The Experts below are selected from a list of 327 Experts worldwide ranked by ideXlab platform
Maxime Zavidovique - One of the best experts on this subject based on the ideXlab platform.
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Regularization of Subsolutions in Discrete Weak KAM Theory
Canadian Journal of Mathematics, 2013Co-Authors: Patrick Bernard, Maxime ZavidoviqueAbstract:We expose different methods of regularizations of Subsolutions in the context of discrete weak KAM theory that allow us to prove the existence and the density of C 1,1 Subsolutions. Moreover, these Subsolutions can be made strict and smooth outside of the Aubry set.
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Aubry sets for weakly coupled systems of Hamilton--Jacobi equations
2012Co-Authors: Andrea Davini, Maxime ZavidoviqueAbstract:We introduce a notion of Aubry set for weakly coupled systems of Hamilton--Jacobi equations on the torus and characterize it as the region where the obstruction to the existence of globally strict critical Subsolutions concentrates. As in the case of a single equation, we prove the existence of critical Subsolutions which are strict and smooth outside the Aubry set. This allows us to derive in a neat way a comparison result among critical sub and supersolutions with respect to their boundary data on the Aubry set, showing in particular that the latter is a uniqueness set for the critical system. Furthermore, we show that the trace of any critical subsolution on this set can be extended to the whole torus in such a way that the output is a critical solution. We also highlight some rigidity phenomena taking place on the Aubry set: first, the values taken by the differences of the components of a critical subsolution, on this set, are independent of the specific subsolution chosen; second, for each point $y$ in the Aubry set, there exists a vector which is a reachable gradient at $y$ of any critical subsolution.
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Weak KAM theoretic aspects for nonregular commuting Hamiltonians
2012Co-Authors: Andrea Davini, Maxime ZavidoviqueAbstract:In this paper we consider the notion of commutation for a pair of continuous and convex Hamiltonians, given in terms of commutation of their Lax- Oleinik semigroups. This is equivalent to the solvability of an associated multi- time Hamilton-Jacobi equation. We examine the weak KAM theoretic aspects of the commutation property and show that the two Hamiltonians have the same weak KAM solutions and the same Aubry set, thus generalizing a result recently obtained by the second author for Tonelli Hamiltonians. We make a further step by proving that the Hamiltonians admit a common critical subsolution, strict outside their Aubry set. This subsolution can be taken of class C^{1,1} in the Tonelli case. To prove our main results in full generality, it is crucial to establish suitable differentiability properties of the critical Subsolutions on the Aubry set. These latter results are new in the purely continuous case and of independent interest.
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Regularization of Subsolutions in discrete weak KAM theory
arXiv: Analysis of PDEs, 2012Co-Authors: Patrick Bernard, Maxime ZavidoviqueAbstract:We expose different methods of regularizations of Subsolutions in the context of discrete weak KAM theory. They allow to prove the existence and the density of $C^{1,1}$ Subsolutions. Moreover, these Subsolutions can be made strict and smooth outside of the Aubry set.
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Existence of $C^{1,1}$ critical Subsolutions in discrete weak KAM theory
Journal of Modern Dynamics, 2011Co-Authors: Maxime ZavidoviqueAbstract:In this article, following [29], we study critical Subsolutions in discrete weak KAM theory. In particular, we establish that if the cost function $c: M \times M\to \R$ defined on a smooth connected manifold is locally semiconcave and satisfies twist conditions, then there exists a $C^{1,1}$ critical subsolution strict on a maximal set (namely, outside of the Aubry set). We also explain how this applies to costs coming from Tonelli Lagrangians. Finally, following ideas introduced in [18] and [26], we study invariant cost functions and apply this study to certain covering spaces, introducing a discrete analog of Mather's $\alpha$ function on the cohomology.
Paul Dupuis - One of the best experts on this subject based on the ideXlab platform.
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Examples of Subsolutions and Their Application
Analysis and Approximation of Rare Events, 2019Co-Authors: Amarjit Budhiraja, Paul DupuisAbstract:In this chapter we present examples to illustrate the importance sampling and splitting techniques developed in Chaps. 14, 15, and 16. There are many different types of problems one might consider, and the interested reader can find additional examples in the references [76, 77, 101, 103, 105, 110, 112, 113, 116, 117]. As mentioned in Chaps. 14 and 16, an important distinction is that in the case of importance sampling, we use a smooth classical-sense subsolution, while in the case of splitting, we use a continuous but not necessarily smooth weak-sense solution. For many of the examples presented, the construction of Subsolutions can be carried out in arbitrary dimension.
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Moderate deviations-based importance sampling for stochastic recursive equations
Advances in Applied Probability, 2017Co-Authors: Paul Dupuis, Dane JohnsonAbstract:Abstract Subsolutions to the Hamilton–Jacobi–Bellman equation associated with a moderate deviations approximation are used to design importance sampling changes of measure for stochastic recursive equations. Analogous to what has been done for large deviations subsolution-based importance sampling, these schemes are shown to be asymptotically optimal under the moderate deviations scaling. We present various implementations and numerical results to contrast their performance, and also discuss the circumstances under which a moderate deviation scaling might be appropriate.
