A finite difference technique for solving optimization problems governed by linear functional differential equations

Aspects of the approximation and optimal control of systems governed by linear retarded nonautonomous functional differential equations (FDE) are considered. First, certain FDE are shown to be equivalent to corresponding abstract ordinary differential equations (ODE). Next, it is demonstrated that these abstract ODE may be approximated by difference equations in finite dimensional spaces. The optimal control problem for systems governed by FDE is then reduced to a sequence of mathematical programming problems. Finally, numerical results for two examples are presented and discussed.     

Spline approximations for linear nonautonomous delay systems

We develop a semi-discrete approximation framework for linear nonautonomous nonhomogeneous functional differential equations of retarded type. The approximation schemes are constructed and convergence results are obtained through the approximation of an associated abstract evolution operator. Computer implementation of the schemes is outlined and a spline-based method included in the framework is constructed. Extensions of the semi-discrete methods to schemes incorporating full discretization and difference equation approximation are also discussed. Numerical results for several examples demonstrating the feasibility of the schemes are presented.     

Approximation schemes for nonlinear neutral optimal control systems

We discuss methods of approximating stable neutral functional differential equations and associated optimal control problems by sequences of optimal control problems for ordinary differential equations. By introducing a class of “mollified” neutral functional differential equations, convergence of the linear interpolating spline and the averaging approximation scheme is proved. A number of numerical examples are included.     

Finite-difference analysis of fully dynamic problems for saturated porous media

Finite-difference methods, using staggered grids in space, are considered for the numerical approximation of fully dynamic poroelasticity problems. First, a family of second-order schemes in time is analyzed. A priori estimates for displacements in discrete energy norms are obtained and the corresponding convergence results are proved. Numerical examples are given to illustrate the convergence properties of these methods. As in the case of an incompressible fluid and small permeability, these schemes suffer from spurious oscillations in time, a first order scheme is proposed and analyzed. For this new scheme a priori estimates and convergence results are also given. Finally, numerical examples in one and two dimensions are presented to show the good monotonicity properties of this method.     

Finite-difference analysis of fully dynamic problems for saturated porous media

Finite-difference methods, using staggered grids in space, are considered for the numerical approximation of fully dynamic poroelasticity problems. First, a family of second-order schemes in time is analyzed. A priori estimates for displacements in discrete energy norms are obtained and the corresponding convergence results are proved. Numerical examples are given to illustrate the convergence properties of these methods. As in the case of an incompressible fluid and small permeability, these schemes suffer from spurious oscillations in time, a first order scheme is proposed and analyzed. For this new scheme a priori estimates and convergence results are also given. Finally, numerical examples in one and two dimensions are presented to show the good monotonicity properties of this method.     

Numerical treatment of oscillatory functional differential equations

In this paper we are concerned with oscillatory functional differential equations (that is, those equations where all the solutions oscillate) under a numerical approximation. Our interest is in the preservation of qualitative properties of solutions under a numerical discretisation. We give conditions under which an equation is oscillatory, and consider whether the discrete schemes derived using linear ?-methods will also be oscillatory. We conclude with some general theory.     

High-order computational methods for option valuation under multifactor models

Many of the different numerical techniques in the partial differential equations framework for solving option pricing problems have employed only standard second-order discretization schemes. A higher-order discretization has the advantage of producing low size matrix systems for computing sufficiently accurate option prices and this paper proposes new computational schemes yielding high-order convergence rates for the solution of multi-factor option problems. These new schemes employ Galerkin finite element discretizations with quadratic basis functions for the approximation of the spatial derivatives in the pricing equations for stochastic volatility and two-asset option problems and time integration of the resulting semi-discrete systems requires the computation of a single matrix exponential. The computations indicate that this combination of high-order finite elements and exponential time integration leads to efficient algorithms for multi-factor problems. Highly accurate European prices are obtained with relatively coarse meshes and high-order convergence rates are also observed for options with the American early exercise feature. Various numerical examples are provided for illustrating the accuracy of the option prices for Heston’s and Bates stochastic volatility models and for two-asset problems under Merton’s jump-diffusion model.     

