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Signal recovery by proximal forward-backward splitting

MULTISCALE MODELING & SIMULATION 2005年第4期4卷 1168-1200页

作者： Combettes, PL Wajs, VR Univ Paris 06 Lab Jacques Louis Lions F-75005 Paris France

We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forward-backward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry/texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s.

关键词： denoising forward-backward algorithm image decomposition image restoration multiresolution analysis inverse problem signal recovery iterative soft-thresholding proximity operator proximal Landweber method

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On Inexact Relative-Error Hybrid Proximal Extragradient, forward-backward and Tseng's Modified forward-backward Methods with Inertial Effects

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SET-VALUED AND VARIATIONAL ANALYSIS 2020年第2期28卷 301-325页

作者： Alves, M. Marques Marcavillaca, Raul T. Univ Fed Santa Catarina Dept Matemat BR-88040900 Florianopolis SC Brazil

For solving monotone inclusion problems, we propose an inertial under-relaxed version of the relative-error hybrid proximal extragradient method. We study the asymptotic convergence of the method, as well as its nonasymptotic global convergence rates in terms of iteration complexity. We analyze the new method under more flexible assumptions than existing ones, both on the extrapolation and on the relative-error parameters. The approach is applied to two types of forward-backward type methods for solving structured monotone inclusions.

关键词： Inertial Relaxed Proximal point method HPE method Pointwise Ergodic Iteration-complexity forward-backward algorithm Tseng's modified forward-backward algorithm

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Maximum Principle for Optimal Control Problems of forward-backward Regime-Switching Systems Involving Impulse Controls

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MATHEMATICAL PROBLEMS IN ENGINEERING 2015年第1期2015卷 1-13页

作者： Wang, Shujun Wu, Zhen Shandong Univ Sch Math Jinan 250100 Shandong Peoples R China

This paper is concerned with optimal control problems of forward-backward Markovian regime-switching systems involving impulse controls. Here the Markov chains are continuous-time and finite-state. We derive the stochastic maximum principle for this kind of systems. Besides the Markov chains, the most distinguishing features of our problem are that the control variables consist of regular and impulsive controls, and that the domain of regular control is not necessarily convex. We obtain the necessary and sufficient conditions for optimal controls. Thereafter, we apply the theoretical results to a financial problem and get the optimal consumption strategies.

关键词： MAXIMUM principles (Mathematics) OPTIMAL control theory forward-backward algorithm MARKOV processes PROBLEM solving

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A forward-backward SPLITTING METHOD FOR MONOTONE INCLUSIONS WITHOUT COCOERCIVITY

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SIAM JOURNAL ON OPTIMIZATION 2020年第2期30卷 1451-1472页

作者： Malitsky, Yura Tam, Matthew K. Univ Gottingen Inst Numer & Appl Math D-37083 Gottingen Germany Univ Melbourne Sch Math & Stat Parkville Vic 3010 Australia

In this work, we propose a simple modification of the forward-backward splitting method for finding a zero in the sum of two monotone operators. Our method converges under the same assumptions as Tseng's forward-backward-forward method, namely, it does not require cocoercivity of the single-valued operator. Moreover, each iteration only uses one forward evaluation rather than two as is the case for Tseng's method. Variants of the method incorporating a linesearch, relaxation and inertia, or a structured three operator inclusion are also discussed.

关键词： forward-backward algorithm Tseng's method operator splitting

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Classical Solutions of Path-Dependent PDEs and Functional forward-backward Stochastic Systems

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MATHEMATICAL PROBLEMS IN ENGINEERING 2013年第1期2013卷 1-11页

作者： Ji, Shaolin Yang, Shuzhen Shandong Univ Inst Financial Studies Jinan 250100 Shandong Peoples R China Shandong Univ Inst Math Jinan 250100 Shandong Peoples R China Shandong Univ Sch Math Jinan 250100 Shandong Peoples R China

In this paper we study the relationship between functional forward-backward stochastic systems and path-dependent PDEs. In the framework of functional Ito calculus, we introduce a path-dependent PDE and prove that its... 详细信息

关键词： PARTIAL differential equations FUNCTIONAL analysis forward-backward algorithm STOCHASTIC processes CALCULUS MATHEMATICAL proofs

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A Generalized forward-backward Splitting

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SIAM JOURNAL ON IMAGING SCIENCES 2013年第3期6卷 1199-1226页

作者： Raguet, Hugo Fadili, Jalal Peyre, Gabriel Univ Paris 09 CNRS CEREMADE F-75775 Paris 16 France Univ Caen 6 CNRS ENSICAEN GREYC F-14050 Caen France

This paper introduces a generalized forward-backward splitting algorithm for finding a zero of a sum of maximal monotone operators B + Sigma(n)(i=1) A(i), where B is cocoercive. It involves the computation of B in an explicit (forward) step and the parallel computation of the resolvents of the A(i)'s in a subsequent implicit (backward) step. We prove the algorithm's convergence in infinite dimension and its robustness to summable errors on the computed operators in the explicit and implicit steps. In particular, this allows efficient minimization of the sum of convex functions f + Sigma(n)(i=1) g(i), where f has a Lipschitz-continuous gradient and each g(i) is simple in the sense that its proximity operator is easy to compute. The resulting method makes use of the regularity of f in the forward step, and the proximity operators of the g(i) is are applied in parallel in the backward step. While the forward-backward algorithm cannot deal with more than n = 1 nonsmooth function, we generalize it to the case of arbitrary n. Examples on inverse problems in imaging demonstrate the advantage of the proposed methods in comparison to other splitting algorithms.

