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Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications

2013-05-24 Mathematics
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Release : 2013-05-24
Genre : Mathematics
Kind : eBook
Book Rating : 330/5 ( reviews)

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Book Synopsis Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications by : Yves Achdou

Download or read book Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications written by Yves Achdou. This book was released on 2013-05-24. Available in PDF, EPUB and Kindle. Book excerpt: These Lecture Notes contain the material relative to the courses given at the CIME summer school held in Cetraro, Italy from August 29 to September 3, 2011. The topic was "Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications". The courses dealt mostly with the following subjects: first order and second order Hamilton-Jacobi-Bellman equations, properties of viscosity solutions, asymptotic behaviors, mean field games, approximation and numerical methods, idempotent analysis. The content of the courses ranged from an introduction to viscosity solutions to quite advanced topics, at the cutting edge of research in the field. We believe that they opened perspectives on new and delicate issues. These lecture notes contain four contributions by Yves Achdou (Finite Difference Methods for Mean Field Games), Guy Barles (An Introduction to the Theory of Viscosity Solutions for First-order Hamilton-Jacobi Equations and Applications), Hitoshi Ishii (A Short Introduction to Viscosity Solutions and the Large Time Behavior of Solutions of Hamilton-Jacobi Equations) and Grigory Litvinov (Idempotent/Tropical Analysis, the Hamilton-Jacobi and Bellman Equations).

Hamilton-Jacobi-Bellman Equations

2018-08-06 Mathematics
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Release : 2018-08-06
Genre : Mathematics
Kind : eBook
Book Rating : 714/5 ( reviews)

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Book Synopsis Hamilton-Jacobi-Bellman Equations by : Dante Kalise

Download or read book Hamilton-Jacobi-Bellman Equations written by Dante Kalise. This book was released on 2018-08-06. Available in PDF, EPUB and Kindle. Book excerpt: Optimal feedback control arises in different areas such as aerospace engineering, chemical processing, resource economics, etc. In this context, the application of dynamic programming techniques leads to the solution of fully nonlinear Hamilton-Jacobi-Bellman equations. This book presents the state of the art in the numerical approximation of Hamilton-Jacobi-Bellman equations, including post-processing of Galerkin methods, high-order methods, boundary treatment in semi-Lagrangian schemes, reduced basis methods, comparison principles for viscosity solutions, max-plus methods, and the numerical approximation of Monge-Ampère equations. This book also features applications in the simulation of adaptive controllers and the control of nonlinear delay differential equations. Contents From a monotone probabilistic scheme to a probabilistic max-plus algorithm for solving Hamilton–Jacobi–Bellman equations Improving policies for Hamilton–Jacobi–Bellman equations by postprocessing Viability approach to simulation of an adaptive controller Galerkin approximations for the optimal control of nonlinear delay differential equations Efficient higher order time discretization schemes for Hamilton–Jacobi–Bellman equations based on diagonally implicit symplectic Runge–Kutta methods Numerical solution of the simple Monge–Ampere equation with nonconvex Dirichlet data on nonconvex domains On the notion of boundary conditions in comparison principles for viscosity solutions Boundary mesh refinement for semi-Lagrangian schemes A reduced basis method for the Hamilton–Jacobi–Bellman equation within the European Union Emission Trading Scheme

Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations

2014-01-31 Mathematics
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Release : 2014-01-31
Genre : Mathematics
Kind : eBook
Book Rating : 04X/5 ( reviews)

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Book Synopsis Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations by : Maurizio Falcone

Download or read book Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations written by Maurizio Falcone. This book was released on 2014-01-31. Available in PDF, EPUB and Kindle. Book excerpt: This largely self-contained book provides a unified framework of semi-Lagrangian strategy for the approximation of hyperbolic PDEs, with a special focus on Hamilton-Jacobi equations. The authors provide a rigorous discussion of the theory of viscosity solutions and the concepts underlying the construction and analysis of difference schemes; they then proceed to high-order semi-Lagrangian schemes and their applications to problems in fluid dynamics, front propagation, optimal control, and image processing. The developments covered in the text and the references come from a wide range of literature.

Approximation of Hamilton Jacobi Equations on Irregular Data

2012-05
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Release : 2012-05
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Book Rating : 532/5 ( reviews)

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Book Synopsis Approximation of Hamilton Jacobi Equations on Irregular Data by : Adriano Festa

Download or read book Approximation of Hamilton Jacobi Equations on Irregular Data written by Adriano Festa. This book was released on 2012-05. Available in PDF, EPUB and Kindle. Book excerpt: This book deals with the development and the analysis of numerical methods for the resolution of first order nonlinear differential equations of Hamilton-Jacobi type on irregular data. These equations arises for example in the study of front propagation via the level set methods, the Shape-from-Shading problem and, in general, in Control theory. Our contribution to the numerical approximation of Hamilton-Jacobi equations consists in the proposal of some semiLagrangian schemes for different kind of discontinuous Hamiltonian and in an analysis of their convergence and a comparison of the results on some test problems. In particular we will approach with an eikonal equation with discontinuous coefficients in a well posed case of existence of Lipschitz continuous solutions. Furthermore, we propose a semiLagrangian scheme also for a Hamilton-Jacobi equation of a eikonal type on a ramified space, for example a graph. This is a not classical domain and only in last years there are developed a systematic theory about this. We present, also, some applications of our results on several problems arise from applied sciences.

Variational Methods for Hamilton-Jacobi Equations and Applications

2021
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Release : 2021
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Book Synopsis Variational Methods for Hamilton-Jacobi Equations and Applications by : Hamza Ennaji

Download or read book Variational Methods for Hamilton-Jacobi Equations and Applications written by Hamza Ennaji. This book was released on 2021. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis we propose some variational methods for the mathematical and numerical analysis of a class of HJ equations. Thanks to the metric character of these equations, the set of subsolution corresponds to the set of 1-Lipschitz functions with respect to the Finsler metric associated to the Hamiltonian. Equivalently, it corresponds to the set of functions whose gradient belongs to a Finsler ball. The solution we are looking for is the maximal one, which can be described via a Hopf-Lax formula, solves a maximization problem under gradient constraint. We derive the associated dual problem which involves the Finsler total variation of vector measures under a divergence constraint. We take advantage of this saddle-point structure to use the augmented Lagrangian method for the numerical approximation of HJ equation. This characterization of the HJ equation allows making the link with some optimal transport problems. This link with optimal transport leads us to generalize the Evans-Gangbo approach. In fact, we show that the maximal viscosity subsolution of the HJ equation can be recovered by taking p→ ∞ in a class of Finslerp-Laplace problems with boundary obstacles. In addition, this allows us to construct the optimal flow for the associated Beckmann problem. As an application, we use our variational approach for the Shape from Shading problem.

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