Paul Dupuis

Dynamical systems and variational inequalities

The variational inequality problem has been utilized to formulate and study a plethora of competitive equilibrium problems in different disciplines, ranging from oligopolistic market equilibrium problems to traffic network equilibrium problems. In this paper we consider for a given variational inequality a naturally related ordinary differential equation. The ordinary differential equations that arise are nonstandard because of discontinuities that appear in the dynamics. These discontinuities are due to the constraints associated with the feasible region of the variational inequality problem. The goals of the paper are two-fold. The first goal is to demonstrate that although non-standard, many of the important quantitative and qualitative properties of ordinary differential equations that hold under the standard conditions, such as Lipschitz continuity type conditions, apply here as well. This is important from the point of view of modeling, since it suggests (at least under some appropriate conditions) that these ordinary differential equations may serve as dynamical models. The second goal is to prove convergence for a class of numerical schemes designed to approximate solutions to a given variational inequality. This is done by exploiting the equivalence between the stationary points of the associated ordinary differential equation and the solutions of the variational inequality problem. It can be expected that the techniques described in this paper will be useful for more elaborate dynamical models, such as stochastic models, and that the connection between such dynamical models and the solutions to the variational inequalities will provide a deeper understanding of equilibrium problems.

On lipschitz continuity of the solution mapping to the skorokhod problem, with applications

The solution m the Skorokhoci Problem defines a deieiminisiic mapping of paths that has been found to be useful in several areas of application. Typical uses of the mapping are construction and analysis of deterministic and stochastic processes that are constrained to remain in a given fixed set, such as stochastic differential equations with reflection and stochastic approximation schemes for problems with constraints In this paper we focus on the case where the set is a convex polyhedron and where the directions along which the constraint mechanism is applied arc possibly oblique and multivalued at corner points. Our goal is to characterize as completely as possible those situations in which the solution mapping is Lipschitz continuous. Our approach is geometric in nature, and shows that the Lipschitz continuity holds when a certain convex set, defined in terms of the normal directions to the faces of the polyhedron and the directions of the constraint mechanism, can be shown to exist. All previous inst...

A Weak Convergence Approach to the Theory of Large Deviations: Dupuis/A Weak

Formulation of Large Deviation Theory in Terms of the Laplace Principle. First Example: Sanovs Theorem. Second Example: Mogulskiis Theorem. Representation Formulas for Other Stochastic Processes. Compactness and Limit Properties for the Random Walk Model. Laplace Principle for the Random Walk Model with Continuous Statistics. Laplace Principle for the Random Walk Model with Discontinuous Statistics. Laplace Principle for the Empirical Measures of a Markov Chain. Extensions of the Laplace Principle for the Empirical Measures of a Markov Chain. Laplace Principle for Continuous-Time Markov Processes with Continuous Statistics. Appendices. Bibliography. Indexes.

Markov Chain Approximations for Deterministic Control Problems with Affine Dynamics and Quadratic Cost in the Control

We consider the construction of Markov chain approximations for an important class of deterministic control problems. The emphasis is on the construction of schemes that can be easily implemented and which possess a number of highly desirable qualitative properties. The class of problems covered is that for which the control is affine in the dynamics and with quadratic running cost. This class covers a number of interesting areas of application, including problems that arise in large deviations, risk-sensitive and robust control, robust filtering, and certain problems in computer vision. Examples are given as well as a proof of convergence.

Direct method for reconstructing shape from shading

An approach to shape-from-shading that is based on a connection with a calculus of variations/optimal control problem is proposed. An explicit representation corresponding to a shaded image is given for the surface; uniqueness of the surface (under suitable conditions) is an immediate consequence. The approach leads naturally to an algorithm for shape reconstruction that is simple, fast, provably convergent (in many cases, provably convergent to the correct solution), and does not require regularization. Given a continuous image, the algorithm can be proved to converge to the continuous surface solution as the image sampling frequency is taken to infinity. Experimental results are presented for synthetic and real images.<<ETX>>

