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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 paper constructs mapped Chebyshev sparse grid methods which enjoy the following properties and applies the new method to solve the electronic Schrodinger equation.
Abstract: This is the second part in a series of papers on using spectral sparse grid methods for solving higher-dimensional PDEs. We extend the basic idea in the first part [J. Shen and H. Yu, SIAM J. Sci. Comp., 32 (2010), pp. 3228-3250] for solving PDEs in bounded higher-dimensional domains to unbounded higher-dimensional domains and apply the new method to solve the electronic Schrodinger equation. By using modified mapped Chebyshev functions as basis functions, we construct mapped Chebyshev sparse grid methods which enjoy the following properties: (i) the mapped Chebyshev approach enables us to build sparse grids with Smolyak's algorithms based on nested, spectrally accurate quadratures and allows us to build fast transforms between the values at the sparse grid points and the corresponding expansion coefficients; (ii) the mapped Chebyshev basis functions lead to identity mass matrices and very sparse stiffness matrices for problems with constant coefficients and allow us to construct a matrix-vector product algorithm with quasi-optimal computational cost even for problems with variable coefficients; and (iii) the resultant linear systems for elliptic equations with constant or variable coefficients can be solved efficiently by using a suitable iterative scheme. Ample numerical results are presented to demonstrate the efficiency and accuracy of the proposed algorithms.
41 citations
TL;DR: In this paper, coordinate transformation techniques in combination with grid stretching for pricing basket options in a sparse grid setting were evaluated for multi-asset examples with up to five underlying assets in the basket.
40 citations
TL;DR: The present work generalizes 49 to account for the impact of the PG discretization in the forward maps on the convergence rates of the Quantities of Interest (QoI) and proposes to accelerate Bayesian estimation by first offline construction of reduced basis surrogates of the Bayesian posterior density.
40 citations
TL;DR: This paper develops a numerical technique which makes Ulam's approach applicable to systems with higher dimensional long term dynamics, and uses a sparse Haar tensor basis as the underlying approximation space.
Abstract: The global macroscopic behavior of a dynamical system is encoded in the eigenfunctions of the associated Frobenius-Perron operator For systems with low dimensional long term dynamics, efficient techniques exist for a numerical approximation of the most important eigenfunctions; cf [M Dellnitz and O Junge, SIAM J Numer Anal, 36 (1999), pp 491-515] They are based on a projection of the operator onto a space of piecewise constant functions supported on a neighborhood of the attractor—Ulam's method In this paper we develop a numerical technique which makes Ulam's approach applicable to systems with higher dimensional long term dynamics It is based on ideas for the treatment of higher dimensional partial differential equations using sparse grids [C Zenger, Sparse grids, in Parallel Algorithms for Partial Differential Equations (Kiel, 1990), Vieweg, Braunschweig, 1991, pp 241-251; H-J Bungartz and M Griebel, Acta Numer, 13 (2004), pp 147-269] Here, we use a sparse Haar tensor basis as the underlying approximation space We develop the technique, establish statements about its complexity and convergence, and present two numerical examples
40 citations