### Abstract

The linear least squares problem, min_{x} ∥ 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 language | English (US) |
---|---|

Pages (from-to) | 11-31 |

Number of pages | 21 |

Journal | Numerical Linear Algebra with Applications |

Volume | 5 |

Issue number | 1 |

State | Published - 1998 |

### Fingerprint

### 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.

Research output: Contribution to journal › Article

*Numerical Linear Algebra with Applications*, vol. 5, no. 1, pp. 11-31.

}

TY - JOUR

T1 - A Parallel Multisplitting Solution of the Least Squares Problem

AU - Renaut, Rosemary

PY - 1998

Y1 - 1998

N2 - 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.

AB - 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.

KW - Iterative solvers

KW - Least squares

KW - Multisplitting

KW - Parallel algorithms

KW - QR factorization

UR - http://www.scopus.com/inward/record.url?scp=0032395761&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032395761&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0032395761

VL - 5

SP - 11

EP - 31

JO - Numerical Linear Algebra with Applications

JF - Numerical Linear Algebra with Applications

SN - 1070-5325

IS - 1

ER -