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

@article{Xu2002TheMO,
  title={The method of alternating projections and the method of subspace corrections in Hilbert space},
  author={Jinchao Xu and Ludmil T. Zikatanov},
  journal={Journal of the American Mathematical Society},
  year={2002},
  volume={15},
  pages={573-597}
}
The method of alternating projections and the method of subspace corrections are general iterative methods that have a variety of applications. The method of alternating projections, first proposed by von Neumann (1933) (see [31]), is an algorithm for finding the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces. The method of subspace corrections, an abstraction of general linear iterative methods such as multigrid and domain… 
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References

SHOWING 1-10 OF 64 REFERENCES
Accelerating the convergence of the method of alternating projections
The powerful von Neumann-Halperin method of alternating projections (MAP) is an algorithm for determining the best approximation to any given point in a Hilbert space from the intersection of a
Iterative Methods by Space Decomposition and Subspace Correction
TLDR
A unified theory for a diverse group of iterative algorithms, such as Jacobi and Gauss–Seidel iterations, diagonal preconditioning, domain decomposition methods, multigrid methods,Multilevel nodal basis preconditionsers and hierarchical basis methods, is presented by using the notions of space decomposition and subspace correction.
On the abstract theory of additive and multiplicative Schwarz algorithms
TLDR
A modification of the abstract convergence theory of the additive and multiplicative Schwarz methods that makes the relation to traditional iteration methods more explicit, making convergence proofs of multilevel and domain decomposition methods clearer, or, at least, more classical.
Error bounds for the method of alternating projections
TLDR
The method of alternating projections produces a sequence which converges to the orthogonal projection onto the intersection of the subspaces, and the sharpest known upper bound for more than two subspaced is obtained.
Convergence estimates for product iterative methods with applications to domain decomposition
In this paper, we consider iterative methods for the solution of symmetric positive definite problems on a space % which are defined in terms of products of operators defined with respect to a number
New convergence estimates for multigrid algorithms
TLDR
New convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems and a generalized ..nu.. cycle algorithm which involves exponentially increasing the number of smoothings on successively coarser grids is defined.
Iterative Solution of Large Sparse Systems of Equations
In the second edition of this classic monograph, complete with four new chapters and updated references, readers will now have access to content describing and analysing classical and modern methods
Some Nonoverlapping Domain Decomposition Methods
TLDR
A unified investigation of a class of nonoverlapping domain decomposition methods for solving second-order elliptic problems in two and three dimensions and several new variants of the algorithms are derived.
Multilevel additive methods for elliptic finite element problems in three dimensions
TLDR
A general framework is developed, which is useful in the design and analysis of a variety of domain decomposition methods as well as certain multigrid methods, and works with nite element approximations of linear, self-adjoint, elliptic problems.
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