The semi-analytical method for time-dependent wave problems with uncertainties

This paper provides a constructive procedure for the computation of approximate solutions of random time-dependent hyperbolic mean square partial differential problems. Based on the theoretical representation of the solution as an infinite random improper integral, obtained via the random Fourier transform method, a double approximation process is implemented. Firstly, a random Gauss-Hermite quadrature is applied, and then, the evaluations at the nodes of the integrand are approximated by using a random Störmer numerical method. Numerical results are illustrated with examples.     

The domain decomposition method for parabolic transmission problems

An estimate of the rate of convergence is given for the domain decomposition method for the second-order parabolic transmission problem. A brief discussion of the method and some of its applications are presented.     

A forward scheme for backward SDEs

We introduce a forward scheme for simulating backward SDEs. Compared to existing schemes, ours avoids high order nestings of conditional expectations backwards in time. In this way the error, when approximating the conditional expectation, depending on the time partition, is significantly reduced. Besides this generic result, we present an implementable algorithm and prove its convergence. Finally, we demonstrate the strength of the new algorithm by solving a financial problem numerically.     

The decomposition method for Cauchy reaction–diffusion problems

In this paper, the solution of Cauchy problems for the reaction–diffusion equation is obtained using the decomposition method. In the case when the reaction parameter is time-dependent only, an analytical solution in series form can be derived, otherwise symbolic numerical computations may need to be performed.     

The backward group preserving scheme for multi-dimensional nonhomogeneous and nonlinear backward wave problems

In this study, we utilize a backward group preserving scheme (BGPS) to cope with the nonhomogeneous as well as nonlinear backward wave problems (BWPs). Because the solution does not continuously count on the given information, the BWP is well-known to be seriously ill-posed. When six numerical instances are examined, we reveal that the BGPS is capable of tackling the nonhomogeneous and nonlinear BWPs. Besides, the BGPS is also robust enough against the perturbation even with the boisterous final data, of which the numerical results are rather accurate, effective and stable.     

The Adomian decomposition method in turning point problems

In this paper, the decomposition method is applied to boundary-value problems of ordinary differential equations with a parameter exhibiting turning points.     

A step-by-step method with Laguerre functions for solving time-dependent problems

A step-by-step modification of the well-known approach proposed by Mikhaylenko and Konyukh to solving dynamic problems is proposed. The approach is based on the Laguerre transform with respect to time. In this modification the Laguerre transform is applied to a sequence of finite time intervals. The solution obtained at the end of a time interval is used as the initial data for solving the problem on the next time interval. The method is illustrated by examples for the harmonic oscillator problem and the 1D wave equation. Accuracy and stability of the method are analyzed. This approach allows obtaining a solution of high accuracy on large time intervals.     

A proximal-based decomposition method for convex minimization problems

This paper presents a decomposition method for solving convex minimization problems. At each iteration, the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables. We derive this algorithm from Rockafellar's proximal method of multipliers, which involves an augmented Lagrangian with an additional quadratic proximal term. The algorithm preserves the good features of the proximal method of multipliers, with the additional advantage that it leads to a decoupling of the constraints, and is thus suitable for parallel implementation. We allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem's data, the method is globally convergent and at a linear rate. The method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets.Corresponding author. Partially supported by Air Force Office of Scientific Research Grant 91-0008 and National Science Foundation Grant DMS-9201297.     

A convergent decomposition method for box-constrained optimization problems

In this work we consider the problem of minimizing a continuously differentiable function over a feasible set defined by box constraints. We present a decomposition method based on the solution of a sequence of subproblems. In particular, we state conditions on the rule for selecting the subproblem variables sufficient to ensure the global convergence of the generated sequence without convexity assumptions. The conditions require to select suitable variables (related to the violation of the optimality conditions) to guarantee theoretical convergence properties, and leave the degree of freedom of selecting any other group of variables to accelerate the convergence.     

正倒向随机微分方程组的数值解法

1990年,Pardoux和Peng(彭实戈)解决了非线性倒向随机微分方程(backward stochastic differential equation,BSDE)解的存在唯一性问题,从而建立了正倒向随机微分方程组(forward backward stochastic differential equations,FBSDEs)的理论基础;之后,正倒向随机微分方程组得到了广泛研究,并被应用于众多研究领域中,如随机最优控制、偏微分方程、金融数学、风险度量、非线性期望等.近年来,正倒向随机微分方程组的数值求解研究获得了越来越多的关注,本文旨在基于正倒向随机微分方程组的特性,介绍正倒向随机微分方程组的主要数值求解方法.我们将重点介绍讨论求解FBSDEs的积分离散法和微分近似法,包括一步法和多步法,以及相应的数值分析和理论分析结果.微分近似法能构造出求解全耦合FBSDEs的高效高精度并行数值方法,并且该方法采用最简单的Euler方法求解正向随机微分方程,极大地简化了问题求解的复杂度.文章最后,我们尝试提出关于FBSDEs数值求解研究面临的一些亟待解决和具有挑战性的问题.     

