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

## TL;DR

This paper presents the MDFEM, a novel method combining multivariate decomposition, higher-order QMC, and FEM to efficiently approximate expectations of solutions to elliptic PDEs with lognormal coefficients, achieving higher-order convergence.

## Contribution

The paper introduces the MDFEM, integrating multivariate decomposition with higher-order QMC and FEM, for improved efficiency in high-dimensional elliptic PDEs with lognormal coefficients.

## Key findings

- Achieves higher-order convergence rates in error versus cost.
- Develops higher-order QMC rules for Gaussian integrals.
- Provides complexity estimates for the MDFEM algorithm.

## 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 $Z(\boldsymbol{y}) = \sum_{j\ge1} y_j\,\phi_j$ with $y_j\sim\mathcal{N}(0,1)$ and a given sequence of functions $\{\phi_j\}_{j\ge1}$. 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 $O(\epsilon^{-1/\lambda-d'/\lambda}) = O(\epsilon^{-(p^*+d'/\tau)/(1-p^*)})$ where $\epsilon^{-1/\lambda}$ and $\epsilon^{-d'/\lambda}$ are respectively the cost of the quasi-Monte Carlo cubature and the finite element approximations, with $d' = d \, (1+\delta')$ for some $\delta' \ge 0$ and $d$ the physical dimension, and $0 < p^* \le (2+d'/\tau)^{-1}$ is a parameter representing the sparsity of $\{\phi_j\}_{j\ge1}$.