Generating functionology
Ten lectures on wavelets
Explicit cost bounds of algorithms for multivariate tensor product problems
Journal of Complexity
High-Order Collocation Methods for Differential Equations with Random Inputs
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
Sparse grid collocation schemes for stochastic natural convection problems
Journal of Computational Physics
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
Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs
Foundations of Computational Mathematics
Sparse Tensor Discretization of Elliptic sPDEs
SIAM Journal on Scientific Computing
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This work presents a stochastic collocation method for solving elliptic PDEs with random coefficients and forcing term which are assumed to depend on a finite number of random variables. The method consists of a hierarchic wavelet discretization in space and a sequence of hierarchic collocation operators in the probability domain to approximate the solution's statistics. The selection of collocation points is based on a Smolyak construction of zeros of orthogonal polynomials with respect to the probability density function of each random input variable. A sparse composition of levels of spatial refinements and stochastic collocation points is then proposed and analyzed, resulting in a substantial reduction of overall degrees of freedom. Like in the Monte Carlo approach, the algorithm results in solving a number of uncoupled, purely deterministic elliptic problems, which allows the integration of existing fast solvers for elliptic PDEs. Numerical examples on two-dimensional domains will then demonstrate the superiority of this sparse composite collocation finite element method compared to the “full composite” collocation finite element method and the Monte Carlo method.