Coupling dual reciprocity BEM and differential quadrature method for time-dependent diffusion problems

This paper presents the very first combined application of dual reciprocity BEM (DRBEM) and differential quadrature (DQ) method to time-dependent diffusion problems. In this study, the DRBEM is employed to discretize the spatial partial derivatives. The DQ method is then applied to analogize temporal derivatives. The resulting algebraic formulation is the known Lyapunov matrix equation, which can be very efficiently solved by the Bartels–Stewart algorithms. The mixed scheme combines strong geometry flexibility and boundary-only feature of the BEM and high accuracy and efficiency of the DQ method. Its superiority is demonstrated through the solution of some benchmark diffusion problems. The DQ method is shown to be numerically accurate, stable and computationally efficient in computing dynamic problems. In particular, the present study reveals that the DRBEM is also very efficient for transient diffusion problems with Dirichlet boundary conditions by coupling the DQ method in time discretization.     

A decomposition method for convex minimization problems and its application

1. IntroductionConsider the following special convex programming problem(P) adn{f(~) g(z); Ax = z},where f: Re - (--co, co] and g: Re - (--co, co] are closed proper convex functions andA is an m x n matrix. The Lagrangian for problem (P) is defined by L: Rad x Re x Re -- (~co, co] as follows:L(x, z, y) = f(x) g(z) (y, Ax ~ z), (1.1)where (., .) denotes the inner product in the general sense and 'y is the Lagrangian multiplierassociated with the constraint Ax = z. The augmented L…     

Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems

This paper is concerned with Kalman-Bucy filtering problems of a forward and backward stochastic system which is a Hamiltonian system arising from a stochastic optimal control problem. There are two main contributions worthy pointing out. One is that we obtain the Kalman-Bucy filtering equation of a forward and backward stochastic system and study a kind of stability of the aforementioned filtering equation. The other is that we develop a backward separation technique, which is different to Wonham's separation theorem, to study a partially observed recursive optimal control problem. This new technique can also cover some more general situation such as a partially observed linear quadratic non-zero sum differential game problem is solved by it. We also give a simple formula to estimate the information value which is the difference of the optimal cost functionals between the partial and the full observable information cases.     

Overlapping Schwarz waveform relaxation method for the solution of the forward–backward heat equation

In this article we analyzed the convergence of the Schwarz waveform relaxation method for solving the forward–backward heat equation. Numerical results are presented for a specific type of model problem.     

An efficient domain decomposition method for three-dimensional parabolic problems

A non-overlapping domain decomposition algorithm to solve three-dimensional parabolic partial differential equations is presented. It has been shown in this paper that the algorithm is unconditionally stable and efficient. Spectral radii for the interface and interior region are provided. Unlike two-dimensional problem, it has been found out that estimating the values of the points of the interface in three-dimensional problem is no longer negligible.     

On a finite element method for a certain class of time-dependent problems

The finite element method is applied to solve a linear initial-boundary value problem. The basic idea is to combine this method for a disretization in space variables with the Laplace transform technique for a time variable. Formulation, existence and uniqueness of a weak solution is investigated. The convergence and the rate of convergence of the proposed approximate solution is discussed     

A simpler analysis of a hybrid numerical method for time-dependent convection-diffusion problems

A finite difference method for a time-dependent convection-diffusion problem in one space dimension is constructed using a Shishkin mesh. In two recent papers (Clavero et al. (2005) [2] and Mukherjee and Natesan (2009) [3]), this method has been shown to be convergent, uniformly in the small diffusion parameter, using somewhat elaborate analytical techniques and under a certain mesh restriction. In the present paper, a much simpler argument is used to prove a higher order of convergence (uniformly in the diffusion parameter) than in [2] and [3] and under a slightly less restrictive condition on the mesh.     

Stability improvement of the Euler forward method for initial value problems

The Euler forward method is transformed into a highly stable two-step explicit algorithm by use of a relaxation paramenter α. The stability of the new explicit family can be very high and linearly proportional to the absolute value of α. An adaptive stability approach with a matrix of parameters α is used that strongly increases the effeciency of the method. The one-dimensional finite element solutions for linear and nonlinear transient heat conduction problems show the stability, accuracy, and computational efficiency characteristics of the proposed algorithms.     

A spatial domain decomposition method for parabolic optimal control problems

We present a non-overlapping spatial domain decomposition method for the solution of linear–quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space–time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space–time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space–time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann–Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou.     

A domain decomposition method for a kind of optimization problems

In this paper, we are concerned with the monotone convergence of a multiplicative method for solving a kind of optimization problems. We show that the iterate sequence produced by the method converges to the solution of the problem monotonically.