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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 focuses on the Stochastic Galerkin approximation of the solution u of an elliptic stochastic PDE, and uses the previous estimates to introduce a new effective class of Sparse Grids that features better convergence properties compared to standard Smolyak or tensor product grids.
Abstract: In this work we first focus on the Stochastic Galerkin approximation of the solution u of an elliptic stochastic PDE. We rely on sharp estimates for the decay of the coefficients of the spectral expansion of u on orthogonal polynomials to build a sequence of polynomial subspaces that features better convergence properties compared to standard polynomial subspaces such as Total Degree or Tensor Product. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new effective 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.
18 citations
01 Jan 2012
TL;DR: Fastsg is presented, an optimized library for the sparse grid technique with support for dimensional truncation with optimizations for best cache use and vectorization, which improves the performance on one processor core up to a factor of 10.
Abstract: In a complex processor landscape dominated by multi-and many-core processors, simplifying programming plays a crucial role in enhancing developers’ productivity. One way is to use highly tuned library functions. In this paper we present fastsg, an optimized library for the sparse grid technique with support for dimensional truncation. With optimizations for best cache use and vectorization, we improve the performance on one processor core up to a factor of 10. Parallelization using OpenMP scales almost linearly on a 12-core system.
18 citations
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TL;DR: In this paper, the authors give a convergence proof for the approximation by sparse collocation of Hilbert-space-valued functions depending on countably many Gaussian random variables, such functions appear as solutions of elliptic PDEs with lognormal diffusion coefficients.
Abstract: We give a convergence proof for the approximation by sparse collocation of Hilbert-space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions of elliptic PDEs with lognormal diffusion coefficients. We outline a general $L^2$-convergence theory based on previous work by Bachmayr et al. (2016) and Chen (2016) and establish an algebraic convergence rate for sufficiently smooth functions assuming a mild growth bound for the univariate hierarchical surpluses of the interpolation scheme applied to Hermite polynomials. We verify specifically for Gauss-Hermite nodes that this assumption holds and also show algebraic convergence w.r.t. the resulting number of sparse grid points for this case. Numerical experiments illustrate the dimension-independent convergence rate.
18 citations
TL;DR: The numerical results show the ability of the proposal to provide smooth and clearly defined structural boundaries and show that the method provides structural designs satisfying a trade-off between conflicting objectives in the RTO problem.
18 citations
TL;DR: Extensive numerical simulations in both linear and nonlinear systems in high dimensions, along with applications of diffusive logistic equations and Fokker-Planck equations, demonstrate the accuracy, efficiency, and robustness of the new methods, indicating potential broad applications of the sparse grid-based integration factor method.
18 citations