Ten lectures on wavelets
The C++ Programming Language
Introduction to Algorithms
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
Multi-Element Generalized Polynomial Chaos for Arbitrary Probability Measures
SIAM Journal on Scientific Computing
Karhunen-Loève approximation of random fields by generalized fast multipole methods
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
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
Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
A Sparse Composite Collocation Finite Element Method for Elliptic SPDEs.
SIAM Journal on Numerical Analysis
Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems
SIAM Journal on Matrix Analysis and Applications
Computers & Mathematics with Applications
High-order methods as an alternative to using sparse tensor products for stochastic Galerkin FEM
Computers & Mathematics with Applications
Hi-index | 0.00 |
We propose and analyze sparse deterministic-stochastic tensor Galerkin finite element methods (sparse sGFEMs) for the numerical solution of elliptic partial differential equations (PDEs) with random coefficients in a physical domain $D\subset\mathbb{R}^d$. In tensor product sGFEMs, the variational solution to the boundary value problem is approximated in tensor product finite element spaces $V^\Gamma\otimes V^D$, where $V^\Gamma$ and $V^D$ denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces, respectively. These approaches lead to sGFEM algorithms of complexity $O(N_\Gamma N_D)$, where $N_\Gamma=\dim V^\Gamma$ and $N_D=\dim V^D$. In this work, we use hierarchic sequences $V^\Gamma_1\subset V^\Gamma_2\subset\ldots$ and $V^D_1\subset V^D_2\subset\ldots$ of finite dimensional spaces to approximate the law of the random solution. The hierarchies of approximation spaces allow us to define sparse tensor product spaces $V^\Gamma_\ell\hat{\otimes}V^D_\ell$, $\ell=1,2,\dots$, yielding algorithms of $O(N_\Gamma\log N_D+N_D\log N_\Gamma)$ work and memory. We estimate the convergence rate of sGFEM for algebraic decay of the input random field Karhunen-Loève coefficients. We give an algorithm for an input adapted a-priori selection of deterministic and stochastic discretization spaces. The convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to Monte Carlo approximations. Numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach.