刚性多滞量积分微分方程的Runge—Kutta方法

本文研究了求解刚性多滞量积分微分方程的Runge-Kutta方法的非线性稳定性和计算有效性.经典Runge—Kutta方法连同复合求积公式和Pouzet求积公式被改造用于求解一类刚性多滞量Volterra型积分微分方程.其分析导出了:在适当条件下,扩展的Runge-Kutta方法是渐近稳定和整体稳定的.此外,数值试验表明所给出的方法是高度有效的.     

一种抑制激波计算中数值振荡现象的双重小波收缩方法

在激波数值计算中,容易出现数值振荡的问题,振荡激烈时会掩盖真实解,为此提出了许多高精度复杂计算格式或采用人工粘性抑制数值振荡.从信号处理的角度,提出双重小波收缩方法,它能自适应提取激波数值振荡解中的真实物理解.先用局部微分求积法求解浅水波方程和理想流体Euler运动方程中的激波问题,发现其数值振荡现象严重,然后采用双重小波收缩方法对其处理,获得了无数值振荡解,它能准确捕捉激波的位置并且保持激波结构.相比于复杂的Riemann(黎曼)求解格式,借助小波收缩方法,可以采用相对简单的计算格式如微分求积法求解激波问题.     

求解二维不可压缩Navier-Stokes方程的混合型微分求积法

微分求积法（ＤＱＭ）能以较少的网格点求得微分方程的高精度数值解,但采用单纯的微分求积法求解二维不可压缩Ｎａｖｉｅｒ＿Ｓｔｏｋｅｓ 方程时,只能对低雷诺数流动获得较好的数值解,当雷诺数较高时会导致数值解不收敛·　为此,提出了一种微分求积法与迎风差分法混合求解二维不可压缩Ｎａｖｉｅｒ＿Ｓｔｏｋｅｓ 方程的预估＿校正数值格式,用伪时间相关算法以较少的网格点获得了较高雷诺数流动的数值解·　作为算例,对１∶１ 和１∶２ 驱动方腔内的流动进行了计算,得到了较好的数值结果·     

时滞积分微分方程的Rosenbrock方法

本文研究时滞积分微分方程的数值方法.通过改造现有常及离散型延迟微分方程的数值方法,并匹配以适当数值求积公式,构造了求解时滞积分微分方程的Rosenbrock方法,导出了其稳定性准则.数值例子阐明了所获方法的计算有效性.     

用“升阶法”求解一类一阶线性微分方程——兼谈逆向思维能力的培养

"升阶法"能够把一类特殊的一阶线性微分方程化为二阶常系数齐次线性微分方程求解,而一般的一阶线性微分方程的求解问题可以转化为二元函数全微分的求积问题.利用"升阶法"和"全微分法"对学生进行逆向思维训练,培养学生的创新思维能力.     

线性常微分方程的全局误差估计和优化求解方法

线性常微分方程初值问题求解在许多应用中起着重要作用.目前,已存在很多的数值方法和求解器用于计算离散网格点上的近似解,但很少有对全局误差(global error)进行估计和优化的方法.本文首先通过将离散数值解插值成为可微函数用来定义方程的残差;再给出残差与近似解的关系定理并推导出全局误差的上界;然后以最小化残差的二范数为目标将方程求解问题转化为优化求解问题;最后通过分析导出矩阵的结构,提出利用共轭梯度法对其进行求解.之后将该方法应用于滤波电路和汽车悬架系统等实际问题.实验分析表明,本文估计方法对线性常微分方程的初值问题的全局误差具有比较好的估计效果,优化求解方法能够在不增加网格点的情形下求解出线性常微分方程在插值解空间中的全局最优解.     

任意变系数微分方程的精确解析法

工程中的许多问题归结为求解任意变系数微分方程的解.本文首次提出精确解析法,用以求解任意变系数微分方程在任意边界条件下的解.文中还给出精确解析法的一般计算格式,得到了一致收敛于精确解及其任意阶导数的解析表达式,并给出收敛性证明.文末给出四个算例,均得到较好的结果,证明了本文理论的正确性.     

