Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Optimized general sparse grid approximation spaces for operator equations


Authors: M. Griebel and S. Knapek
Journal: Math. Comp. 78 (2009), 2223-2257
MSC (2000): Primary 41A17, 41A25, 41A30, 41A65, 45L10, 65D99, 65N12, 65N30, 65N38, 65N55
Published electronically: April 23, 2009
MathSciNet review: 2521287
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the construction of optimized sparse grid approximation spaces for elliptic pseudodifferential operators of arbitrary order. Based on the framework of tensor-product biorthogonal wavelet bases and stable subspace splittings, we construct operator-adapted subspaces with a dimension smaller than that of the standard full grid spaces but which have the same approximation order as the standard full grid spaces, provided that certain additional regularity assumptions on the solution are fulfilled. Specifically for operators of positive order, their dimension is $ O(2^{J})$ independent of the dimension $ n$ of the problem, compared to $ O(2^{Jn})$ for the full grid space. Also, for operators of negative order the overall cost is significantly in favor of the new approximation spaces. We give cost estimates for the case of continuous linear information. We show these results in a constructive manner by proposing a Galerkin method together with optimal preconditioning. The theory covers elliptic boundary value problems as well as boundary integral equations.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 41A17, 41A25, 41A30, 41A65, 45L10, 65D99, 65N12, 65N30, 65N38, 65N55

Retrieve articles in all journals with MSC (2000): 41A17, 41A25, 41A30, 41A65, 45L10, 65D99, 65N12, 65N30, 65N38, 65N55


Additional Information

M. Griebel
Affiliation: Institut für Numerische Simulation, Universität Bonn, D-53115 Bonn, Germany
Email: griebel@ins.uni-bonn.de

S. Knapek
Affiliation: Institut für Numerische Simulation, Universität Bonn, D-53115 Bonn, Germany

DOI: http://dx.doi.org/10.1090/S0025-5718-09-02248-0
PII: S 0025-5718(09)02248-0
Received by editor(s): April 10, 2008
Received by editor(s) in revised form: December 4, 2008
Published electronically: April 23, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.