The method of alternating projections and the method of subspace corrections in Hilbert space

Authors:
Jinchao Xu and Ludmil Zikatanov

Journal:
J. Amer. Math. Soc. **15** (2002), 573-597

MSC (2000):
Primary 47A58, 47N10, 47N40, 49M20, 65F10, 65J05, 65N22, 65N55

DOI:
https://doi.org/10.1090/S0894-0347-02-00398-3

Published electronically:
April 8, 2002

MathSciNet review:
1896233

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Abstract: A new identity is given in this paper for estimating the norm of the product of nonexpansive operators in Hilbert space. This identity can be applied for the design and analysis of the method of alternating projections and the method of subspace corrections. The method of alternating projections is an iterative algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces by alternatively computing the best approximations from the individual subspaces which make up the intersection. The method of subspace corrections is an iterative algorithm for finding the solution of a linear equation in a Hilbert space by approximately solving equations restricted on a number of closed subspaces which make up the entire space. The new identity given in the paper provides a sharpest possible estimate for the rate of convergence of these algorithms. It is also proved in the paper that the method of alternating projections is essentially equivalent to the method of subspace corrections. Some simple examples of multigrid and domain decomposition methods are given to illustrate the application of the new identity.

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Additional Information

**Jinchao Xu**

Affiliation:
Center for Computational Mathematics and Applications, Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802

Email:
xu@math.psu.edu

**Ludmil Zikatanov**

Affiliation:
Center for Computational Mathematics and Applications, Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802

Email:
ltz@math.psu.edu

DOI:
https://doi.org/10.1090/S0894-0347-02-00398-3

Keywords:
Alternating projections,
subspace corrections,
nonexpansive operators,
multigrid,
domain decomposition

Received by editor(s):
July 11, 2000

Published electronically:
April 8, 2002

Additional Notes:
The authors were supported in part by NSF Grant #DMS-0074299 and the Center for Computational Mathematics and Applications, The Pennsylvania State University.

Article copyright:
© Copyright 2002
American Mathematical Society