Topic
Sparse grid
About: Sparse grid is a research topic. Over the lifetime, 1013 publications have been published within this topic receiving 20664 citations.
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TL;DR: This work describes three possible piecewise multilinear hierarchical interpolation schemes in detail, and documents the features of the sparse grid interpolation software package spinterp for MATLAB.
Abstract: To recover or approximate smooth multivariate functions, sparse grids are superior to full grids due to a significant reduction of the required support nodes. The order of the convergence rate in the maximum norm is preserved up to a logarithmic factor. We describe three possible piecewise multilinear hierarchical interpolation schemes in detail and conduct a numerical comparison. Furthermore, we document the features of our sparse grid interpolation software package spinterp for MATLAB.
233 citations
TL;DR: The proposed method provides accurate results for stochastic dimensionality as high as 500 even with large-input variability and the efficiency of the proposed method is examined by comparing with Monte Carlo (MC) simulation.
221 citations
TL;DR: A sparse pseudospectral approximation method that accurately reproduces the coefficients for basis functions that naturally correspond to the sparse grid integration rule by reexamining Smolyak's algorithm and exploiting the connections between interpolation and projection in tensor product spaces.
212 citations
01 Jan 2011
TL;DR: Numerical results for linear elliptic SPDEs indicate a slight computational work advantage of isotropic SC over SG, with SC-SM and SG-TD being the best choices of approximation spaces for each method.
Abstract: Much attention has recently been devoted to the development of Stochastic Galerkin (SG) and Stochastic Collocation (SC) methods for uncertainty quantification. An open and relevant research topic is the comparison of these two methods. By introducing a suitable generalization of the classical sparse grid SC method, we are able to compare SG and SC on the same underlying multivariate polynomial space in terms of accuracy vs. computational work. The approximation spaces considered here include isotropic and anisotropic versions of Tensor Product (TP), Total Degree (TD), Hyperbolic Cross (HC) and Smolyak (SM) polynomials. Numerical results for linear elliptic SPDEs indicate a slight computational work advantage of isotropic SC over SG, with SC-SM and SG-TD being the best choices of approximation spaces for each method. Finally, numerical results corroborate the optimality of the theoretical estimate of anisotropy ratios introduced by the authors in a previous work for the construction of anisotropic approximation spaces.
211 citations
01 Apr 2008
TL;DR: Experience with non-intrusive PCE methods for algebraic and PDE-based benchmark test problems is presented, demonstrating the need for accurate, efficient coefficient estimation approaches that can be used for problems with significant numbers of random variables.
Abstract: Polynomial chaos expansions (PCE) are an attractive technique for uncertainty quantification (UQ) due to their strong mathematical basis and ability to produce functional representations of stochastic variability. When tailoring the orthogonal polynomial bases to match the forms of the input uncertainties in a Wiener-Askey scheme, excellent convergence properties can be achieved for general probabilistic analysis problems. Non-intrusive PCE methods allow the use of simulations as black boxes within UQ studies, and involve the calculation of chaos expansion coefficients based on a set of response function evaluations. These methods may be characterized as being either Galerkin projection methods, using sampling or numerical integration, or regression approaches (also known as point collocation or stochastic response surfaces), using linear least squares. Numerical integration methods may be further categorized as either tensor product quadrature or sparse grid Smolyak cubature and as either isotropic or anisotropic. Experience with these approaches is presented for algebraic and PDE-based benchmark test problems, demonstrating the need for accurate, efficient coefficient estimation approaches that sca le for problems with significant numbers of random variables.
185 citations