ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS
Copyright policy )ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS
In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids.
- Polytechnic University of Milan Italy
- École Normale Supérieure France
- King Abdullah University of Science and Technology Saudi Arabia
Numerical solutions to stochastic differential and integral equations, best M-terms polynomial approximation, numerical examples, convergence, uncertainty quantification, sparse grids, elliptic equations, PDEs with random data, Uncertainty quantification; PDEs with random data; elliptic equations; multivariate polynomial approximation; best M-terms polynomial approximation; stochastic Galerkin methods; Smolyak approximation; sparse grids; stochastic collocation methods, Stability and convergence of numerical methods for boundary value problems involving PDEs, Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, stochastic Galerkin methods, Stochastic partial differential equations (aspects of stochastic analysis), stochastic coefficients, stochastic collocation methods, Spectral, collocation and related methods for boundary value problems involving PDEs, PDEs with randomness, stochastic partial differential equations, linear elliptic equations, Uncertainty quantification, Computational methods for stochastic equations (aspects of stochastic analysis), multivariate polynomial approximation, Smolyak approximation
Numerical solutions to stochastic differential and integral equations, best M-terms polynomial approximation, numerical examples, convergence, uncertainty quantification, sparse grids, elliptic equations, PDEs with random data, Uncertainty quantification; PDEs with random data; elliptic equations; multivariate polynomial approximation; best M-terms polynomial approximation; stochastic Galerkin methods; Smolyak approximation; sparse grids; stochastic collocation methods, Stability and convergence of numerical methods for boundary value problems involving PDEs, Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, stochastic Galerkin methods, Stochastic partial differential equations (aspects of stochastic analysis), stochastic coefficients, stochastic collocation methods, Spectral, collocation and related methods for boundary value problems involving PDEs, PDEs with randomness, stochastic partial differential equations, linear elliptic equations, Uncertainty quantification, Computational methods for stochastic equations (aspects of stochastic analysis), multivariate polynomial approximation, Smolyak approximation
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