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The design and analysis of a generalized RESTART/DPR algorithm for rare event simulation
Annals of Operations Research, 2009Co-Authors: Thomas Dean, Paul DupuisAbstract:We consider a general class of branching methods with killing for the estimation of rare events. The class includes a number of existing schemes, including RESTART and DPR (Direct Probability Redistribution). A method for the design and analysis is developed when the quantity of interest can be embedded in a sequence whose limit is determined by a large deviation principle. A notion of subsolution for the related calculus of variations problem is introduced, and two main results are proved. One is that the number of particles and the total work scales subexponentially in the large deviation parameter when the branching process is constructed according to a subsolution. The second is that the asymptotic performance of the schemes as measured by the variance of the estimate can be characterized in terms of the subsolution. Some examples are given to demonstrate the performance of the method.
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Splitting for rare event simulation : A large deviation approach to design and analysis
Stochastic Processes and their Applications, 2009Co-Authors: Thomas Dean, Paul DupuisAbstract:Particle splitting methods are considered for the estimation of rare events. The probability of interest is that a Markov process first enters a set B before another set A, and it is assumed that this probability satisfies a large deviation scaling. A notion of subsolution is defined for the related calculus of variations problem, and two main results are proved under mild conditions. The first is that the number of particles generated by the algorithm grows subexponentially if and only if a certain scalar multiple of the importance function is a subsolution. The second is that, under the same condition, the variance of the algorithm is characterized (asymptotically) in terms of the subsolution. The design of asymptotically optimal schemes is discussed, and numerical examples are presented.
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Subsolutions of an Isaacs Equation and Efficient Schemes for Importance Sampling
Mathematics of Operations Research, 2007Co-Authors: Paul Dupuis, Hui WangAbstract:It was established in Dupuis and Wang [Dupuis, P., H. Wang. 2004. Importance sampling, large deviations, and differential games. Stoch. Stoch. Rep.76 481--508, Dupuis, P., H. Wang. 2005. Dynamic importance sampling for uniformly recurrent Markov chains. Ann. Appl. Probab.15 1--38] that importance sampling algorithms for estimating rare-event probabilities are intimately connected with two-person zero-sum differential games and the associated Isaacs equation. This game interpretation shows that dynamic or state-dependent schemes are needed in order to attain asymptotic optimality in a general setting. The purpose of the present paper is to show that classical Subsolutions of the Isaacs equation can be used as a basic and flexible tool for the construction and analysis of efficient dynamic importance sampling schemes. There are two main contributions. The first is a basic theoretical result characterizing the asymptotic performance of importance sampling estimators based on Subsolutions. The second is an explicit method for constructing classical Subsolutions as a mollification of piecewise affine functions. Numerical examples are included for illustration and to demonstrate that simple, nearly asymptotically optimal importance sampling schemes can be obtained for a variety of problems via the subsolution approach.
Antonio Siconolfi - One of the best experts on this subject based on the ideXlab platform.
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Scalar reduction techniques for weakly coupled Hamilton–Jacobi systems
Nodea-nonlinear Differential Equations and Applications, 2018Co-Authors: Antonio Siconolfi, Sahar ZabadAbstract:We study a class of weakly coupled systems of Hamilton–Jacobi equations at the critical level. We associate to it a family of scalar discounted equation. Using control-theoretic techniques we construct an algorithm which allows obtaining a critical solution to the system as limit of a monotonic sequence of Subsolutions. We moreover get a characterization of isolated points of the Aubry set and establish semiconcavity properties for critical Subsolutions.
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Scalar Reduction Tchniques for Weakly Coupled Hamilton-Jacobi Systems
arXiv: Analysis of PDEs, 2017Co-Authors: Antonio Siconolfi, Sahar ZabadAbstract:We study a class of weakly coupled systems of Hamilton{Jacobi equations at the critical level. We associate to it a family of scalar discounted equation. Using control{theoretic tech- niques we construct an algorithm which allows obtaining a critical solution to the system as limit of a monotonic sequence of Subsolutions. We moreover get a characterization of isolated points of the Aubry set and establish semiconcavity properties for critical Subsolutions.
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a lagrangian approach to weakly coupled hamilton jacobi systems
Siam Journal on Mathematical Analysis, 2016Co-Authors: Hiroyoshi Mitake, Antonio Siconolfi, Hung V Tran, Naoki YamadaAbstract:We perform a qualitative analysis of a class of weakly coupled Hamilton--Jacobi systems in the spirit of weak KAM theory. We define a family of related action functionals containing the Lagrangians associated with the Hamiltonians of the system. We use them to characterize the Subsolutions of the system and to provide explicit representation formulae for Subsolutions enjoying an additional maximality property. A crucial step for our analysis is to put the problem in a suitable random frame. The presentation is accessible to readers without a background in probability; only some basic knowledge of measure theory is required.