非线性抛物方程耦合的离散化及其误差分析

0　引　　言对于线性抛物型初边值问题的有限元与边界元耦合法,我们已作过研究(可参见[2],[3]).然而,许多实际问题,如流体、对流扩散、热辐射及热传输等涉及到非线性问题,因此研究非线性问题的数值方法显得尤为重要.近年来,G.N.Gatica与G.C.Hsiao已将有限元(FEM)与边界元(BEM)耦合法拓广到非线性椭圆问题(如[5],[6]),但如何应用FEM与BEM耦合法来处理非线性抛物型初边值问题,就作者所知迄今为止尚属空白.这里我们试图对此进行研究.设Ω为R2中一有界单连通区域,其边界为Γ:=Ω.Ωc:=R2\Ω为闭区域Ω的补区域,I:=(0,T].我们考虑如…     

Binary weighted essentially non-oscillatory (BWENO) approximation

Weighted essentially non-oscillatory (WENO) schemes have been mainly used for solving hyperbolic partial differential equations (PDEs). Such schemes are capable of high order approximation in smooth regions and non-oscillatory sharp resolution of discontinuities. The base of the WENO schemes is a non-oscillatory WENO approximation procedure, which is not necessarily related to PDEs. The typical WENO procedures are WENO interpolation and WENO reconstruction. The WENO algorithm has gained much popularity but the basic idea of approximation did not change much over the years. In this paper, we first briefly review the idea of WENO interpolation and propose a modification of the basic algorithm. New approximation should improve basic characteristics of the approximation and provide a more flexible framework for future applications. New WENO procedure involves a binary tree weighted construction that is based on key ideas of WENO algorithm and we refer to it as the binary weighted essentially non-oscillatory (BWENO) approximation. New algorithm comes in a rational and a polynomial version. Furthermore, we describe the WENO reconstruction procedure, which is usually involved in the numerical schemes for hyperbolic PDEs, and propose the new reconstruction procedure based on the described BWENO interpolation. The obtained numerical results show that the newly proposed procedures perform very well on the considered test examples.     

Second-Order Methods for Solving Stochastic Differential Equations

In this paper we discuss the numerical methods with second-order accuracy for solving stochastic differential equations. An unbiased sample approximation method for $I_n=\int ^{t_{n+1}}_{t_n}(B_u-B_{t_n})^2du$ is proposed, where {$B_u$} is a Brownian motion. Then second-order schemes are derived both for scalar cases and for system cases. The errors are measured in the mean square sense. Several numerical examples are included, and numerical results indicate that second-order schemes compare favorably with Euler's schemes and 1.5th-order schemes.     

On finite element schemes of the dirichlet problem

In this paper, we consider finite element schemes applied to the Dirichlet problem for the system of nonlinear elliptic equations, based on piecewise linear polynomials, and present iterative methods for solving algebraic nonlinear equations, which construct monotone sequences. Furthermore, we derive error estimates which imply uniform convergence. Our results are based on the discrete maximum principle. Finally, some typical numerical examples are given to demonstrate the usefulness of convergence results.     

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstra     

On the positivity of linear weights in WENO approximations

High order accurate weighted essentially non-oscillatory （WENO） schemes have been used extensively in numerical solutions of hyperbolic partial differential equations and other convection dominated problems. However the WENO procedure can not be applied directly to obtain a stable scheme when negative linear weights are present. In this paper, we first briefly review the WENO framework and the role of linear weights, and then present a detailed study on the positivity of linear weights in a few typical WENO procedures, including WENO interpolation, WENO reconstruction and WENO approximation to first and second derivatives, and WENO integration. Explicit formulae for the linear weights are also given for these WENO procedures. The results of this paper should be useful for future design of WENO schemes involving interpolation, reconstruction, approximation to first and second derivatives, and integration procedures.     

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     

Booster Method for Singularly-Perturbed One-Dimensional Reaction-Diffusion Neumann Problems

A numerical method for singularly-perturbed self-adjoint boundary-value problems for second-order ordinary differential equations subject to Neumann boundary conditions is proposed. In this method (booster method), an asymptotic approximation is incorporated into a finite-difference scheme to improve the numerical solution. Uniform error estimates are derived for this method when implemented in known difference schemes. Numerical examples are presented to illustrate the present method.     

