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

By Jinchao Xu, Ludmil T. Zikatanov. 2002

306 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/1e87ae873cae1b49ef4c61b71af89a87d1994f07

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In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Rows vs. Columns: Randomized Kaczmarz or Gauss-Seidel for Ridge Regression (2015, background): “Recently, Strohmer and Vershynin [25] showed that the RK method converges linearly to the solution β ◦ of Xβ = y in expectation, with a rate that depends on natural geometric properties of the system, improving upon previous convergence analyses (e.g. [26]).”
Paved with Good Intentions: Analysis of a Randomized Block Kaczmarz Method (2012, methodology): “is an extensive literature on the convergence behavior of block projection methods, a class of algorithms that includes block Kaczmarz schemes. In particular, we call out the work of Xu and Zikatanov [XZ02], which contains a reﬁned convergence analysis that applies to a wide range of algorithms. Nev”
Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case (2007, methodology): “However, although the analysis and development of DD techniques for classical discretisation methods (finite difference, finite volume and finite element – conforming, nonconforming and mixed – methods) is by now well understood and rather mature (see the pioneering work [31] or [40,43] and [46–48])”
A review of numerical methods for nonlinear partial differential equations (2012, unknown): “As we noted before in the context of ﬁnite-diﬀerence approximations, the solution strategy is often tied to the speciﬁc nature of the underlying PDEs; we mention in particular preconditioning techniques, conjugate gradient and multi-level methods [24, 92, 215, 216, 220, 179].”
Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems (2006, methodology): “Our convergence proof depends on a powerful identity of Xu and Zikatanov [35] for estimating the norm of the product of non-expansive operators and the following property of a properly defined Scott-Zhang interpolation operator Πj : XJ → Xj (see Lemma 3.3 below) J∑ j=1 ∑ z∈ eNj |(Πjv −Πj−1v)(z)|2 6 ”
Multiplicative Schwarz Methods for Discontinuous Galerkin Approximations ofElliptic Problems (2007, methodology): “Furthermore, due to the close relation between multiplicative Schwarz and multigrid methods [38,41] (both are product type iterative methods), the construction and analysis presented here might throw some new light in the research of multigrid preconditioners for DG approximations, a field where few”