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a lagrangian approach to weakly coupled hamilton jacobi systems
arXiv: Analysis of PDEs, 2015Co-Authors: Hiroyoshi Mitake, Antonio Siconolfi, Hung V Tran, Naoki YamadaAbstract:We study a class of weakly coupled Hamilton-Jacobi systems with a specific aim to perform a qualitative analysis in the spirit of weak KAM theory. Our main achievement is the definition of a family of related action functionals containing the Lagrangians obtained by duality from the Hamiltonians of the system. We use them to characterize, by means of a suitable estimate, all the Subsolutions of the system, and to explicitly represent some Subsolutions enjoying an additional maximality property. A crucial step for our analysis is to put the problem in a suitable random frame. Only some basic knowledge of measure theory is required, and the presentation is accessible to readers without background in probability.
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Existence and regularity of strict critical Subsolutions in the stationary ergodic setting
arXiv: Analysis of PDEs, 2012Co-Authors: Andrea Davini, Antonio SiconolfiAbstract:We prove that any continuous and convex stationary ergodic Hamiltonian admits critical Subsolutions, which are strict outside the random Aubry set. They make up, in addition, a dense subset of all critical Subsolutions with respect to a suitable metric. If the Hamiltonian is additionally assumed of Tonelli type, then there exist strict Subsolutions of class $\CC^{1,1}$ in $\R^N$. The proofs are based on the use of Lax--Oleinik semigroups and their regularizing properties in the stationary ergodic environment, as well as on a generalized notion of Aubry set.
Maria Paluch - One of the best experts on this subject based on the ideXlab platform.
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Influence of some aqueous Subsolutions on behaviour of stearic acid monolayers
Colloids and Surfaces A: Physicochemical and Engineering Aspects, 1995Co-Authors: Beata Korchowiec, Maria PaluchAbstract:Abstract Spread monolayers of stearic acid were studied as a function of aqueous subsolution compositions. Subsolutions containing n -dodecyltrimethylammonium bromide (DTMAB), n -dodecylpyridinium chloride (DPC) and n -dodecyl sulphonic acid sodium salt (DSS) were used. The concentration of Subsolutions was constant in the 0.000001–0.0005 M range. At these concentrations the micellization process did not occur. A strong influence of the subsolution concentration on surface pressure-area (π- A ) isotherms of stearic acid monolayers was observed. From these isotherms, adsorptions of DTMAB, DPC and DSS into a monolayer-covered surface have been calculated. The plots of these adsorptions against the inverse of the area per stearic acid molecule are linear, as required by Barnes' theory. Values of the effective cross-sectional area of monolayer molecule and surface concentration of surfactant in the monolayer-free surface are obtained from these plots and compared with the known properties of stearic acid monolayers on DTMAB, DPC and DSS aqueous solutions.
Ahmed Zeriahi - One of the best experts on this subject based on the ideXlab platform.
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Pluripotential solutions versus viscosity solutions to complex Monge-Amp{è}re flows
arXiv: Complex Variables, 2019Co-Authors: Vincent Guedj, Chinh H. Lu, Ahmed ZeriahiAbstract:We compare various notions of weak Subsolutions to degenerate complex Monge-Amp{e}re flows, showing that they all coincide. This allows us to show that the viscosity solution coincides with the envelope of pluripotential Subsolutions. Dedicated to Duong Hong Phong on the occasion of his 65th birthday.
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Pluripotential solutions versus viscosity solutions to complex Monge-Ampère flows
2019Co-Authors: Vincent Guedj, Ahmed ZeriahiAbstract:We compare various notions of weak Subsolutions to degenerate complex Monge-Ampère flows, showing that they all coincide. This allows us to show that the viscosity solution coincides with the envelope of pluripotential Subsolutions. Dedicated to Duong Hong Phong on the occasion of his 65th birthday.
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Weak Subsolutions to complex Monge–Ampère equations
Journal of the Mathematical Society of Japan, 2019Co-Authors: Vincent Guedj, Ahmed ZeriahiAbstract:We compare various notions of weak Subsolutions to degenerate complex Monge–Ampere equations, showing that they all coincide. This allows us to give an alternative proof of mixed Monge–Ampere inequalities due to Kolodziej and Dinew.
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Weak Subsolutions to complex Monge-Amp\`ere equations
arXiv: Complex Variables, 2017Co-Authors: Vincent Guedj, Chinh H. Lu, Ahmed ZeriahiAbstract:We compare various notions of weak Subsolutions to degenerate complex Monge-Amp\`ere equations, showing that they all coincide. This allows us to give an alternative proof of mixed Monge-Amp\`ere inequalities due to Kolodziej and Dinew.
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weak Subsolutions to complex monge amp ere equations
arXiv: Complex Variables, 2017Co-Authors: Vincent Guedj, Ahmed ZeriahiAbstract:We compare various notions of weak Subsolutions to degenerate complex Monge-Amp\`ere equations, showing that they all coincide. This allows us to give an alternative proof of mixed Monge-Amp\`ere inequalities due to Kolodziej and Dinew.
