The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific community. Copyright Statement This electronic document is in PDF format. One needs Acrobat Reader (available freely for most platforms from the Adobe web site) to benefit from the full interactive machinery: using the package hyperref by Sebastian Rahtz, the table of contents and all LATEX cross-references are automatically converted into clickable hyperlinks, bookmarks are generated automatically, etc.. So, do not hesitate to click on references to equation or section numbers, on items of thetableofcontents and of the index, etc.. One may freely use and print this document for one’s own purpose or even distribute it freely, but not commercially, provided it is distributed in its entirety and without modifications, including this preface and copyright statement. Any use of thecontents should be acknowledged according to the standard scientific practice. The

We propose a new sweeping algorithm which discretizes the Legendre transform of the numerical Hamiltonian using an explicit formula. This formula yields the numerical solution at a grid point using only its immediate neighboring grid values and is easy to implement numerically. The minimization that is related to the Legendre transform in our sweeping scheme can either be solved analytically or numerically. We illustrate the efficiency and accuracy approach with several numerical examples in 2D and 3D. 1.

this paper is to derive high-order numerical schemes for the approximation of the value function, this being done using the Dynamic Programming approach.

. On montre l'analogie existant entre le calcul des probabilites et la programmation dynamique. Dans la premiere situation les convolutions iterees de lois de probabilite jouent un role central, dans la seconde les inf-convolutions de fonctions couts ont un role similaire. L'outil d'analyse privilegiedelapremiere situation est la transformee de Fourier, celui de la seconde devrait etre la transform ee de Fenchel. Aux lois gaussiennes --- stables par convolution --- correspondent les formes quadratiques --- stables par inf-convolution. A la loi des grands des nombres et au theoreme de la limite centrale correspondent des theoremes asymptotiques pour la programmation dynamique --- convergence de la fonction valeur de l'etat moyenne vers la fonction caracteristique du minimum du cout instantane, convergencede la fonction valeur de l'ecart au minimun renormalise vers une forme quadratique. ABSTRACT. Asymptotic Theorems in Dynamic Programming We show the analogy between probability calculu...

Burgers turbulence subject to a force f(x, t) = j f j (x)(t t j ), where the t_j's are "kicking times" and the "impulses" f_j(x) have arbitrary space dependence, combines features of the purely decaying and the continuously forced cases. With large-scale forcing this "kicked" Burgers turbulence presents many of the regimes proposed by E, Khanin, Mazel and Sinai (1997) for the case of random white-intime forcing. It is also amenable to efficient numerical simulations in the inviscid limit, using a modification of the Fast Legendre Transform method developed for decaying Burgers turbulence by Noullez and Vergassola (1994). For the kicked case, concepts such as "minimizers" and "main shock", which play crucial roles in recent developments for forced Burgers turbulence, become elementary since everything can be constructed from simple two-dimensional area-preserving Euler-Lagrange maps. The main results...

This article proposes a new approach for computing a semi-explicit form of the solution to a class of Hamilton–Jacobi (HJ) partial differential equations (PDEs), using control techniques based on viability theory. We characterize the epigraph of the value function solving the HJ PDE as a capture basin of a target through an auxiliary dynamical system, called “characteristic system”. The properties of capture basins enable us to define components as building blocks of the solution to the HJ PDE in the Barron/Jensen-Frankowska sense. These components can encode initial conditions, boundary conditions, and internal “boundary” conditions, which are the topic of this article. A generalized Lax-Hopf formula is derived, and enables us to formulate the necessary and sufficient conditions for a mixed initial and boundary conditions problem with multiple internal boundary conditions to be well posed. We illustrate the capabilities of the method with a data assimilation problem for reconstruction of highway traffic flow using Lagrangian measurements generated from Next Generation Simulation (NGSIM) traffic data.

This paper is devoted to a new method which allows to compute the convex envelope of a given function, by an evolution equation and techniques of curve evolution. We study the problem in the context of viscosity solutions and we propose numerical algorithms, to convexify a function, in one and two dimensions. In the end, we validate the model by presenting various numerical results.

Abstract. Computational Convex Analysis algorithms have been rediscovered several times in the past by researchers from different fields. To further communications between practitioners, we review the field of computational convex analysis, which focuses on the numerical computation of fundamental transforms arising from convex analysis. Current models use symbolic, numeric, and hybrid symbolic-numeric algorithms. Our objective is to disseminate widely the most efficient numerical algorithms, and to further communications between several fields benefiting from the same techniques. We survey applications of the algorithms which have been applied to problems arising from image processing (distance transform, generalized distance transform, mathematical morphology), partial differential equations (solving Hamilton-Jacobi equations, and using differential equations numerical schemes to compute the convex envelope), max-plus algebra, multi-fractal analysis, and several others. They span a wide range of applications in computer vision, robot navigation, phase transition in thermodynamics, electrical networks,

Convex optimization is a branch of mathematics dealing with non-linear optimization problems with additional geometric structure. This area has been the focus of considerable research due to the fact that convex optimization problems are scalable and can be efficiently solved by interior-point methods. Additionally, convex optimization problems are much more prevalent than previously thought as existing problems are constantly being recast in a convex framework. Over the last ten years or so, convex optimization has found applications in many new areas including control theory, signal processing, communications and networks, circuit design, data analysis and finance. As with any new problem, of key concern is visualization of the problem space in order to help develop intuition. In this thesis we develop and explore tools for the visualization of convex functions and related objects. We provide symbolic functionality where possible and appropriate, and proceed numerically otherwise. Of critical importance in convex optimization are the operations of Fenchel conjugation and subdifferentiation of convex functions. The algorithms for solving convex optimization