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ON CONVERGENCE PROPERTY OF THE LANCZOS METHOD FOR SOLVING A COMPLEX SHIFTED HERMITIAN LINEAR SYSTEM

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Journal of Computational Mathematics 2013年第3期31卷 326-334页

作者： guiding gu Department of Applied Mathematics Shanghai University of Finance and Economics Shanghai 200433 China

We discuss the convergence property of the Lanczos method for solving a complex shifted Hermitian linear system （αI ＋ H）x = f. By showing the colinear coefficient of two system＇s residuals, our convergence analysis reveals that under the condition Re（α）＋ λmin（H）〉 0, the method converges faster than that for the real shifted Hermitian linear system （Re（α）I ＋ H）x = f. Numerical experiments verify such convergence property.

关键词： Hermitian matrix Complex shifted linear system Lanczos method.

GENERAL FULL IMPLICIT STRONG TAYLOR APPROXIMATIONS FOR STIFF STOCHASTIC DIFFERENTIAL EQUATIONS

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Journal of Computational Mathematics 2022年第4期40卷 541-569页

作者： Kai Liu guiding gu School of Mathematics Shanghai University of Finance and EconomicsShanghai 200433China College of Science Hunan University of Technology and BusinessChangsha 410205China

In this paper,we present the backward stochastic Taylor expansions for a Ito process,including backward Ito-Taylor expansions and backward Stratonovich-Taylor *** construct the general full implicit strong Taylor approximations(including Ito-Taylor and Stratonovich-Taylor schemes)with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations(SSDE)by employing truncations of backward stochastic Taylor *** demonstrate that these schemes will converge strongly with corresponding order 1,2,3,....Mean-square stability has been investigated for full implicit strong Stratonovich-Taylor scheme with order 2,and it has larger meansquare stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order *** can improve the stability of simulations considerably without too much additional computational effort by using our full implicit *** full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift and the diffusion *** numerical experiment show these points.

关键词： Stiff stochastic differential equations Approximations Backward Stochastic Taylor expansions Full implicit Taylor methods

IMPROVED PMHSS ITERATION METHODS FOR COMPLEX SYMMETRIC LINEAR SYSTEMS

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Journal of Computational Mathematics 2019年第2期37卷 278-296页

作者： Kai Liu guiding gu School of Mathematics Shanghai University of Finance and Economics Shanghai 200433 China

Based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method for the complex symmetrie linear system, two improved iterative methods, namely, the modified PMHSS (MPMHSS) method and the double modified PMHSS (DMPMHSS) method, are proposed in this paper. The spectra] radii of the iteration matrices of two methods are given. We show that by choosing an appropriate parameter, MPMHSS could speed up the convergence on PMHSS. The DMPMHSS method is a four-step alternating iteration that is developed upon the two-step alternating iteration of MPMHSS. We discuss the choice of the parameters and establish the convergence of DMPMHSS. In particular, we give an analysis of the spectral radius of PMHSS and DMPMHSS at the parameter free situation, and we show that DMPMHSS converges faster than PMHSS in most cases. Our numerical experiments show these points.

关键词： Complex symmetrie linear system PMHSS

Restarted FOM Augmented with Ritz Vectors for Shifted Linear Systems

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Numerical Mathematics A Journal of Chinese Universities(English Series) 2006年第1期15卷 40-49页

作者： Zhanwen Li guiding gu Department of Mathematics Hebei University of Science and Technology Shift-azhuang 050018 China. Department of Applied Mathematics Shanghai University of Finance and Eco-nomics Shanghai 200433 China.

The restarted FOM method presented by Simoncini[7]according to the natural collinearity of all residuals is an efficient method for solving shifted systems,which generates the same Krylov subspace when the shifts are handled ***,restarting slows down the *** present a practical method for solving the shifted systems by adding some Ritz vectors into the Krylov subspace to form an augmented Krylov subspace. Numerical experiments illustrate that the augmented FOM approach(restarted version)can converge more quickly than the restarted FOM method.

关键词： Krylov空间 FOM 重新启动移动线性系统质量因数

成组BFGS修正的紧凑形式与成组记忆修正算法

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计算数学 1999年第4期21卷 417-428页

作者：顾桂定王德人复旦大学数学所上海大学数学系

In 1994, O’leary and Yeremin extended the quasi-Newton method for minimizing a collection of functions with a common Hessian matrix to the block version,and discussed some algebraic properties of this block quasi-Newton method. In thispaper, we derive compact representations of the block BFGS’s updating *** representations allow us to efficiently implement limited memory methods,e.g., the limited memory BFGS method, for minimizing a collection of functionswith a common Hessian matrix. The method relieves the requirement for the storage counts and has the savings in the operation counts, in particular, for large scaleproblems. The numerical experiments for the multiple unconstrained optimizationproblems show that the method works efficiently. Compared with O’Leary’s multiple version of BFGS method, our multiple version of the limited memory BFGSmethod is more efficient in the total operation counts and the storage counts.

关键词： BFGS算法成组型算法有记忆算法拟牛顿法

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块交替拟Newton法与超定方程组求解

上海大学学报（自然科学版） 1996年第1期2卷 6-11页

作者：顾桂定上海大学理学院数学系

本文推广了交替拟Ｎｅｗｔｏｎ方法至块的形式，并将它应用于成组超定方程组的求解，证明了对于给出的ｐ组ｍ×ｎ的线性超定方程组，方法具有至多（ｍ＋１）／Ｐ步迭代的有限终止性．

关键词：超定方程组有限终止性块交替拟牛顿法

随机波动率模型下基于精确模拟算法的期权计算理论

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同济大学学报（自然科学版） 2017年第10期45卷 1539-1548页

作者：马俊美杨宇婷顾桂定徐承龙上海财经大学数学学院上海200433 应用数学福建省高校重点实验室(莆田学院) 福建莆田351100 上海市金融信息技术研究重点实验室上海200433

基于两类随机波动率模型研究了欧式期权的价格和敏感性估计问题.在Broadie和Kaya的精确模拟算法基础上,讨论了舍取抽样技术在精确模拟算法中的有效应用.在此基础上研究条件蒙特卡罗、对偶变量技术等方差减小技术在欧式期权定价和敏感性G... 详细信息

基于两类随机波动率模型研究了欧式期权的价格和敏感性估计问题.在Broadie和Kaya的精确模拟算法基础上,讨论了舍取抽样技术在精确模拟算法中的有效应用.在此基础上研究条件蒙特卡罗、对偶变量技术等方差减小技术在欧式期权定价和敏感性Greeks计算中的加速问题.数值结果表明,相比欧拉离散和原始的蒙特卡罗模拟算法,基于精确模拟算法的条件蒙特卡罗加速技术能得到无偏且方差更小的估计值,具有较好的误差减小效果.该算法可以很方便地解决其他更加复杂的金融产品的计算问题,如障碍期权的定价和敏感性估计问题、篮子期权的计算问题等.

关键词：随机波动率精确模拟加速条件蒙特卡罗 Greeks