Hamilton-Jacobi-Bellman方程的区域分解算法

对离散Hamilton-Jacobi-Bellman方程提出了一类区域分解算法,并在合理的假设下证明了该算法的单调收敛性,数值结果表明该算法的有效性与准确性．     

Multi-grid methods for Hamilton-Jacobi-Bellman equations

Summary In this paper we develop multi-grid algorithms for the numerical solution of Hamilton-Jacobi-Bellman equations. The proposed schemes result from a combination of standard multi-grid techniques and the iterative methods used by Lions and mercier in [11]. A convergence result is given and the efficiency of the algorithms is illustrated by some numerical examples.     

Numerical Analysis of a Problem Involving a Viscoelastic Body with Double Porosity

We study from a numerical point of view a multidimensional problem involving a viscoelastic body with two porous structures. The mechanical problem leads to a linear system of three coupled hyperbolic partial differential equations. Its corresponding variational formulation gives rise to three coupled parabolic linear equations. An existence and uniqueness result, and an energy decay property, are recalled. Then, fully discrete approximations are introduced using the finite element method and the implicit Euler scheme. A discrete stability property and a priori error estimates are proved, from which the linear convergence of the algorithm is derived under suitable additional regularity conditions. Finally, some numerical simulations are performed in one and two dimensions to show the accuracy of the approximation and the behaviour of the solution.     

A Crank-Nicolson scheme for the Landau-Lifshitz equation without damping

An accurate and efficient numerical approach, based on a finite difference method with Crank-Nicolson time stepping, is proposed for the Landau-Lifshitz equation without damping. The phenomenological Landau-Lifshitz equation describes the dynamics of ferromagnetism. The Crank-Nicolson method is very popular in the numerical schemes for parabolic equations since it is second-order accurate in time. Although widely used, the method does not always produce accurate results when it is applied to the Landau-Lifshitz equation. The objective of this article is to enumerate the problems and then to propose an accurate and robust numerical solution algorithm. A discrete scheme and a numerical solution algorithm for the Landau-Lifshitz equation are described. A nonlinear multigrid method is used for handling the nonlinearities of the resulting discrete system of equations at each time step. We show numerically that the proposed scheme has a second-order convergence in space and time.     

An adaptive grid scheme for the discrete Hamilton-Jacobi-Bellman equation

Summary. In this paper an adaptive finite difference scheme for the solution of the discrete first order Hamilton-Jacobi-Bellman equation is presented. Local a posteriori error estimates are established and certain properties of these estimates are proved. Based on these estimates an adapting iteration for the discretization of the state space is developed. An implementation of the scheme for two-dimensional grids is given and numerical examples are discussed. Received January 23, 1995 / Revised version December 6, 1995     

求解HJB方程的区域分解法

本文提出了求解HJB方程的一种区域分解法,并证明了算法的收敛性,这种算法将[3]提出的两子域区域分解法推广到多子域的情形.     

Numerical Solution of Hamilton-Jacobi-Bellman Equations by an Upwind Finite Volume Method

In this paper we present a finite volume method for solving Hamilton-Jacobi-Bellman(HJB) equations governing a class of optimal feedback control problems. This method is based on a finite volume discretization in state space coupled with an upwind finite difference technique, and on an implicit backward Euler finite differencing in time, which is absolutely stable. It is shown that the system matrix of the resulting discrete equation is an M-matrix. To show the effectiveness of this approach, numerical experiments on test problems with up to three states and two control variables were performed. The numerical results show that the method yields accurate approximate solutions to both the control and the state variables.     

Numerical analysis and simulations of a quasistatic frictional contact problem with damage in viscoelasticity

We consider a quasistatic problem which models the bilateral contact between a viscoelastic body and a foundation, taking into account the damage and the friction. The damage which results from tension or compression is then involved in the constitutive law and it is modelled using a nonlinear parabolic inclusion. The variational problem is formulated as a coupled system of evolutionary equations for which we state the existence of a unique solution. Then, we introduce a fully discrete scheme using the finite element method to approximate the spatial variable and the Euler scheme to discretize the time derivatives. Error estimates are derived and, under suitable regularity hypotheses, the convergence of the numerical scheme obtained. Finally, a numerical algorithm and results are presented for some two-dimensional examples.     

Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations

Summary. A general method for constructing high-order approximation schemes for Hamilton-Jacobi-Bellman equations is given. The method is based on a discrete version of the Dynamic Programming Principle. We prove a general convergence result for this class of approximation schemes also obtaining, under more restrictive assumptions, an estimate in of the order of convergence and of the local truncation error. The schemes can be applied, in particular, to the stationary linear first order equation in . We present several examples of schemes belonging to this class and with fast convergence to the solution. Received July 4, 1992 / Revised version received July 7, 1993     

Global optimization of arborescent multilevel inventory systems

We consider the numerical resolution of hierarchical inventory problems under global optimization. First we describe the model as well as the dynamical stochastic system and the impulse controls involved. Next we characterize the optimal cost function and we formulate the Hamilton-Jacobi-Bellman equations. We present a numerical scheme and a fast algorithm of resolution, with a result on the speed of convergence. Finally, we apply the discretization method to some examples where we show the usefulness of the proposed numerical method as well as the advantages of operating under global optimization.     

