New iterative schemes for nonlinear fixed point problems, with applications to problems with bifurcations and incomplete-data problems

Roland, Ch.; Varadhan, Ravi

doi:10.1016/j.apnum.2005.02.006

Cited by 16 publications

(12 citation statements)

References 10 publications

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“…This method is at the origin of the squaring technique. Indeed, the encouraging numerical results obtained by Raydan et al motivated its adaptation to the nonlinear case by Roland and Varadhan [11], who noted the following relation between the Cauchy and CBB error equations…”

Section: Linear Fixed Point Problemmentioning

confidence: 90%

“…The term squared is to be understood in the sense of the following proposition: [11], with respect to the error propagation equations.…”

Section: Generalization and Propertiesmentioning

confidence: 99%

“…Recently, Roland and Varadhan [11] proposed an efficient alternative to the Lemaréchal's method, called the adjusted method. It's defined as the method which satisfies the following error equation…”

Section: Introductionmentioning

confidence: 99%

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Squared polynomial extrapolation methods with cycling: an application to the positron emission tomography problem

2007

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Roland and Varadhan (Appl. Numer. Math., 55:215-226, 2005) presented a new idea called "squaring" to improve the convergence of Lemaréchal's scheme for solving nonlinear fixed-point problems. Varadhan and Roland (Squared extrapolation methods: A new class of simple and efficient numerical schemes for accelerating the convergence of the EM algorithm, Department of Biostatistics Working Paper. Johns Hopkins University, http://www.bepress.com/jhubiostat/paper63, 2004) noted that Lemaréchal's scheme can be viewed as a member of the class of polynomial extrapolation methods with cycling that uses two fixed-point iterations per cycle. Here we combine these two ideas, cycled extrapolation and squaring, and construct a new class of methods, called squared polynomial methods (SQUAREM), for accelerating the convergence of fixed-point iterations. Our main goal is to evaluate whether the squaring device is effective in improving the rate of convergence of cycled extrapolation methods that use more than two fixed- 160 Numer Algor (2007) 44:159-172 point iterations per cycle. We study the behavior of the new schemes on an image reconstruction problem for positron emission tomography (PET) using simulated data. Our numerical experiments show the effectiveness of first-and higher-order squared polynomial extrapolation methods in accelerating image reconstruction, and also their relative superiority compared to the classical, "unsquared" vector polynomial methods.

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Section: Linear Fixed Point Problemmentioning

confidence: 90%

“…The term squared is to be understood in the sense of the following proposition: [11], with respect to the error propagation equations.…”

Section: Generalization and Propertiesmentioning

confidence: 99%

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Squared polynomial extrapolation methods with cycling: an application to the positron emission tomography problem

2007

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“…A spectral approach has been considered to solve large-scale nonlinear systems of equations using only the residual vector as search direction (Grippo and Sciandrone 2007;La Cruz and Raydan 2003;La Cruz et al 2006;Varadhan and Gilbert 2009;Zhang and Zhou 2006). Spectral variations have also been developed for accelerating the convergence of fixed-point iterations, in connection with the well-known EM algorithm which is frequently used in computational statistics (Roland and Varadhan 2005;Roland, Varadhan, and Frangakis 2007;Varadhan and Roland 2008).…”

Section: Applications and Extensionsmentioning

confidence: 99%

Spectral Projected Gradient Methods: Review and Perspectives

Birgin¹,

Martínez²,

Raydan³

2014

J. Stat. Soft.

148

112

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Over the last two decades, it has been observed that using the gradient vector as a search direction in large-scale optimization may lead to efficient algorithms. The effectiveness relies on choosing the step lengths according to novel ideas that are related to the spectrum of the underlying local Hessian rather than related to the standard decrease in the objective function. A review of these so-called spectral projected gradient methods for convex constrained optimization is presented. To illustrate the performance of these low-cost schemes, an optimization problem on the set of positive definite matrices is described.

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“…Extensive numerical experiments showed that the non monotone CBB method is remarkably stable and outperforms the steepest descent method and the Barzilai-Borwein method. This justifies its adaptation to the non linear case by Roland and Varadhan [10]. Although the non linear version is performing well, it can suffer from a stability problem called stagnation: progressively the parameter α n vanishes and therefore the value of the current iterate x n no longer evolves.…”

Section: Introductionmentioning

confidence: 99%

Geometric interpretation of some Cauchy related methods

Berlinet

Roland

2011

Numer. Math.

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This paper shows that a class of methods for solving linear equations, including the Cauchy-Barzilai-Borwein method, can be interpreted by means of a simple geometric object, the Bézier parabola. This curve is built from the current iterate using a transformation characterizing the system to be solved. The localization of the next iterates in the plane of the parabola sheds some light on the behavior of the methods and provides some new understanding of their relative efficiency.

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New iterative schemes for nonlinear fixed point problems, with applications to problems with bifurcations and incomplete-data problems

Cited by 16 publications

References 10 publications

Squared polynomial extrapolation methods with cycling: an application to the positron emission tomography problem

Squared polynomial extrapolation methods with cycling: an application to the positron emission tomography problem

Spectral Projected Gradient Methods: Review and Perspectives

Geometric interpretation of some Cauchy related methods

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