Multivariate Decomposition Finite Element Method for Elliptic PDEs with Random Diffusion Coefficients
We develop a multivariate decomposition finite element method (MDFEM) for solving elliptic PDEs with random diffusion coefficients where the diffusion coefficient is parameterized by a countably infinite number of random variables and an appropriate system of functions. We consider two models of parametric diffusion coefficients: uniform and lognormal.
We show that the MDFEM reduces 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 lend itself to easier parallelization than to solve a single large dimensional problem.
We adjust the analysis of the MDM to incorporate the log-factor which may or may not appear in error bounds for multivariate quadrature, i.e., cubature, methods; as this is needed for our analysis.
For the uniform case we specialize the cubature rules to be two types of quasi-Monte Carlo (QMC) rules, being digitally shifted polynomial lattice rules and higher-order polynomial lattice rules.
We then present a bound on the error of the MDFEM and show higher-order convergence w.r.t.\ the total computational cost in case of higher-order polynomial lattice rules in combination with higher-order finite element methods.
We extend the MDFEM algorithm to the lognormal case. The chief difficulty is that in this case the MDFEM algorithm requires cubature methods over the unbounded Euclidean space with respect to the Gaussian distribution. The idea is to truncate the Euclidean domain and then transform the resulting integrals into the unit cube in such a way that they can be approximated by higher-order QMC rules, in particular, by interlaced polynomial lattice rules. Applying such cubature rules for the MDFEM algorithm, we achieve higher-order convergence in terms of error versus total computational cost, hence, considerably improve upon existing results which only use first order cubature rules.
Finally, we use a similar truncation strategy to approximate multivariate integrals over the Euclidean space for functions which are analytic, in fact, this was studied first. We use an infinite grid with different mesh sizes in each direction to sample the function, and then truncate it. In our analysis, the mesh sizes and the truncated domain are chosen by optimally balancing the truncation error and the discretization error. We show explicit upper bounds which attain the exponential rate of convergence for a space of weighted analytic functions.
Numerical tests confirm our theoretical findings.