Generalized Gearhart-Koshy acceleration for the Kaczmarz method
HTML articles powered by AMS MathViewer
- by Janosch Rieger;
- Math. Comp. 92 (2023), 1251-1272
- DOI: https://doi.org/10.1090/mcom/3818
- Published electronically: February 2, 2023
- HTML | PDF | Request permission
Abstract:
The Kaczmarz method is an iterative numerical method for solving large and sparse rectangular systems of linear equations. Gearhart, Koshy and Tam have developed an acceleration technique for the Kaczmarz method that minimizes the distance to the desired solution in the direction of a full Kaczmarz step.
The present paper generalizes this technique to an acceleration scheme that minimizes the Euclidean norm error over an affine subspace spanned by a number of previous iterates and one additional cycle of the Kaczmarz method. The key challenge is to find a formulation in which all parameters of the least-squares problem defining the unique minimizer are known, and to solve this problem efficiently.
When only a single Kaczmarz cycle is considered, the proposed affine search is more effective than the Gearhart-Koshy/Tam line-search, which in turn is more effective than the underlying Kaczmarz method. A numerical experiment from the context of computerized tomography suggests that the proposed affine search has the potential to outperform the the Gearhart-Koshy/Tam line-search and the underlying Kaczmarz method in terms of the computational cost that is needed to achieve a given error tolerance.
References
- Zhong-Zhi Bai and Wen-Ting Wu, On greedy randomized Kaczmarz method for solving large sparse linear systems, SIAM J. Sci. Comput. 40 (2018), no. 1, A592–A606. MR 3765917, DOI 10.1137/17M1137747
- Heinz H. Bauschke, Frank Deutsch, Hein Hundal, and Sung-Ho Park, Accelerating the convergence of the method of alternating projections, Trans. Amer. Math. Soc. 355 (2003), no. 9, 3433–3461. MR 1990157, DOI 10.1090/S0002-9947-03-03136-2
- Åke Björck, Numerical methods in matrix computations, Texts in Applied Mathematics, vol. 59, Springer, Cham, 2015. MR 3288840, DOI 10.1007/978-3-319-05089-8
- L. M. Brègman, Finding the common point of convex sets by the method of successive projection, Dokl. Akad. Nauk SSSR 162 (1965), 487–490 (Russian). MR 198341
- Yair Censor, Row-action methods for huge and sparse systems and their applications, SIAM Rev. 23 (1981), no. 4, 444–466. MR 636080, DOI 10.1137/1023097
- Frank Deutsch, Best approximation in inner product spaces, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, vol. 7, Springer-Verlag, New York, 2001. MR 1823556, DOI 10.1007/978-1-4684-9298-9
- William B. Gearhart and Mathew Koshy, Acceleration schemes for the method of alternating projections, J. Comput. Appl. Math. 26 (1989), no. 3, 235–249. MR 1010558, DOI 10.1016/0377-0427(89)90296-3
- R. Gordon, R. Bender, and G. T. Herman, Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography, J. Theor. Biol. 29 (1970), no. 3, 471–481.
- Robert M. Gower and Peter Richtárik, Randomized iterative methods for linear systems, SIAM J. Matrix Anal. Appl. 36 (2015), no. 4, 1660–1690. MR 3432148, DOI 10.1137/15M1025487
- Jamie Haddock and Anna Ma, Greed works: an improved analysis of sampling Kaczmarz-Motzkin, SIAM J. Math. Data Sci. 3 (2021), no. 1, 342–368. MR 4222189, DOI 10.1137/19M1307044
- Jamie Haddock and Deanna Needell, On Motzkin’s method for inconsistent linear systems, BIT 59 (2019), no. 2, 387–401. MR 3974045, DOI 10.1007/s10543-018-0737-6
- Per Christian Hansen and Jakob Sauer Jørgensen, AIR Tools II: algebraic iterative reconstruction methods, improved implementation, Numer. Algorithms 79 (2018), no. 1, 107–137. MR 3846961, DOI 10.1007/s11075-017-0430-x
- S. Kaczmarz, Angenäherte Auflösung von Systemen linearer Gleichungen, Bull. Int. Acad. Polon. Sci. A 35 (1937), 355–357.
- T. S. Motzkin and I. J. Schoenberg, The relaxation method for linear inequalities, Canad. J. Math. 6 (1954), 393–404. MR 62787, DOI 10.4153/cjm-1954-038-x
- Ion Necoara, Faster randomized block Kaczmarz algorithms, SIAM J. Matrix Anal. Appl. 40 (2019), no. 4, 1425–1452. MR 4036092, DOI 10.1137/19M1251643
- Deanna Needell and Joel A. Tropp, Paved with good intentions: analysis of a randomized block Kaczmarz method, Linear Algebra Appl. 441 (2014), 199–221. MR 3134343, DOI 10.1016/j.laa.2012.12.022
- Stefan Steinerberger, A weighted randomized Kaczmarz method for solving linear systems, Math. Comp. 90 (2021), no. 332, 2815–2826. MR 4305370, DOI 10.1090/mcom/3644
- Thomas Strohmer and Roman Vershynin, A randomized Kaczmarz algorithm with exponential convergence, J. Fourier Anal. Appl. 15 (2009), no. 2, 262–278. MR 2500924, DOI 10.1007/s00041-008-9030-4
- Matthew K. Tam, Gearhart-Koshy acceleration for affine subspaces, Oper. Res. Lett. 49 (2021), no. 2, 157–163. MR 4197301, DOI 10.1016/j.orl.2020.12.007
- Kunio Tanabe, Projection method for solving a singular system of linear equations and its applications, Numer. Math. 17 (1971), 203–214. MR 293824, DOI 10.1007/BF01436376
Bibliographic Information
- Janosch Rieger
- Affiliation: School of Mathematics, Monash University, VIC 3800, Australia
- MR Author ID: 829842
- Email: janosch.rieger@monash.edu
- Received by editor(s): January 26, 2022
- Received by editor(s) in revised form: January 26, 2022, July 12, 2022, October 24, 2022, and October 24, 2022
- Published electronically: February 2, 2023
- © Copyright 2023 American Mathematical Society
- Journal: Math. Comp. 92 (2023), 1251-1272
- MSC (2020): Primary 65F10, 65F20, 68W20
- DOI: https://doi.org/10.1090/mcom/3818
- MathSciNet review: 4550325