Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Physica D - Special issue originating from the 18th Annual International Conference of the Center for Nonlinear Studies, Los Alamos, NM, May 11&mdash ;15, 1998
The Wiener--Askey Polynomial Chaos for Stochastic Differential Equations
SIAM Journal on Scientific Computing
Journal of Computational Physics
Uncertainty propagation using Wiener-Haar expansions
Journal of Computational Physics
High-Order Collocation Methods for Differential Equations with Random Inputs
SIAM Journal on Scientific Computing
Multi-Resolution-Analysis Scheme for Uncertainty Quantification in Chemical Systems
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
Approximating Infinity-Dimensional Stochastic Darcy's Equations without Uniform Ellipticity
SIAM Journal on Numerical Analysis
Uncertainty Quantification and Weak Approximation of an Elliptic Inverse Problem
SIAM Journal on Numerical Analysis
Multilevel Monte Carlo methods for highly heterogeneous media
Proceedings of the Winter Simulation Conference
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We consider the problem of numerically approximating the solution of an elliptic partial differential equation with random coefficients and homogeneous Dirichlet boundary conditions. We focus on the case of a lognormal coefficient and deal with the lack of uniform coercivity and uniform boundedness with respect to the randomness. This model is frequently used in hydrogeology. We approximate this coefficient by a finite dimensional noise using a truncated Karhunen-Loève expansion. We give estimates of the corresponding error on the solution, both a strong error estimate and a weak error estimate, that is, an estimate of the error commited on the law of the solution. We obtain a weak rate of convergence which is twice the strong one. In addition, we give a complete error estimate for the stochastic collocation method in this case, where neither coercivity nor boundedness is stochastically uniform. To conclude, we apply these results of strong and weak convergence to two classical cases of covariance kernel choices, the case of an exponential covariance kernel on a box and the case of an analytic covariance kernel, yielding explicit weak and strong convergence rates.