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Monotone polygon

About: Monotone polygon is a research topic. Over the lifetime, 14573 publications have been published within this topic receiving 270286 citations.


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Journal ArticleDOI
TL;DR: The convergence of a wide class of approximation schemes to the viscosity solution of fully nonlinear second-order elliptic or parabolic, possibly degenerate, partial differential equations is studied in this paper.
Abstract: The convergence of a wide class of approximation schemes to the viscosity solution of fully nonlinear second-order elliptic or parabolic, possibly degenerate, partial differential equations is studied. It is proved that any monotone, stable, and consistent scheme converges (to the correct solution), provided that there exists a comparison principle for the limiting equation. Several examples are given where the result applies. >

1,063 citations

Journal ArticleDOI
TL;DR: A prox-type method with efficiency estimate O(\epsilon^{-1}) for approximating saddle points of convex-concave C$^{1,1}$ functions and solutions of variational inequalities with monotone Lipschitz continuous operators is proposed.
Abstract: We propose a prox-type method with efficiency estimate $O(\epsilon^{-1})$ for approximating saddle points of convex-concave C$^{1,1}$ functions and solutions of variational inequalities with monotone Lipschitz continuous operators. Application examples include matrix games, eigenvalue minimization, and computing the Lovasz capacity number of a graph, and these are illustrated by numerical experiments with large-scale matrix games and Lovasz capacity problems.

980 citations

Journal ArticleDOI
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

Journal ArticleDOI
TL;DR: In this article, it was shown that for any two monotone operators Tx and T2 from X to X*, the operator F», + T2 is again monotonous.
Abstract: is called the effective domain of F, and F is said to be locally bounded at a point x e D(T) if there exists a neighborhood U of x such that the set (1.4) T(U) = (J{T(u)\ueU} is a bounded subset of X. It is apparent that, given any two monotone operators Tx and T2 from X to X*, the operator F», + T2 is again monotone, where (1 5) (Ti + T2)(x) = Tx(x) + T2(x) = {*? +x% I xf e Tx(x), xt e T2(x)}. If Tx and F2 are maximal, it does not necessarily follow, however, that F», + T2 is maximal—some sort of condition is needed, since for example the graph of Tx + T2 can even be empty (as happens when D(Tx) n D(T2)= 0). The problem of determining conditions under which Tx + T2 is maximal turns out to be of fundamental importance in the theory of monotone operators. Results in this direction have been proved by Lescarret [9] and Browder [5], [6], [7]. The strongest result which is known at present is :

922 citations

Journal ArticleDOI
TL;DR: In this paper, the problem of obtaining convergence to steady state solutions of the Euler equations when limiters are used in conjunction with upwind schemes on unstructured grids is addressed.

885 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
2023687
20221,632
2021833
2020766
2019740
2018602