浅谈常微分方程中的换元思想

在解决数学问题时,换元思想是转化能力的一种体现,它渗透到数学领域的各个方面,在培养学生解决数学问题能力方面有着非常重要的意义.通过常微分方程参数解的推导以及Riccati方程求解过程,可以充分展示和体会到换元思想方法的妙用.     

机械求积法及其外推算法求解Helmholtz非线性边界积分方程

当Helmholtz微分方程转化为非线性边界积分方程后,可以利用机械求积法求得近似解,此方法具有较高的收敛精度阶O(h3)和较低的计算复杂度.构造机械求积法时,一个非线性方程系统通过离散非线性积分方程得到.此外,每个矩阵元素的值都不需要计算任何奇异积分.根据渐近紧理论和Stepleman定理,整个系统的稳定性和收敛性得到了证明.利用h3-Richardson外推算法,收敛精度阶可以提高到O(h5).为了求解非线性方程组,利用Ostrowski不动点定理研究了Newton的解的收敛性.几个算例从数值上说明了本算法的有效性.     

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.     

A Projected Algebraic Multigrid Method for Linear Complementarity Problems

We present an algebraic version of an iterative multigrid method for obstacle problems, called projected algebraic multigrid (PAMG) here. We show that classical algebraic multigrid algorithms can easily be extended to deal with this kind of problem. This paves the way for efficient multigrid solution of obstacle problems with partial differential equations arising, for example, in financial engineering.     

First passage time for multivariate jump‐diffusion processes in finance and other areas of applications

The first passage time (FPT) problem is an important problem with a wide range of applications in science, engineering, economics, and industry. Mathematically, such a problem can be reduced to estimating the probability of a stochastic process first to reach a boundary level. In most important applications in the financial industry, the FPT problem does not have an analytical solution and the development of efficient numerical methods becomes the only practical avenue for its solution. Most of our examples in this contribution are centered around the evaluation of default correlations in credit risk analysis, where we are concerned with the joint defaults of several correlated firms, the task that is reducible to a FPT problem. This task represents a great challenge for jump‐diffusion processes (JDP). In this contribution, we develop further our previous fast Monte Carlo method in the case of multivariate (and correlated) JDP. This generalization allows us, among other things, to evaluate the default events of several correlated assets based on a set of empirical data. The developed technique is an efficient tool for a number of financial, economic, and business applications, such as credit analysis, barrier option pricing, macroeconomic dynamics, and the evaluation of risk, as well as for a number of other areas of applications in science and engineering, where the FPT problem arises. Copyright © 2008 John Wiley & Sons, Ltd.     

Refined Hyers-Ulam approximation of approximately Jensen type mappings

In 1940 S.M. Ulam proposed the famous Ulam stability problem. In 1941 D.H. Hyers solved this problem for additive mappings subject to the Hyers condition on approximately additive mappings. In this paper we generalize the Hyers result for the Ulam stability problem for Jensen type mappings, by considering approximately Jensen type mappings satisfying conditions weaker than the Hyers condition, in terms of products of powers of norms. This process leads to a refinement of the well-known Hyers-Ulam approximation for the Ulam stability problem. Besides we introduce additive mappings of the first and second form and investigate pertinent stability results for these mappings. Also we introduce approximately Jensen type mappings and prove that these mappings can be exactly Jensen type, respectively. These stability results can be applied in stochastic analysis, financial and actuarial mathematics, as well as in psychology and sociology.     

Modelling credit rating by fuzzy adaptive network

Human judgment plays an important role in the rating of enterprise financial conditions. The recently developed fuzzy adaptive network (FAN), which can handle systems whose behaviour is influenced by human judgment, appears to be ideally suited for the modelling of this credit rating problem. In this paper, FAN is used to model the credit rating of small financial enterprises. To illustrate the approach, the data of the credit rating problem is first represented by the use of fuzzy numbers. Then, the FAN network based on inference rules is constructed. And finally, the network is trained or learned by using the fuzzy number training data. The main advantages of the proposed network are the ability for linguistic representation, linguistic aggregation and the learning ability of the neural network.     

