NDLI: Fast Multilevel Augmentation Methods for Solving Hammerstein Equations

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Chen, Zhongying Wu, Bin Xu, Yuesheng
Copyright Year	2009
Abstract	Fast multilevel augmentation methods based on projections for solving Hammerstein equations are developed. Each of these methods requires availability of a multilevel decomposition of the solution space and a projection from the solution space onto a finite dimensional subspace. A subspace at a level is obtained from the subspace at the previous level by adding a difference subspace. Accordingly, the projection equation at the present level is obtained by augmenting the projection equation at the previous level. Using this idea recursively, we solve the nonlinear equation (by an iteration method) only at an initial lower resolution level while obtaining its solution at a higher resolution level. With the help of the multiscale analysis, we separate the procedure of solving the nonlinear operator equation at a high level into two major components: (1) solving the nonlinear equation only at an initial lower level, and (2) compensating the error by solving a linear system at the high level. We prove that the proposed methods require only linear (up to a logarithmic factor) computational complexity and have the optimal convergence order. A relationship between the proposed method and the multigrid method is discussed. Two specific fast methods based on the Galerkin projection and the collocation projection are developed. Numerical results are presented to confirm the theoretical estimates, with comparisons to the two-grid method.
Starting Page	2321
Ending Page	2346
Page Count	26
File Format	PDF
ISSN	00361429
DOI	10.1137/080734157
e-ISSN	10957170
Issue Number	3
Volume Number	47
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2009-06-25
Access Restriction	Subscribed
Subject Keyword	Equations with nonlinear operators Integral equations Hammerstein equations fast multilevel augmentation methods
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Numerical Analysis Computational Mathematics

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