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A modified quasi‐Newton diagonal update algorithm for total variation denoising problems and nonlinear monotone equations with applications in compressive sensing
Author(s) -
Zhang Benxin,
Zhu Zhibin
Publication year - 2015
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1968
Subject(s) - hessian matrix , mathematics , diagonal , monotone polygon , newton's method , nonlinear system , diagonal matrix , quasi newton method , matrix (chemical analysis) , convergence (economics) , algorithm , compressed sensing , mathematical analysis , mathematical optimization , geometry , economics , composite material , economic growth , physics , materials science , quantum mechanics
Summary In this paper, we present a new algorithm to accelerate the Chambolle gradient projection method for total variation image restoration. The new proposed method considers an approximation of the Hessian based on the secant equation. Combined with the quasi‐Cauchy equations and diagonal updating, we can obtain a positive definite diagonal matrix. In the proposed minimization method model, we use the positive definite diagonal matrix instead of the constant time stepsize in Chambolle's method. The global convergence of the proposed scheme is proved. Some numerical results illustrate the efficiency of this method. Moreover, we also extend the quasi‐Newton diagonal updating method to solve nonlinear systems of monotone equations. Performance comparisons show that the proposed method is efficient. A practical application of the monotone equations is shown and tested on sparse signal reconstruction in compressed sensing. Copyright © 2015 John Wiley & Sons, Ltd.