Sparse grid

About: Sparse grid is a research topic. Over the lifetime, 1013 publications have been published within this topic receiving 20664 citations.

Parallel Multigrid Methods on Sparse Grids

Abstract: This paper deals with multigrid methods on two-dimensional sparse grids and their vectorization. First we will introduce sparse grids and discuss briefly their properties. Sparse grids contain only O(n ld n) grid points in contrast to the usually used O(n2)-grids whereas, for a sufficiently smooth function, the accuracy of the representation is only slightly deteriorated from O(n−2) to O(n−2 ld n). We sketch the main features of a multigrid method that works on these sparse grids and discuss its vectorization and parallelization aspects. Additionally, we present the results of numerical experiments for an implementation of this algorithm on the CRAY-Y-MP.

Adaptive Smolyak Pseudospectral Approximations

Abstract: Polynomial approximations of computationally intensive models are central to uncertainty quantification. This paper describes an adaptive method for non-intrusive pseudospectral approximation, based on Smolyak's algorithm with generalized sparse grids. We rigorously analyze and extend the non-adaptive method proposed in [6], and compare it to a common alternative approach for using sparse grids to construct polynomial approximations, direct quadrature. Analysis of direct quadrature shows that O(1) errors are an intrinsic property of some configurations of the method, as a consequence of internal aliasing. We provide precise conditions, based on the chosen polynomial basis and quadrature rules, under which this aliasing error occurs. We then establish theoretical results on the accuracy of Smolyak pseudospectral approximation, and show that the Smolyak approximation avoids internal aliasing and makes far more effective use of sparse function evaluations. These results are applicable to broad choices of quadrature rule and generalized sparse grids. Exploiting this flexibility, we introduce a greedy heuristic for adaptive refinement of the pseudospectral approximation. We numerically demonstrate convergence of the algorithm on the Genz test functions, and illustrate the accuracy and efficiency of the adaptive approach on a realistic chemical kinetics problem.

Solution of Time-dependent Advection-Diffusion Problems with the Sparse-grid Combination Technique and a Rosenbrock Solver

Abstract: In the current paper the e-ciency of the sparse-grid combination tech- nique applied to time-dependent advection-difiusion problems is investigated. For the time-integration we employ a third-order Rosenbrock scheme implemented with adap- tive step-size control and approximate matrix factorization. Two model problems are considered, a scalar 2D linear, constant-coe-cient problem and a system of 2D non- linear Burgers' equations. In short, the combination technique proved more e-cient than a single grid approach for the simpler linear problem. For the Burgers' equations this gain in e-ciency was only observed if one of the two solution components was set to zero, which makes the problem more grid-aligned. 2000 Mathematics Subject Classiflcation: 65G99, 65M20, 65M55, 65L06, 76R99.

Epi-convergent scenario generation method for stochastic problems via sparse grid

Abstract: One central problem in solving stochastic programming problems is to generate moderate-sized scenario trees which represent well the risk faced by a decision maker. In this paper we propose an efficient scenario generation method based on sparse grid, and prove it is epi-convergent. Furthermore, we show numerically that the proposed method converges to the true optimal value fast in comparison with Monte Carlo and Quasi Monte Carlo methods.