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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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An accelerated randomized Kaczmarz algorithm
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by Ji Liu and Stephen J. Wright PDF
Math. Comp. 85 (2016), 153-178 Request permission

Abstract:

The randomized Kaczmarz ($\rm {RK}$) algorithm is a simple but powerful approach for solving consistent linear systems $Ax=b$. This paper proposes an accelerated randomized Kaczmarz ($\rm {ARK}$) algorithm with better convergence than the standard $\rm {RK}$ algorithm on ill-conditioned problems. The per-iteration cost of $\rm {RK}$ and $\rm {ARK}$ are similar if $A$ is dense, but $\rm {RK}$ is much more able to exploit sparsity in $A$ than is $\rm {ARK}$. To deal with the sparse case, an efficient implementation for $\rm {ARK}$, called $\rm {SARK}$, is proposed. A comparison of convergence rates and average per-iteration complexities among $\rm {RK}$, $\rm {ARK}$, and $\rm {SARK}$ is given, taking into account different levels of sparseness and conditioning. Comparisons with the leading deterministic algorithm — conjugate gradient applied to the normal equations — are also given. Finally, the analysis is validated via computational testing.
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Additional Information
  • Ji Liu
  • Affiliation: Department of Computer Sciences, University of Wisconsin-Madison, Madison, Wisconsin 53706-1685
  • Address at time of publication: Department of Computer Sciences, 1210 W. Dayton St., Madison, Wisconsin 53706-1685
  • Email: ji.liu.uwisc@gmail.com
  • Stephen J. Wright
  • Affiliation: Department of Computer Sciences, University of Wisconsin-Madison, Madison, Wisconsin 53706-1685
  • MR Author ID: 213980
  • Email: swright@cs.wisc.edu
  • Received by editor(s): October 8, 2013
  • Received by editor(s) in revised form: May 25, 2014
  • Published electronically: May 29, 2015
  • Additional Notes: The first author was supported in part by NSF Awards DMS-0914524 and DMS-1216318 and ONR Award N00014-13-1-0129.
    The second author was supported in part by NSF Awards DMS-0914524 and DMS-1216318, ONR Award N00014-13-1-0129, DOE Award DE-SC0002283, and Subcontract 3F-30222 from Argonne National Laboratory.
  • © Copyright 2015 American Mathematical Society
  • Journal: Math. Comp. 85 (2016), 153-178
  • MSC (2010): Primary 65F10; Secondary 68W20
  • DOI: https://doi.org/10.1090/mcom/2971
  • MathSciNet review: 3404446