Méthodes itératives pour les équations et inéquations aux dérivées partielles non linéaires de type monotone
TL;DR: In this article, the duality relation between a fixed element and a closed convex subset of a real-banach space is defined, and the main method is based on the calculation of the operatorPX (the projector on the closed setX).
Abstract: LetV be a reflexive Real Banach space and letA be an operator (not necessarily linear) fromV into its dualV′. We shall suppose thatX is a closed convex subset ofV. We shall denote by (,) the duality relation betweenV andV′. For a fixed elementf∈V′ we shall try to determineu∈X such that
$$(Au,v - u) \geqslant (f,v - u)\rlap{--} \vee v\varepsilon X.$$
Problems of type (1) occur in optimal control theory and contain a fairly large class of type of linear and non linear partial differential equations (and inequations). The main method is based on the calculation of the operatorPX (the projector on the closed convex setX)
i)
the operatorPX is accessible, in which case we indicate how to obtain a numerical solution.
ii)
the operatorPX is not accessible, in which case we indicate a variaty of methods to approximate the solution
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Abstract: The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework. Applications of proximal methods in signal recovery and synthesis are discussed.
2,095 citations
01 Jan 2011
TL;DR: The basic properties of proximity operators which are relevant to signal processing and optimization methods based on these operators are reviewed and proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework.
Abstract: The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of inverse problems and, especially, in signal processing, where it has become increasingly important. In this paper, we review the basic properties of proximity operators which are relevant to signal processing and present optimization methods based on these operators. These proximal splitting methods are shown to capture and extend several well-known algorithms in a unifying framework. Applications of proximal methods in signal recovery and synthesis are discussed.
1,942 citations
Cites methods from "Méthodes itératives pour les équati..."
...(43) Convergence is guaranteed by Proposition 3.3....
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...…described so far are designed for m = 2 functions, we can attempt to reformulate (53) as a 2-function problem in the m-fold product space H = RN × · · · × RN (54) (such techniques were introduced in [110, 111] and have been used in the context of convex feasibility problems in [10, 43, 45])....
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01 Jan 1975
1,054 citations
TL;DR: A modification to the forward-backward splitting method for finding a zero of the sum of two maximal monotone mappings is proposed, under which the method converges assuming only the forward mapping is (Lipschitz) continuous on some closed convex subset of its domain.
Abstract: We consider the forward-backward splitting method for finding a zero of the sum of two maximal monotone mappings. This method is known to converge when the inverse of the forward mapping is strongly monotone. We propose a modification to this method, in the spirit of the extragradient method for monotone variational inequalities, under which the method converges assuming only the forward mapping is (Lipschitz) continuous on some closed convex subset of its domain. The modification entails an additional forward step and a projection step at each iteration. Applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed.
935 citations
Cites methods from "Méthodes itératives pour les équati..."
...In the case where B = NC , with C a nonempty closed convex set in H, this method reduces to a projection method proposed by Sibony [49] for monotone variational inequalities and, in the further case where F is the gradient of a di erentiable convex function, it reduces to a gradient projection method of Goldstein and of Levitin and Polyak [1]....
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...Other projection-type methods, including the method ofSibony, typically require F 1 to be strongly monotone for convergence (see [2, 25, 29, 49]and references therein)....
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...In the case where B = NC , with C anonempty closed convex set in H, this method reduces to a projection method proposed bySibony [49] for monotone variational inequalities and, in the further case where F is thegradient of a di erentiable convex function, it reduces to a gradient projection method ofGoldstein and of Levitin and Polyak [1]....
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...Other projection-type methods, including the method of Sibony, typically require F 1 to be strongly monotone for convergence (see [2, 25, 29, 49] and references therein)....
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...The modi -cation is motivated by the extragradient method of Korpelevich [19] for monotone variationalinequalities, which modi es the projection method of Sibony by performing an additionalforward step and a projection step at each iteration....
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TL;DR: This paper presents a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection, Wiener-Hopf equations, updating the solution, auxiliary principle, inertial proximal, penalty function, dynamical system and well-posedness.
541 citations
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Abstract: © Bulletin de la S. M. F., 1968, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
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TL;DR: In this paper, the authors define the space of the fonctions (uniformification) lipschitziennes of a torsion and define a set of notations suivant: H A (t2) = v ~ L 2 (f2); L 2(f2) and v = 0 sur F and H A(t 2) = {veH 2 (2); I v(x)p.p.
Abstract: Soit f2 un ouvert born6 de ]R n de fronti6re r6guli6re F. On utilisera dans la suite les notations suivantes: H A (t2) = v ~ L 2 (f2); ~ e L 2 (f2) et v = 0 sur F et Hl'~~ est l'espace des fonctions (uniform6ment) lipschitziennes sur et H~'~176176176 Si v~Hl'~176 on pose IV(Xl)-V(Xz)l Ilvl[1,~= Sup x,,x2~O [Xl-Xz[ Xl ~X2 Enfin on d6signe par 6(x) la distance de xEf2 ~ F. On consid~re le probl~me suivant (qui intervient dans l'6tude de la torsion 61asto-plastique cf. [5], [11]): Etant donn6s f6L2(~'2) et 2>0, trouver u~Kl= {v~H~' ~176 Ilvll 1, c~ ~ 1} tel que (av aU) dx<_O V vEK1. 0x,-Le probl~me (1) est un exemple particulier d'in6quation variationnelle (of. [4], [6], [8]) et admet une solution unique. On montre au w I que, sous certaines hypoth&es, le probEme (1) est 6quivalent au probl~me suivant: trouver u eK2 = {veH~ (~2); I v(x)[ < 6 (x)p.p. sur ~} tel que (2) (avau) ,x o i=1 \" (on notera que (2) admet aussi une solution unique). On indique au w II diverses techniques permettant de calculer la solution de (2). I1 est clair qu'en analyse num6rique les contraintes de type K 2 sont plus simples/t repr6senter que celles de type Kt ; d'ofi l'int6rSt de cette re&bode pour la r6solution de (1).
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