Classification of companies using maximal margin ellipsoidal surfaces

We recently proposed a data mining approach for classifying companies into several groups using ellipsoidal surfaces. This problem can be formulated as a semi-definite programming problem, which can be solved within a practical amount of computation time by using a state-of-the-art semi-definite programming software. It turned out that this method performs better for this application than earlier methods based on linear and general quadratic surfaces. In this paper we will improve the performance of ellipsoidal separation by incorporating the idea of maximal margin hyperplane developed in the field of support vector machine. It will be demonstrated that the new method can very well simulate the rating of a leading rating company of Japan by using up to 18 financial attributes of 363 companies. This paper is expected to provide another evidence of the importance of ellipsoidal separation approach in credit risk analysis.     

Exact methods for large-scale multi-period financial planning problems

A relevant financial planning problem is the periodical rebalance of a portfolio of assets such that the portfolio’s total value exhibits certain characteristics. This problem can be modelled using a transition graph G to represent the future state space evolution of the corresponding economy and mathematically formulated as a linear programming problem. We present two different mathematical formulations of the problem. The first considers explicitly the set of the possible scenarios (scenario-based approach), while the second considers implicitly the whole set of scenarios provided by the graph G (graph-based approach). Unfortunately, for both the formulations the size of the corresponding linear programs can be huge even for simple financial problems. However, the graph-based approach seems to be a more powerful model, since it allows to consider a huge number of scenarios in a very compact formulation. The purpose of this paper is to present both heuristic and exact methods for the solution of large-scale multi-period financial planning problems using the graph-based model. In particular, in this paper we propose lower and upper bounds and three exact methods based on column, row and column/row generation, respectively. Since the methods based on column/row generation exploits simultaneously both the primal and the dual structure of the problem we call it Criss-Cross generation method. Computational results are given to prove the effectiveness of the proposed methods.      

Fast resolution of a single factor Heath–Jarrow–Morton model with stochastic volatility

This paper considers the single factor Heath–Jarrow–Morton model for the interest rate curve with stochastic volatility. Its natural formulation, described in terms of stochastic differential equations, is solved through Monte Carlo simulations, that usually involve rather large computation time, inefficient from a practical (financial) perspective. This model turns to be Markovian in three dimensions and therefore it can be mapped into a 3D partial differential equations problem. We propose an optimized numerical method to solve the 3D PDE model in both low computation time and reasonable accuracy, a fundamental criterion for practical purposes. The spatial and temporal discretizations are performed using finite-difference and Crank–Nicholson schemes respectively, and the computational efficiency is largely increased performing a scale analysis and using Alternating Direction Implicit schemes. Several numerical considerations such as convergence criteria or computation time are analyzed and discussed.     

Fast resolution of a single factor Heath-Jarrow-Morton model with stochastic volatility

This paper considers the single factor Heath-Jarrow-Morton model for the interest rate curve with stochastic volatility. Its natural formulation, described in terms of stochastic differential equations, is solved through Monte Carlo simulations, that usually involve rather large computation time, inefficient from a practical (financial) perspective. This model turns to be Markovian in three dimensions and therefore it can be mapped into a 3D partial differential equations problem. We propose an optimized numerical method to solve the 3D PDE model in both low computation time and reasonable accuracy, a fundamental criterion for practical purposes. The spatial and temporal discretizations are performed using finite-difference and Crank-Nicholson schemes respectively, and the computational efficiency is largely increased performing a scale analysis and using Alternating Direction Implicit schemes. Several numerical considerations such as convergence criteria or computation time are analyzed and discussed.     

Asymptotic behavior of alternative Jensen and Jensen type functional equations

In 1941 D.H. Hyers solved the well-known Ulam stability problem for linear mappings. In 1951 D.G. Bourgin was the second author to treat the Ulam problem for additive mappings. In 1982-2005 we established the Hyers-Ulam stability for the Ulam problem of linear and nonlinear mappings. In 1998 S.-M. Jung and in 2002-2005 the authors of this paper investigated the Hyers-Ulam stability of additive and quadratic mappings on restricted domains. In this paper we improve our bounds and thus our results obtained, in 2003 for Jensen type mappings and establish new theorems about the Ulam stability of additive mappings of the second form on restricted domains. Besides we introduce alternative Jensen type functional equations and investigate pertinent stability results for these alternative equations. Finally, we apply our recent research results to the asymptotic behavior of functional equations of these alternative types. These stability results can be applied in stochastic analysis, financial and actuarial mathematics, as well as in psychology and sociology.