Author: Dante Kalise Publisher: Walter de Gruyter GmbH & Co KG ISBN: 3110543591 Category : Mathematics Languages : en Pages : 210
Book Description
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
Author: Yves Achdou Publisher: Springer ISBN: 3642364330 Category : Mathematics Languages : en Pages : 316
Book Description
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).
Author: Yangang Chen Publisher: ISBN: Category : Differential equations, Nonlinear Languages : en Pages : 197
Book Description
Hamilton-Jacobi-Bellman (HJB) equations are nonlinear controlled partial differential equations (PDEs). In this thesis, we propose various numerical methods for HJB equations arising from three specific applications. First, we study numerical methods for the HJB equation coupled with a Kolmogorov-Fokker-Planck (KFP) equation arising from mean field games. In order to solve the nonlinear discretized systems efficiently, we propose a multigrid method. The main novelty of our approach is that we subtract artificial viscosity from the direct discretization coarse grid operators, such that the coarse grid error estimations are more accurate. The convergence rate of the proposed multigrid method is mesh-independent and faster than the existing methods in the literature. Next, we investigate numerical methods for the HJB formulation that arises from the mass transport image registration model. We convert the PDE of the model (a Monge-Ampère equation) to an equivalent HJB equation, propose a monotone mixed discretization, and prove that it is guaranteed to converge to the viscosity solution. Then we propose multigrid methods for the mixed discretization, where we set wide stencil points as coarse grid points, use injection at wide stencil points as the restriction, and achieve a mesh-independent convergence rate. Moreover, we propose a novel periodic boundary condition for the image registration PDE, such that when two images are related by a combination of a translation and a non-rigid deformation, the numerical scheme recovers the underlying transformation correctly. Finally, we propose a deep neural network framework for the HJB equations emerging from the study of American options in high dimensions. We convert the HJB equation to an equivalent Backward Stochastic Differential Equation (BSDE), introduce the least squares residual of the BSDE as the loss function, and propose a new neural network architecture that utilizes the domain knowledge of American options. Our proposed framework yields American option prices and deltas on the entire spacetime, not only at a given point. The computational cost of the proposed approach is quadratic in dimension, which addresses the curse of dimensionality issue that state-of-the-art approaches suffer.
Author: Maurizio Falcone Publisher: World Scientific ISBN: 9789812799807 Category : Mathematics Languages : en Pages : 256
Book Description
Geometrical optics and viscosity solutions / A.-P. Blanc, G. T. Kossioris and G. N. Makrakis -- Computation of vorticity evolution for a cylindrical Type-II superconductor subject to parallel and transverse applied magnetic fields / A. Briggs ... [et al.] -- A characterization of the value function for a class of degenerate control problems / F. Camilli -- Some microstructures in three dimensions / M. Chipot and V. Lecuyer -- Convergence of numerical schemes for the approximation of level set solutions to mean curvature flow / K. Deckelnick and G. Dziuk -- Optimal discretization steps in semi-lagrangian approximation of first-order PDEs / M. Falcone, R. Ferretti and T. Manfroni -- Convergence past singularities to the forced mean curvature flow for a modified reaction-diffusion approach / F. Fierro -- The viscosity-duality solutions approach to geometric pptics for the Helmholtz equation / L. Gosse and F. James -- Adaptive grid generation for evolutive Hamilton-Jacobi-Bellman equations / L. Grune -- Solution and application of anisotropic curvature driven evolution of curves (and surfaces) / K. Mikula -- An adaptive scheme on unstructured grids for the shape-from-shading problem / M. Sagona and A. Seghini -- On a posteriori error estimation for constant obstacle problems / A. Veeser.
Author: Martino Bardi Publisher: Springer Science & Business Media ISBN: 0817647554 Category : Science Languages : en Pages : 588
Book Description
This softcover book is a self-contained account of the theory of viscosity solutions for first-order partial differential equations of Hamilton–Jacobi type and its interplay with Bellman’s dynamic programming approach to optimal control and differential games. It will be of interest to scientists involved in the theory of optimal control of deterministic linear and nonlinear systems. The work may be used by graduate students and researchers in control theory both as an introductory textbook and as an up-to-date reference book.
