y})={\sum }_{j\ge 1}{y}_j{\varphi }_j$ with $y_j\sim \mathcal{N}(0,1)$ and a given sequence of functions $\{{\varphi }_j{\}}_{j\ge 1}$. We use the MDFEM to approximate the expected value of a linear functional of the solution of the PDE which is an infinite-dimensional integral over the parameter space. The proposed algorithm uses the multivariate decomposition method (MDM) to compute the infinite-dimensional integral by a decomposition into finite-dimensional integrals, which we resolve using quasi-Monte Carlo (QMC) methods, and for which we use the finite element method (FEM) to solve different instances of the PDE. We develop higher-order quasi-Monte Carlo rules for integration over the finite-dimensional Euclidean space with respect to the Gaussian distribution by use of a truncation strategy. By linear transformations of interlaced polynomial lattice rules from the unit cube to a multivariate box of the Euclidean space we achieve higher-order convergence rates for functions belonging to a class of anchored Gaussian Sobolev spaces, taking into account the truncation error. Under appropriate conditions, the MDFEM achieves higher-order convergence rates in term of error versus cost, i.e., to achieve an accuracy of $O(\epsilon )$ the computational cost is $O({\epsilon }^{-1/\lambda -d^{\prime }/\lambda })=O({\epsilon }^{-(p^{\ast }+d^{\prime }/\tau )/(1-p^{\ast })})$ where ${\epsilon }^{-1/\lambda }$ and ${\epsilon }^{-d^{\prime }/\lambda }$ are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with $d^{\prime }=d(1+{\delta }^{\prime })$ for some ${\delta }^{\prime }\ge 0$ and $d$ the physical dimension, and $0<p^{\ast }\le (2+d^{\prime }/\tau {)}^{-1}$ is a parameter representing the sparsity of $\{{\varphi }_j{\}}_{j\ge 1}$.

49 pages

References in corpus (1)

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

Cited by in corpus (3)

• Scaled lattice rules for integration on ${\mathbb{R}}^d$ achieving higher-order convergence with error analysis in terms of orthogonal projections onto periodic spaces
• Goal-Oriented Adaptive Finite Element Multilevel Monte Carlo with Convergence Rates
• MDFEM: Multivariate decomposition finite element method for elliptic PDEs with uniform random diffusion coefficients using higher-order QMC and FEM