Convergence of a randomized Douglas-Rachford method for linear system

Leyu Hu; Xingju Cai

doi:10.3934/naco.2020045

Article Contents

2020, Volume 10, Issue 4: 463-474. Doi: 10.3934/naco.2020045

This issue Previous Article An improved algorithm for generalized least squares estimation Next Article A trust region algorithm for computing extreme eigenvalues of tensors

Convergence of a randomized Douglas-Rachford method for linear system

Leyu Hu and
Xingju Cai

School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing, 210023, China

Received: March 2020

Revised: September 2020

Published: September 2020

The research of the second author is partially supported by the NSFC grants 11871279, 11571178.

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Abstract

In this article, we propose a randomized Douglas-Rachford(DR) method for linear system. This algorithm is based on the cyclic DR method. We consider a linear system as a feasible problem of finding intersection of hyperplanes. In each iteration, the next iteration point is determined by a random DR operator. We prove the convergence of the iteration points based on expectation. And the variance of the iteration points declines to zero. The numerical experiment shows that the proposed algorithm performs better than the cyclic DR method.

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Mathematics Subject Classification: 65F10, 65F20, 65K05, 90C25, 90C15, 15A06.

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Figure 1. Comparison of number of projections/reflections between randomized Kaczmarz(RK), cyclic DR(CDR) and randomized DR(RDR) when the size of $ A $ and $ b $ is $ m = 1000 $ and $ n = 100 $

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Table 1. Number of Projections/Reflections(NPR) for $ n = 50 $

$ m $		$ 1000 $	$ 2000 $	$ 3000 $	$ 4000 $	$ 5000 $
NPR	CDR	1264	1404	1282	1488	1390
	RK	725.3	701.1	696.1	671.3	683.8
	RDR	705.2	683.8	696	669.6	698.8

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Table 2. Number of Projections/Reflections(NPR) for $ n = 100 $

$ m $		$ 1000 $	$ 2000 $	$ 3000 $	$ 4000 $	$ 5000 $
NPR	CDR	2620	2604	2692	2770	2770
	RK	1508.4	1419	1425.5	1422.5	1395.1
	RDR	1523.2	1425.8	1409.6	1421.4	1381.8

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Table 3. Number of Projections/Reflections(NPR) for $ n = 150 $

$ m $		$ 1000 $	$ 2000 $	$ 3000 $	$ 4000 $	$ 5000 $
NPR	CDR	4272	4286	4104	4096	3854
	RK	2466.9	2194.4	2175.4	2092.1	2152
	RDR	2574.2	2227.8	2160.8	2133	2113.8

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