A theoretical analysis of a new second order finite volume approximation based on a low-order scheme using general admissible spatial meshes for the one dimensional wave equation

This paper details our note [6] and it is an extension of our previous works  and  which dealt with first order (both in time and space) and second order time accurate (second order in time and first order in space) implicit finite volume schemes for second order hyperbolic equations with Dirichlet boundary conditions on general nonconforming multidimensional spatial meshes introduced recently in [14]. We aim in this work (and some forthcoming studies) to get higher order (both in time and space) finite volume approximations for the exact solution of hyperbolic equations using the class of spatial generic meshes introduced recently in [14] on low order schemes from which the matrices used to compute the discrete solutions are sparse. We focus in the present contribution on the one dimensional wave equation and on one of its implicit finite volume schemes described in [4]. The implicit finite volume scheme approximating the one dimensional wave equation we consider (hereafter referred to as the basic finite volume scheme) yields linear systems to be solved successively. The matrices involved in these linear systems are tridiagonal, symmetric and definite positive. The finite volume approximate solution of the basic finite volume scheme is of order h+k$h+k$, where h (resp. k  ) is the mesh size of the spatial (resp. time) discretization. We construct a new finite volume approximation of order (h+k)²${(h+k)}^2$ in several discrete norms which allow us to get approximations of order two for the exact solution and its first derivatives. This new high-order approximation can be computed using linear systems whose matrices are the same ones used to compute the discrete solution of the basic finite volume scheme while the right hand sides are corrected. The construction of these right hand sides includes the approximation of some high order spatial derivatives of the exact solution. The computation of the approximation of these high order spatial derivatives can be performed using the same matrices stated above with another two tridiagonal matrices. The manner by which this new high-order approximation is constructed can be repeated to compute successively finite volume approximations of arbitrary order using the same matrices stated above. These high-order approximations can be obtained on any one dimensional admissible finite volume mesh in the sense of [13] without any condition. To reach the above results, a theoretical framework is developed and some numerical examples supporting the theory are presented. Some of the tools of this framework are new and interesting and they are stated in the one space dimension but they can be extended to several space dimensions. In particular a new and useful a prior estimate for a suitable discrete problem is developed and proved. The proof of this a prior estimate result is based essentially on the decomposition of the solution of the discrete problem into the solutions of two suitable discrete problems. A new technique is used in order to get a convenient finite volume approximation whose discrete time derivatives of order up to order two are also converging towards the solution of the wave equation and their corresponding time derivatives.     

Discretization of multiparameter eigenvalue problems

Summary Although multiparameter eigenvalue problems, as for example Mathieu's differential equation, have been known for a long time, so far no work has been done on the numerical treatment of these problems. So in this paper we extend the spectral theory for one parameter (cf. [7, II, VII]) to multiparameter eigenvalue problmes, formulate in the framework of discrete approximation a convergent numerical treatment, establish algebraic bifurcation equations for the intersection points of the eigenvalue curves and illustrate this with some numerical examples.     

Qualitatively stable finite difference schemes for advection–reaction equations

A systematic procedure is proposed and implemented for the design of nonstandard finite difference methods as reliable numerical simulations that preserve significant properties inherent to the solutions of advection–reaction equations. In the case of hyperbolic fixed-points, a renormalization of the denominators of the discrete derivatives is performed for the numerical solutions to display the linear stability properties of the exact solutions. Non-hyperbolic fixed-points are described with the help of two new monotonic properties the construction of schemes, which preserve these properties, being done by nonlocal approximation of nonlinear terms in the reaction terms.     

A discrete collocation method for Fredholm integro-differential equations with weakly singular kernels

A fully discrete version of a piecewise polynomial collocation method is constructed to solve initial or boundary value problems of linear Fredholm integro-differential equations with weakly singular kernels. Using an integral equation reformulation and special graded grids, optimal global convergence estimates are derived. For special values of parameters an improvement of the convergence rate of elaborated numerical schemes is established. Some of our theoretical results are illustrated by numerical experiments.     

Nonlinear Implicit One-Step Schemes for Solving Initial Value Problems for Ordinary Differential Equation with Steep Gradients

A general theory for nonlinear implicit one-step schemes for solving initial value problems for ordinary differential equations is presented in this paper. The general expansion of "symmetric" implicit one-step schemes having second-order is derived and stability and convergence are studied. As examples, some geometric schemes are given. Based on previous work of the first author on a generalization of means, a fourth-order nonlinear implicit one-step scheme is presented for solving equations with steep gradients. Also, a hybrid method based on the GMS and a fourth-order linear scheme is discussed. Some numerical results are given.