关键词： forward-backward algorithm monotone operator splitting nonsmooth convex optimization proximal splitting image processing sparsity

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Strong convergence of the forward-backward splitting method with multiple parameters in Hilbert spaces

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OPTIMIZATION 2018年第4期67卷 493-505页

作者： Wang, Yamin Wang, Fenghui Henan Normal Univ Sch Math & Informat Sci Xinxiang Peoples R China Henan Normal Univ Sch Math & Informat Sci Henan Engn Lab Big Data Stat Anal & Optimal Contr Xinxiang Peoples R China Luoyang Normal Univ Dept Math Luoyang Peoples R China

Many problems arising from machine learning, signal & image recovery, and compressed sensing can be casted into a monotone inclusion problem for finding a zero of the sum of two monotone operators. The forward-backward splitting algorithm is one of the most powerful and successful methods for solving such a problem. However, this algorithm has only weak convergence in the infinite dimensional settings. In this paper, we propose a new modification of the FBA so that it possesses a norm convergent property. Moreover, we establish two strong convergence theorems of the proposed algorithms under more general conditions.

关键词： forward-backward algorithm averaged map projection strong convergence

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A generalized forward-backward splitting method for solving quasi inclusion problems in Banach spaces

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NUMERICAL algorithmS 2016年第4期71卷 915-932页

作者： Cholamjiak, Prasit Univ Phayao Sch Sci Phayao 56000 Thailand

We propose a new general type of splitting methods for accretive operators in Banach spaces. We then give the sufficient conditions to guarantee the strong convergence. In the last section, we apply our results to the... 详细信息

关键词： Accretive operator Maximal monotone operator Banach space Splitting method forward-backward algorithm

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Inference in population genetics using forward and backward, discrete and continuous time processes

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JOURNAL OF THEORETICAL BIOLOGY 2018年 439卷 166-180页

作者： Bergman, Juraj Schrempf, Dominik Kosiol, Carolin Vogl, Claus Vetmeduni Vienna Inst Populat Genet Vet Pl 1 A-1210 Vienna Austria Vienna Grad Sch Populat Genet A-1210 Vienna Austria Vetmeduni Vienna Inst Tierzucht & Genet Vet Pl 1 A-1210 Vienna Austria Univ St Andrews Sch Biol Ctr Biol Div St Andrews KY16 9TH Fife Scotland

A central aim of population genetics is the inference of the evolutionary history of a population. To this end, the underlying process can be represented by a model of the evolution of allele frequencies parametrized by e.g., the population size, mutation rates and selection coefficients. A large class of models use forward-in-time models, such as the discrete Wright-Fisher and Moran models and the continuous forward diffusion, to obtain distributions of population allele frequencies, conditional on an ancestral initial allele frequency distribution. backward-in-time diffusion processes have been rarely used in the context of parameter inference. Here, we demonstrate how forward and backward diffusion processes can be combined to efficiently calculate the exact joint probability distribution of sample and population allele frequencies at all times in the past, for both discrete and continuous population genetics models. This procedure is analogous to the forward-backward algorithm of hidden Markov models. While the efficiency of discrete models is limited by the population size, for continuous models it suffices to expand the transition density in orthogonal polynomials of the order of the sample size to infer marginal likelihoods of population genetic parameters. Additionally, conditional allele trajectories and marginal likelihoods of samples from single populations or from multiple populations that split in the past can be obtained. The described approaches allow for efficient maximum likelihood inference of population genetic parameters in a wide variety of demographic scenarios. (c) 2017 Elsevier Ltd. All rights reserved.

关键词： Bi-allelic mutation-drift model Markov chain forward-backward algorithm forward-backward diffusion Exact inference

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forward-backward and Tseng's Type Penalty Schemes for Monotone Inclusion Problems

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SET-VALUED AND VARIATIONAL ANALYSIS 2014年第2期22卷 313-331页

作者： Bot, Radu Ioan Csetnek, Ernoe Robert Univ Vienna Fac Math A-1090 Vienna Austria

We deal with monotone inclusion problems of the form 0 a A x + D x + N (C) (x) in real Hilbert spaces, where A is a maximally monotone operator, D a cocoercive operator and C the nonempty set of zeros of another cocoercive operator. We propose a forward-backward penalty algorithm for solving this problem which extends the one proposed by Attouch et al. (SIAM J. Optim. 21(4): 1251-1274, 2011). The condition which guarantees the weak ergodic convergence of the sequence of iterates generated by the proposed scheme is formulated by means of the Fitzpatrick function associated to the maximally monotone operator that describes the set C. In the second part we introduce a forward-backward-forward algorithm for monotone inclusion problems having the same structure, but this time by replacing the cocoercivity hypotheses with Lipschitz continuity conditions. The latter penalty type algorithm opens the gate to handle monotone inclusion problems with more complicated structures, for instance, involving compositions of maximally monotone operators with linear continuous ones.

关键词： Maximally monotone operator Fitzpatrick function Resolvent Cocoercive operator Lipschitz continuous operator forward-backward algorithm forward-backward-forward algorithm Subdifferential Fenchel conjugate

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