Convex duality and the Skorokhod problem. II

Abstract. The solution to the Skorokhod Problem defines a deterministic mapping, referred to as the Skorokhod Map, that takes unconstrained paths to paths that are confined to live within a given domain G⊂ℝn. Given a set of allowed constraint directions for each point of ∂G and a path ψ, the solution to the Skorokhod Problem defines the constrained version φ of ψ, where the constraining force acts along one of the given boundary directions using the “least effort” required to keep φ in G. The Skorokhod Map is one of the main tools used in the analysis and construction of constrained deterministic and stochastic processes. When the Skorokhod Map is sufficiently regular, and in particular when it is Lipschitz continuous on path space, the study of many problems involving these constrained processes is greatly simplified.We focus on the case where the domain G is a convex polyhedron, with a constant and possibly oblique constraint direction specified on each face of G, and with a corresponding cone of constraint directions at the intersection of faces. The main results to date for problems of this type were obtained by Harrison and Reiman [22] using contraction mapping techniques. In this paper we discuss why such techniques are limited to a class of Skorokhod Problems that is a slight generalization of the class originally considered in [22]. We then consider an alternative approach to proving regularity of the Skorokhod Map developed in [13]. In this approach, Lipschitz continuity of the map is proved by showing the existence of a convex set that satisfies a set of conditions defined in terms of the data of the Skorokhod Problem. We first show how the geometric condition of [13] can be reformulated using convex duality. The reformulated condition is much easier to verify and, moreover, allows one to develop a general qualitative theory of the Skorokhod Map. An additional contribution of the paper is a new set of methods for the construction of solutions to the Skorokhod Problem.These methods are applied in the second part of this paper [17] to particular classes of Skorokhod Problems.

Dynamic importance sampling for queueing networks.

Abstract : Importance sampling is a technique that is commonly used to speed up Monte Carlo simulation of rare events. However, little is known regarding the design of efficient importance sampling algorithms in the context of queueing networks. The standard approach, which simulates the system using an a priori fixed change of measure suggested by large deviation analysis, has been shown to fail in even the simplest network setting (e.g., a two-node tandem network). Exploiting connections between importance sampling, differential games, and classical subsolutions of the corresponding Isaacs equation, we show how to design and analyze simple and efficient dynamic importance sampling schemes for general classes of networks. The models used to illustrate the approach include d-node tandem Jackson networks and a two node network with feedback, and the rare events studied are those of large queueing backlogs, including total population overflow and the overflow of individual buffers.

A global algorithm for shape from shading

A global algorithm for reconstructing shape from shading is described. This algorithm incorporates an earlier local algorithm that has been shown to be capable of fast, robust surface reconstruction for general surfaces if a small amount of information on the surface is provided. The new algorithm is capable of determining this information automatically, and thus can reconstruct a general surface from shading with no a priori information on the surface. In experimental tests on complex synthetic images, this algorithm has produced good surface reconstructions over most of the image. For 128 /spl times/ 128 images, the reconstruction took less than 30 s on a DECstation 5000. The algorithm appears noise resistant, giving good reconstructions even with an added pixel noise of /spl plusmn/10%.<<ETX>>

The large deviation principle for a general class of queueing systems. I

We prove the existence of a rate function and the validity of the large deviation principle for a general class of jump Markov processes that model queueing systems. A key step in the proof is a local large deviation principle for tubes centered at a class of piecewise linear, continuous paths mapping [0,1] into Rd . In order to prove certain large deviation limits, we represent the large deviation probabilities as the minimal cost functions of associated stochastic optimal control problems and use a subadditivity-type argument. We give a characterization of the rate function that can be used either to evaluate it explicitly in the cases where this is possible or to compute it numerically in the cases where an explicit evaluation is not possible.

Subsolutions of an Isaacs Equation and Efficient Schemes for Importance Sampling

Abstract : Previous papers by authors establish the connection between importance sampling algorithms for estimating rare-event probabilities, two-person zero-sum differential games, and the associated Isaacs equation. In order to construct nearly optimal schemes in a general setting, one must consider dynamic schemes, i.e., changes of measure that, in the course of a single simulation, can depend on the outcome of the simulation up till that time. The present paper and a companion paper show that classical sense subsolutions of the Isaacs equation provide a basic and flexible tool for the construction and analysis of nearly optimal schemes. Asymptotic analysis is the topic of the present paper, while the companion paper focuses on explicit methods for the construction of subsolutions, implementation aspects and numerical results.