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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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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.
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Subsolutions of an Isaacs Equation and Efficient Schemes for Importance Sampling: Examples and Numerics
2005Co-Authors: Paul Dupuis, Hui WangAbstract:Abstract : It has been established that importance sampling algorithms for estimating rare-event probabilities are intimately connected with two-person zero-sum differential games and the associated Isaacs equation. The purpose of the present paper and a companion paper is to show that the classical sense Subsolutions of the Isaacs equation can be used as a basic and flexible tool for the construction and analysis of efficient importance sampling schemes. The importance sampling algorithms based on Subsolutions are dynamic in the sense that during the course of a single simulation, the change of measure used at each time step may depend on the outcome of the simulation up until that time. While focused on theoretical aspects, the present paper discusses explicit methods of constructing Subsolutions, implementation issues, and simulation results.
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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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.
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Nonconvex degenerate Hamilton–Jacobi equations
Mathematische Zeitschrift, 2002Co-Authors: Fabio Camilli, Antonio SiconolfiAbstract:In this paper we prove a characterization of the maximal viscosity Subsolution for a class of Hamilton–Jacobi equations coupled with Dirichlet boundary condition for which uniqueness of the viscosity solution does not necessarily hold.
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
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.
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Existence of $C^{1,1}$ critical Subsolutions in discrete weak KAM theory
arXiv: Dynamical Systems, 2010Co-Authors: Maxime ZavidoviqueAbstract:In this article, following a first work of the author, 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 semi-concave and verifies 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 the work of Fathi-Maderna and Mather, we study invariant cost functions and apply this study to certain covering spaces, introducing a discrete analogue of Mather's $\alpha$ function on the cohomology.
Ngoc Cuong Nguyen - One of the best experts on this subject based on the ideXlab platform.
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a remark on the continuous Subsolution problem for the complex monge amp ere equation
arXiv: Complex Variables, 2018Co-Authors: Slawomir Kolodziej, Ngoc Cuong NguyenAbstract:We prove that if the modulus of continuity of a plurisubharmonic Subsolution satisfies a Dini type condition then the Dirichlet problem for the complex Monge-Amp\`ere equation has the continuous solution. The modulus of continuity of the solution is also given if the right hand side is locally dominated by capacity.
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A remark on the continuous Subsolution problem for the complex Monge-Amp\`ere equation.
arXiv: Complex Variables, 2018Co-Authors: Sławomir Kołodziej, Ngoc Cuong NguyenAbstract:We prove that if the modulus of continuity of a plurisubharmonic Subsolution satisfies a Dini type condition then the Dirichlet problem for the complex Monge-Amp\`ere equation has the continuous solution. The modulus of continuity of the solution is also given if the right hand side is locally dominated by capacity.
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on the holder continuous Subsolution problem for the complex monge ampere equation
Calculus of Variations and Partial Differential Equations, 2018Co-Authors: Ngoc Cuong NguyenAbstract:We give a necessary and sufficient condition for positive Borel measures such that the Dirichlet problem, with zero boundary data, for the complex Monge–Ampere equation admits Holder continuous plurisubharmonic solutions. In particular, when the Subsolution has finite Monge–Ampere total mass, we obtain an affirmative answer to a question of Zeriahi et al. (Complex Var. Elliptic Equ. 61(7):902–930, 2016).
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On the Hölder continuous Subsolution problem for the complex Monge–Ampère equation
Calculus of Variations and Partial Differential Equations, 2017Co-Authors: Ngoc Cuong NguyenAbstract:We give a necessary and sufficient condition for positive Borel measures such that the Dirichlet problem, with zero boundary data, for the complex Monge–Ampere equation admits Holder continuous plurisubharmonic solutions. In particular, when the Subsolution has finite Monge–Ampere total mass, we obtain an affirmative answer to a question of Zeriahi et al. (Complex Var. Elliptic Equ. 61(7):902–930, 2016).
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Subsolution theorem for the complex hessian equation
arXiv: Complex Variables, 2012Co-Authors: Ngoc Cuong NguyenAbstract:We prove the Subsolution theorem for the complex Hessian equations in a smoothly bounded strongly $m$-pseudoconvex domain, $1 < m < n$, in $\bC^n$.
Sahar Zabad - 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.
