A Parallel Multisplitting Solution of the Least Squares Problem

Research output: Contribution to journalArticle

26 Citations (Scopus)

Abstract

The linear least squares problem, minx ∥ Ax-b ∥2, is solved by applying a multisplitting(MS) strategy in which the system matrix is decomposed by columns into p blocks. The b and x vectors are partitioned consistently with the matrix decomposition. The global least squares problem is then replaced by a sequence of local least squares problems which can be solved in parallel by MS. In MS the solutions to the local problems are recombined using weighting matrices to pick out the appropriate components of each subproblem solution. A new two-stage algorithm which optimizes the global update each iteration is also given. For this algorithm the updates are obtained by finding the optimal update with respect to the weights of the recombination. For the least squares problem presented, the global update optimization can also be formulated as a least squares problem of dimension p. Theoretical results are presented which prove the convergence of the iterations. Numerical results which detail the iteration behavior relative to subproblem size, convergence criteria and recombination techniques are given. The two-stage MS strategy is shown to be effective for near-separable problems.

Original languageEnglish (US)
Pages (from-to)11-31
Number of pages21
JournalNumerical Linear Algebra with Applications
Volume5
Issue number1
StatePublished - 1998

Fingerprint

Multisplitting
Least Squares Problem
Update
Iteration
Recombination
Global optimization
Matrix Decomposition
Convergence Criteria
Linear Least Squares
Decomposition
Weighting
Optimise
Numerical Results
Optimization

Keywords

  • Iterative solvers
  • Least squares
  • Multisplitting
  • Parallel algorithms
  • QR factorization

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Applied Mathematics

Cite this

A Parallel Multisplitting Solution of the Least Squares Problem. / Renaut, Rosemary.

In: Numerical Linear Algebra with Applications, Vol. 5, No. 1, 1998, p. 11-31.

Research output: Contribution to journalArticle

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