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Title: Space-Time Approximation with Sparse Grids

Abstract

In this article we introduce approximation spaces for parabolic problems which are based on the tensor product construction of a multiscale basis in space and a multiscale basis in time. Proper truncation then leads to so-called space-time sparse grid spaces. For a uniform discretization of the spatial space of dimension d with O(N{sup d}) degrees of freedom, these spaces involve for d > 1 also only O(N{sup d}) degrees of freedom for the discretization of the whole space-time problem. But they provide the same approximation rate as classical space-time Finite Element spaces which need O(N{sup d+1}) degrees of freedoms. This makes these approximation spaces well suited for conventional parabolic and for time-dependent optimization problems. We analyze the approximation properties and the dimension of these sparse grid space-time spaces for general stable multiscale bases. We then restrict ourselves to an interpolatory multiscale basis, i.e. a hierarchical basis. Here, to be able to handle also complicated spatial domains {Omega}, we construct the hierarchical basis from a given spatial Finite Element basis as follows: First we determine coarse grid points recursively over the levels by the coarsening step of the algebraic multigrid method. Then, we derive interpolatory prolongation operators between the respective coarsemore » and fine grid points by a least squares approach. This way we obtain an algebraic hierarchical basis for the spatial domain which we then use in our space-time sparse grid approach. We give numerical results on the convergence rate of the interpolation error of these spaces for various space-time problems with two spatial dimensions. Also implementational issues, data structures and questions of adaptivity are addressed to some extent.« less

Authors:
; ;
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
888615
Report Number(s):
UCRL-JRNL-211686
Journal ID: ISSN 1064-8275; SJOCE3; TRN: US200618%%375
DOE Contract Number:  
W-7405-ENG-48
Resource Type:
Journal Article
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 28; Journal Issue: 2; Journal ID: ISSN 1064-8275
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; CONSTRUCTION; CONVERGENCE; DEGREES OF FREEDOM; DIMENSIONS; INTERPOLATION; OPTIMIZATION; SPACE-TIME

Citation Formats

Griebel, M, Oeltz, D, and Vassilevski, P S. Space-Time Approximation with Sparse Grids. United States: N. p., 2005. Web.
Griebel, M, Oeltz, D, & Vassilevski, P S. Space-Time Approximation with Sparse Grids. United States.
Griebel, M, Oeltz, D, and Vassilevski, P S. 2005. "Space-Time Approximation with Sparse Grids". United States. https://www.osti.gov/servlets/purl/888615.
@article{osti_888615,
title = {Space-Time Approximation with Sparse Grids},
author = {Griebel, M and Oeltz, D and Vassilevski, P S},
abstractNote = {In this article we introduce approximation spaces for parabolic problems which are based on the tensor product construction of a multiscale basis in space and a multiscale basis in time. Proper truncation then leads to so-called space-time sparse grid spaces. For a uniform discretization of the spatial space of dimension d with O(N{sup d}) degrees of freedom, these spaces involve for d > 1 also only O(N{sup d}) degrees of freedom for the discretization of the whole space-time problem. But they provide the same approximation rate as classical space-time Finite Element spaces which need O(N{sup d+1}) degrees of freedoms. This makes these approximation spaces well suited for conventional parabolic and for time-dependent optimization problems. We analyze the approximation properties and the dimension of these sparse grid space-time spaces for general stable multiscale bases. We then restrict ourselves to an interpolatory multiscale basis, i.e. a hierarchical basis. Here, to be able to handle also complicated spatial domains {Omega}, we construct the hierarchical basis from a given spatial Finite Element basis as follows: First we determine coarse grid points recursively over the levels by the coarsening step of the algebraic multigrid method. Then, we derive interpolatory prolongation operators between the respective coarse and fine grid points by a least squares approach. This way we obtain an algebraic hierarchical basis for the spatial domain which we then use in our space-time sparse grid approach. We give numerical results on the convergence rate of the interpolation error of these spaces for various space-time problems with two spatial dimensions. Also implementational issues, data structures and questions of adaptivity are addressed to some extent.},
doi = {},
url = {https://www.osti.gov/biblio/888615}, journal = {SIAM Journal on Scientific Computing},
issn = {1064-8275},
number = 2,
volume = 28,
place = {United States},
year = {Thu Apr 14 00:00:00 EDT 2005},
month = {Thu Apr 14 00:00:00 EDT 2005}
}