关于非定常不可压Navier-Stokes方程的时间高精度隐式差分方法

The incompressible Navier-Stokes equations,upon spatial discretization,become a system of differential algebraic equations,formally of index2.But due to the special forms of the discrete gradient and disrete divergence,its index can be regarded as 1.Thus,in this paper,a systematic approach following the ODE theory and methods is presented for the construction of high-order time-accurate implicit schemes for the incompressible Navier-Stokes equations,with projection methods for efficiency of numerical solution.The 3rd order 3-step BDF with componentconsistent pressure-correction projection method is a first attempt in this direction;the related iterative solution of the auxiliary velocyty,the boundary conditions and the stability of the algorithm are discussed.Results of numerical tests on the incompressible Navier-Stokes equations with an exact solution are presented,confirming the accureacy,stability and component-consistency of the proposed method.     

Numerical solution of a minimax ergodic optimal control problem

In this work we consider an L^∞ minimax ergodic optimal control problem with cumulative cost. We approximate the cost function as a limit of evolutions problems. We present the associated Hamilton-Jacobi-Bellman equation and we prove that it has a unique solution in the viscosity sense. As this HJB equation is consistent with a numerical procedure, we use this discretization to obtain a procedure for the primitive problem. For the numerical solution of the ergodic version we need a perturbation of the instantaneous cost function. We give an appropriate selection of the discretization and penalization parameters to obtain discrete solutions that converge to the optimal cost. We present numerical results. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A New Reduced-Order fve Algorithm Based on POD Method for Viscoelastic Equations

A proper orthogonal decomposition (POD) technique is used to reduce the finite volume element (FVE) method for two-dimensional (2D) viscoelastic equations. A reduced-order fully discrete FVE algorithm with fewer degrees of freedom and sufficiently high accuracy based on POD method is established. The error estimates of the reduced-order fully discrete FVE solutions and the implementation for solving the reduced-order fully discrete FVE algorithm are provided. Some numerical examples are used to illustrate that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order fully discrete FVE algorithm is one of the most effective numerical methods by comparing with corresponding numerical results of finite element formulation and finite difference scheme and that the reduced-order fully discrete FVE algorithm based on POD method is feasible and efficient for solving 2D viscoelastic equations.     

An Efficient Policy Iteration Algorithm for Dynamic Programming Equations

We present a scheme for Hamilton-Jacobi-Bellman equations based on a semi-Lagrangian discretization and an iterative method in the policy space. The scheme exploits the idea that a good initialization of the policy iteration procedure yields a faster numerical convergence to the optimal solution. The scheme features a pre-processing step with value iterations on a coarse grid. Numerical tests assess the efficient performance of the method. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Various passport options and their valuation

The passport option is a call option on the balance of a trading account. The option holder retains the gain from trading, while the writer is liable for the loss. Multi-asset passport options and passport options with discrete constraints are studied. For the first ones the pricing equations are Hamilton-Jacobi-Bellman equations. For those with discrete constraints, a linear complementary problem must be solved in order to price the option. The gain by selling passport options to utility maximizing investors and to investors who guess the market a certain percentage of the time is also examined.     

A mixed kinetic-diffusion surfactant model for the Henry isotherm

This paper focuses on the mathematical modeling and the variational and numerical analyses of surfactants behavior at the air–water interface, taking into account the mixed kinetic-diffusion model evolving to the Henry isotherm. The existence and uniqueness of a weak solution is proved applying classical results for linear parabolic equations and fixed-point techniques. Fully discrete approximations are obtained by using a finite element method and the backward Euler scheme. Error estimates are then proved from which, under adequate additional regularity conditions, the linear convergence of the algorithm is derived. Finally, some numerical simulations are presented in order to demonstrate the accuracy of the algorithm and the behavior of the solution for a commercially available surfactant.     

Finite iterative method for solving coupled Sylvester-transpose matrix equations

This paper is concerned with solutions to the so-called coupled Sylvester-transpose matrix equations, which include the generalized Sylvester matrix equation and Lyapunov matrix equation as special cases. By extending the idea of conjugate gradient method, an iterative algorithm is constructed to solve this kind of coupled matrix equations. When the considered matrix equations are consistent, for any initial matrix group, a solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm solution group of the coupled Sylvester-transpose matrix equations can be derived when a suitable initial matrix group is chosen. By applying the proposed algorithm, the optimal approximation solution group to a given matrix group can be obtained by finding the least Frobenius norm solution group of new general coupled matrix equations. Finally, a numerical example is given to illustrate that the algorithm is effective.     

Asymptotic stability of exact and discrete solutions for neutral multidelay-integro-differential equations

This paper concerns the long-time behavior of the exact and discrete solutions to a class of nonlinear neutral integro-differential equations with multiple delays. Using a generalized Halanay inequality, we give two sufficient conditions for the asymptotic stability of the exact solution to this class of equations. Runge–Kutta methods with compound quadrature rule are considered to discretize this class of equations with commensurate delays. Nonlinear stability conditions for the presented methods are derived. It is found that, under suitable conditions, this class of numerical methods retain the asymptotic stability of the underlying system. Some numerical examples that illustrate the theoretical results are given.     

Numerical computation of the value function of optimally controlled stochastic switching processes by multi-grid techniques

By the dynamic programming principle the value function of an optimally controlled stochasticswitching process can be shown to satisfy a boundary value problem for a fully nonlinear second-order elliptic differential equation of Hamilton-Jacobi-Bellman (HJB-) type. For the numerical solution of that HJB-equation we present a multi-grid algorithm whose main features arethe use of nonlinear Gauss-Seidel iteration in the smoothing process and an adaptive local choice of prolongations and restrictions in the coarse-to-fine and fine-to-coarse transfers. Local convergence is proved by combining nonlinear multi-grid convergence theory and elementarysubdifferential calculus. The efficiency of the algorithm is demonstrated for optimal advertising in stochastic dynamic sales response models of Vidale-Wolfe type.