Preconditioned conjugate gradients for solving singular systems
Journal of Computational and Applied Mathematics - Special issue on iterative methods for the solution of linear systems
Stochastic finite elements: a spectral approach
Stochastic finite elements: a spectral approach
Multilevel algorithms considered as iterative methods on semidefinite systems
SIAM Journal on Scientific Computing
The solution of multidimensional real Helmholtz equations on Sparse Grids
SIAM Journal on Scientific Computing
A Second Degree Method for Nonlinear Inverse Problems
SIAM Journal on Numerical Analysis
Numerical Challenges in the Use of Polynomial Chaos Representations for Stochastic Processes
SIAM Journal on Scientific Computing
Worst case scenario analysis for elliptic problems with uncertainty
Numerische Mathematik
Adaptive Solution of Operator Equations Using Wavelet Frames
SIAM Journal on Numerical Analysis
Numerical Methods for Differential Equations in Random Domains
SIAM Journal on Scientific Computing
A fictitious domain approach to the numerical solution of PDEs in stochastic domains
Numerische Mathematik
A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
SIAM Journal on Numerical Analysis
Sparse second moment analysis for elliptic problems in stochastic domains
Numerische Mathematik
Multilevel frames for sparse tensor product spaces
Numerische Mathematik
Combination technique based k-th moment analysis of elliptic problems with random diffusion
Journal of Computational Physics
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We compute the expectation and the two-point correlation of the solution to elliptic boundary value problems with stochastic loadings. In case of elliptic problems on stochastic domains or with stochastic coefficients analogous expressions hold to leading order in the size of the stochastic perturbation. The solution's two-point correlation satisfies a deterministic tensor product partial differential equation on the twofold product domain. For its numerical solution we apply a sparse tensor product approximation by multilevel frames. This way standard finite element techniques can be used. Numerical examples illustrate feasibility and scope of the method.