Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
SIAM Journal on Scientific Computing
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Parallel computation of flow in heterogeneous media modelled by mixed finite elements
Journal of Computational Physics
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Decoupling Three-Dimensional Mixed Problems Using Divergence-Free Finite Elements
SIAM Journal on Scientific Computing
Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces
SIAM Journal on Numerical Analysis
Strong tractability of multivariate integration using quasi-Monte Carlo algorithms
Mathematics of Computation
Domain decomposition for multiscale PDEs
Numerische Mathematik
Constructing Embedded Lattice Rules for Multivariate Integration
SIAM Journal on Scientific Computing
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Constructing Sobol Sequences with Better Two-Dimensional Projections
SIAM Journal on Scientific Computing
Analysis of FETI methods for multiscale PDEs
Numerische Mathematik
Multi-element probabilistic collocation method in high dimensions
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Spectral element approximation of Fredholm integral eigenvalue problems
Journal of Computational and Applied Mathematics
A one-time truncate and encode multiresolution stochastic framework
Journal of Computational Physics
Hi-index | 31.46 |
We devise and implement quasi-Monte Carlo methods for computing the expectations of nonlinear functionals of solutions of a class of elliptic partial differential equations with random coefficients. Our motivation comes from fluid flow in random porous media, where relevant functionals include the fluid pressure/velocity at any point in space or the breakthrough time of a pollution plume being transported by the velocity field. Our emphasis is on situations where a very large number of random variables is needed to model the coefficient field. As an alternative to classical Monte Carlo, we here employ quasi-Monte Carlo methods, which use deterministically chosen sample points in an appropriate (usually high-dimensional) parameter space. Each realization of the PDE solution requires a finite element (FE) approximation in space, and this is done using a realization of the coefficient field restricted to a suitable regular spatial grid (not necessarily the same as the FE grid). In the statistically homogeneous case the corresponding covariance matrix can be diagonalized and the required coefficient realizations can be computed efficiently using FFT. In this way we avoid the use of a truncated Karhunen-Loeve expansion, but introduce high nominal dimension in parameter space. Numerical experiments with 2-dimensional rough random fields, high variance and small length scale are reported, showing that the quasi-Monte Carlo method consistently outperforms the Monte Carlo method, with a smaller error and a noticeably better than O(N^-^1^/^2) convergence rate, where N is the number of samples. Moreover, the rate of convergence of the quasi-Monte Carlo method does not appear to degrade as the nominal dimension increases. Examples with dimension as high as 10^6 are reported.