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
Karhunen-Loève approximation of random fields by generalized fast multipole methods
Journal of Computational Physics - Special issue: Uncertainty quantification in simulation science
Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics (Scientific Computation)
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
Is Gauss Quadrature Better than Clenshaw-Curtis?
SIAM Review
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
The multi-element probabilistic collocation method (ME-PCM): Error analysis and applications
Journal of Computational Physics
Numerical Approximation of Partial Differential Equations
Numerical Approximation of Partial Differential Equations
Journal of Computational Physics
Convergence Rates of Best N-term Galerkin Approximations for a Class of Elliptic sPDEs
Foundations of Computational Mathematics
Certified Reduced Basis Methods and Output Bounds for the Harmonic Maxwell's Equations
SIAM Journal on Scientific Computing
A Reduced Basis Model with Parametric Coupling for Fluid-Structure Interaction Problems
SIAM Journal on Scientific Computing
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The stochastic collocation method (Babuška et al. in SIAM J Numer Anal 45(3):1005---1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411---2442, 2008a; SIAM J Numer Anal 46(5):2309---2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118---1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289---294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu, 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229---275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41---44):3187---3206, 2009; Arch Comput Methods Eng 17:435---454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions $$O(1)$$O(1) to moderate dimensions $$O(10)$$O(10) and to high dimensions $$O(100)$$O(100). The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs.