Institute Matematica Pura e Aplicada IMPA

Rio de Janeiro, Brazil

Institute Matematica Pura e Aplicada IMPA

Rio de Janeiro, Brazil
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Svaiter B.F.,Institute Matematica Pura e Aplicada IMPA | Svaiter N.F.,Brazilian Center for Research in Physics (CBPF)
International Journal of Modern Physics A | Year: 2016

In this paper, we present a new mathematical rigorous technique for computing the average free energy of a disordered system with quenched randomness, using the replicas. The basic tool of this technique is a distributional zeta-function, a complex function whose derivative at the origin yields the average free energy of the system as the sum of two contributions: the first one is a series in which all the integer moments of the partition function of the model contribute; the second one, which cannot be written as a series of the integer moments, can be made as small as desired. This result supports the use of integer moments of the partition function, computed via replicas, for expressing the average free energy of the system. One advantage of the proposed formalism is that it does not require the understanding of the properties of the permutation group when the number of replicas goes to zero. Moreover, the symmetry is broken using the saddle-point equations of the model. As an application for the distributional zeta-function technique, we obtain the average free energy of the disordered λφ4 model defined in a d-dimensional Euclidean space. © 2016 World Scientific Publishing Company.

Iusem A.N.,Institute Matematica Pura e Aplicada IMPA | Oliveira R.I.,Institute Matematica Pura e Aplicada IMPA | Thompson P.,Institute Matematica Pura e Aplicada IMPA
SIAM Journal on Optimization | Year: 2017

We propose an extragradient method with stepsizes bounded away from zero for stochastic variational inequalities requiring only pseudomonotonicity. We provide convergence and complexity analysis, allowing for an unbounded feasible set, unbounded operator, and nonuniform variance of the oracle, and, also, we do not require any regularization. Alongside the stochastic ap-proximation procedure, we iteratively reduce the variance of the stochastic error. Our method attains the optimal oracle complexity O(1=€2) (up to a logarithmic term) and a faster rate O(1=K) in terms of the mean (quadratic) natural residual and the D-gap function, where K is the number of iterations required for a given tolerance € > 0. Such convergence rate represents an acceleration with respect to the stochastic error. The generated sequence also enjoys a new feature: The sequence is bounded in Lp if the stochastic error has finite p-moment. Explicit estimates for the convergence rate, the oracle complexity, and the p-moments are given depending on problem parameters and distance of the initial iterate to the solution set. Moreover, sharper constants are possible if the variance is uni-form over the solution set or the feasible set. Our results provide new classes of stochastic variational inequalities for which a convergence rate of O(1=K) holds in terms of the mean-squared distance to the solution set. Our analysis includes the distributed solution of pseudomonotone Cartesian variational inequalities under partial coordination of parameters between users of a network. © 2017 Society for Industrial and Applied Mathematics.

Frid H.,Institute Matematica Pura e Aplicada IMPA | Li Y.,Shanghai JiaoTong University
Archive for Rational Mechanics and Analysis | Year: 2017

We consider a mixed type boundary value problem for a class of degenerate parabolic–hyperbolic equations. Namely, we consider a Cartesian product domain and split its boundary into two parts. In one of them we impose a Dirichlet boundary condition; in the other, we impose a Neumann condition. We apply a normal trace formula for L2-divergence-measure fields to prove a new strong trace property in the part of the boundary where the Neumann condition is imposed. We prove the existence and uniqueness of the entropy solution. © 2017 Springer-Verlag GmbH Germany

Balseiro P.,Institute Matematica Pura e Aplicada IMPA | Marrero J.C.,University of La Laguna | De Diego D.M.,Institute Ciencias Matematicas CSIC UAM UC3M UCM | Padron E.,University of La Laguna
Nonlinearity | Year: 2010

In this paper, we construct Hamilton-Jacobi equations for a large variety of mechanical systems (nonholonomic systems subjected to linear or affine constraints, dissipative systems subjected to external forces, time-dependent mechanical systems etc). We recover all these, in principle, different cases, using a unified framework based on skew-symmetric algebroids with a distinguished 1-cocycle. Several examples illustrate the theory. © 2010 IOP Publishing Ltd&London Mathematical Society.

Bueno O.,Institute Matematica Pura e Aplicada IMPA | Svaiter B.F.,Institute Matematica Pura e Aplicada IMPA
Mathematical Programming | Year: 2013

Previous examples of non-type (D) maximal monotone operators were restricted to \ell 1, L1, and Banach spaces containing isometric copies of these spaces. This fact led to the conjecture that non-type (D) operators were restricted to this class of Banach spaces. We present a linear non-type (D) operator in c0. © 2013 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

Behling R.,Institute Matematica Pura e Aplicada IMPA | Iusem A.,Institute Matematica Pura e Aplicada IMPA
Mathematical Programming | Year: 2013

We address the problem of solving a continuously differentiable nonlinear system of equations under the condition of calmness. This property, also called upper Lipschitz-continuity in the literature, can be described by a local error bound and is being widely used as a regularity condition in optimization. Indeed, it is known to be significantly weaker than classic regularity assumptions that imply that solutions are isolated. We prove that under this condition, the rank of the Jacobian of the function that defines the system of equations must be locally constant on the solution set. In addition, we prove that locally, the solution set must be a differentiable manifold. Our results are illustrated by examples and discussed in terms of their theoretical relevance and algorithmic implications. © 2011 Springer and Mathematical Optimization Society.

Iusem A.N.,Institute Matematica Pura e Aplicada IMPA
AIP Conference Proceedings | Year: 2010

Consider a real-valued bifunction f which is concave in its first argument and convex in its second one. We study its subdifferential with respect to the second argument, evaluated at pairs of the form (x,x), and the subdifferential of -f with respect to its first argument, evaluated at the same pairs. The resulting operators are not always monotone, and we analyze additional conditions on f which ensure their monotonicity, and furthermore their maximal monotonicity. Our main result is that these operators are maximal monotone when f is continuous and it vanishes whenever both arguments coincide. Our results have consequences in terms of the reformulation of equilibrium problems as variational inequality ones. © 2010 American Institute of Physics.

Cruz L.M.V.,Institute Matematica Pura e Aplicada IMPA | Velho L.,Institute Matematica Pura e Aplicada IMPA
Proceedings - 23rd Conference on Graphics, Patterns and Images Tutorials, SIBGRAPI-T 2010 | Year: 2010

Geometric Modeling is a widely studied area in computer graphics and methods for constructing 3D models with intuitive interfaces are a topic that has been attracting the interest of many researches. In contrast to the complicated interfaces of modeling softwares, created using the WIMP paradigm (Window, Icon, Menu, Pointer), several studies have shown applications with interfaces based on gestures, which are simpler and more natural. In this scenario, the area of Sketch-Based Interfaces and Modeling (SBIM) emerged. Sketch is a very efficient tool to convey the essence of an object with few strokes, because humans have the ability of inferring 3D models from 2D drawings. However, associating a sketch to a 3D model is not a computationally trivial task. In this paper we present a conceptualization of the area, highlighting opportunities, challenges and trends in SBIM. © 2010 IEEE.

Santos A.S.,Federal University of Sergipe | Santos A.S.,Institute Matematica Pura e Aplicada IMPA
Annales Henri Poincare | Year: 2010

It has been showed by Byde (Indiana Univ. Math. J. 52(5):1147-1199, 2003) that it is possible to attach a Delaunay-type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The resulting manifold is noncompact with the same constant scalar curvature. The main goal of this paper is to generalize this result. We will construct a one-parameter family of solutions to the positive singular Yamabe problem for any compact non-degenerate manifold with Weyl tensor vanishing to sufficiently high order at the singular point. If the dimension is at most 5, no condition on the Weyl tensor is needed. We will use perturbation techniques and gluing methods. © Birkhäuser / Springer Basel AG 2010.

Ambrozio L.C.,Institute Matematica Pura e Aplicada IMPA
Communications in Mathematical Physics | Year: 2015

In this paper we prove the Penrose inequality for metrics that are small perturbations of the Schwarzschild anti-de Sitter metrics of positive mass. We use the existence of a global foliation by weakly stable constant mean curvature spheres and the monotonicity of the Hawking mass. © 2015, Springer-Verlag Berlin Heidelberg.

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