Author: Lindsay Joan Martin Publisher: ISBN: Category : Languages : en Pages : 256
Book Description
The goal of this thesis is to present two frameworks for the computation of the solutions of Hamilton-Jacobi-Bellman (HJB) equations. In Chapter 2, we present a framework for computing solutions to HJB equations on smooth hypersurfaces. It is well known that the viscosity solution of the HJB equation is equivalent to the value function of a corresponding optimal control problem. We extend the optimal control problem given on the surface to an equivalent one defined in a sufficiently thin narrow band of surface. The extension is done appropriately so that the viscosity solution of the extended HJB equation in the narrow band is identical to the constant normal extension of the viscosity solution of the HJB equation on the surface. With this framework, we can easily use efficient, existing (high order) numerical methods developed on Cartesian grids to solve HJB equations on surfaces, with a computational cost that scales with the dimension of the surfaces. This framework also provides a systematic way for solving HJB equations on unstructured point clouds that are sampled from a surface. In Chapter 3, we present a parallelizable domain decomposition algorithm to solve Eikonal equations, a special case of HJB equations. The method is an iterative two-scale method that uses a parareal-like update scheme in combination with standard Eikonal solvers. The purpose of the two scales is to accelerate convergence and maintain accuracy. We adapt a weighted version of the parareal method for stability, and the optimal weights are studied via a model problem. One can view the new method as a general framework where an effective coarse grid solver is computed “on the fly” from coarse and fine grid solutions that are computed in previous iterations. To demonstrate the framework, we develop a specific scheme using Cartesian grids and the fast sweeping method for solving Eikonal equations. Numerical examples are given to demonstrate the method’s effectiveness on a variety of stereotypes of Eikonal equations
Author: Daniela Tonon Publisher: Springer ISBN: 3319607715 Category : Mathematics Languages : en Pages : 399
Book Description
Focusing on applications to science and engineering, this book presents the results of the ITN-FP7 SADCO network’s innovative research in optimization and control in the following interconnected topics: optimality conditions in optimal control, dynamic programming approaches to optimal feedback synthesis and reachability analysis, and computational developments in model predictive control. The novelty of the book resides in the fact that it has been developed by early career researchers, providing a good balance between clarity and scientific rigor. Each chapter features an introduction addressed to PhD students and some original contributions aimed at specialist researchers. Requiring only a graduate mathematical background, the book is self-contained. It will be of particular interest to graduate and advanced undergraduate students, industrial practitioners and to senior scientists wishing to update their knowledge.
Author: Jan Hendrik Witte Publisher: ISBN: Category : Hamilton-Jacobi equations Languages : en Pages : 93
Book Description
We consider the numerical solution of discretised Hamilton-Jacobi-Bellman (HJB) equations with applications in finance. For the discrete linear complementarity problem arising in American option pricing, we study a policy iteration method. We show, analytically and numerically, that, in standard situations, the computational cost of this approach is comparable to that of European option pricing. We also characterise the shortcomings of policy iteration, providing a lower bound for the number of steps required when having inaccurate initial data. For discretised HJB equations with a finite control set, we propose a penalty approach. The accuracy of the penalty approximation is of first order in the penalty parameter, and we present a Newton-type iterative solver terminating after finitely many steps with a solution to the penalised equation. For discretised HJB equations and discretised HJB obstacle problems with compact control sets, we also introduce penalty approximations. In both cases, the approximation accuracy is of first order in the penalty parameter. We again design Newton-type methods for the solution of the penalised equations. For the penalised HJB equation, the iterative solver has monotone global convergence. For the penalised HJB obstacle problem, the iterative solver has local quadratic convergence. We carefully benchmark all our numerical schemes against current state-of-the-art techniques, demonstrating competitiveness.
Author: Martino Bardi Publisher: Springer Science & Business Media ISBN: 9780817640293 Category : Mathematics Languages : en Pages : 404
Book Description
The theory of two-person, zero-sum differential games started at the be ginning of the 1960s with the works of R. Isaacs in the United States and L. S. Pontryagin and his school in the former Soviet Union. Isaacs based his work on the Dynamic Programming method. He analyzed many special cases of the partial differential equation now called Hamilton Jacobi-Isaacs-briefiy HJI-trying to solve them explicitly and synthe sizing optimal feedbacks from the solution. He began a study of singular surfaces that was continued mainly by J. Breakwell and P. Bernhard and led to the explicit solution of some low-dimensional but highly nontriv ial games; a recent survey of this theory can be found in the book by J. Lewin entitled Differential Games (Springer, 1994). Since the early stages of the theory, several authors worked on making the notion of value of a differential game precise and providing a rigorous derivation of the HJI equation, which does not have a classical solution in most cases; we mention here the works of W. Fleming, A. Friedman (see his book, Differential Games, Wiley, 1971), P. P. Varaiya, E. Roxin, R. J. Elliott and N. J. Kalton, N. N. Krasovskii, and A. I. Subbotin (see their book Po sitional Differential Games, Nauka, 1974, and Springer, 1988), and L. D. Berkovitz. A major breakthrough was the introduction in the 1980s of two new notions of generalized solution for Hamilton-Jacobi equations, namely, viscosity solutions, by M. G. Crandall and P. -L.
Author: Dante Kalise
Author: Yves Achdou
Author: Yangang Chen
Author: Dante Kalise
Author: Maurizio Falcone
Author: Martino Bardi
Author: Lindsay Joan Martin
Author: Daniela Tonon
Author: Jan Hendrik Witte
Author: Martino Bardi