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Evaluating accuracy of Hessian-based predictor-corrector integrators

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Journal of Central South University 2017年第7期24卷 1696-1702页

作者：陆绍飞伍恒 Eduardo Colmenares 刘绪崇 College of Computer Science and Electronic Engineering Hunan UniversityChangsha 410082China Department of Computer Science Texas Tech UniversityLubbock 79409USA Department of Computer Science Midwestern State UniversityWichita Falls 76308USA Hunan Provincial Key Laboratory of Network Investigational Technology Hunan Police AcademyChangsha 410138China

Direct dynamics simulations are a useful and general approach for studying the atomistic properties of complex chemical systems because they do not require fitting an analytic potential energy function.Hessian-based predictor-corrector integrators are a widely used approach for calculating the trajectories of moving atoms in direct dynamics simulations.We employ a monodromy matrix to propose a tool for evaluating the accuracy of integrators in the trajectory calculation.We choose a general velocity Verlet as a different object.We also simulate molecular with hydrogen(CO_2) and molecular with hydrogen(H_2O) motions.Comparing the eigenvalues of monodromy matrix,many simulations show that Hessian-based predictor-corrector integrators perform well for Hessian updates and non-Hessian updates.Hessian-based predictor-corrector integrator with Hessian update has a strong performance in the H_2O simulations.Hessian-based predictor-corrector integrator with Hessian update has a strong performance when the integrating step of the velocity Verlet approach is tripled for the predicting step.In the CO_2 simulations,a strong performance occurs when the integrating step is a multiple of five.

关键词： monodromy matrix eigenvalue Hessian-based predictor-corrector velocity Verlet

A Split-Step predictor-corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions

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Communications on Applied Mathematics and Computation 2019年第4期1卷 525-544页

作者： Kamran Kazmi Abdul Khaliq Department of Mathematics University of Wisconsin OshkoshOshkoshWI 54901USA Department of Mathematical Sciences and Center for Computational Sciences Middle Tennessee State UniversityMurfreesboroTN 37132USA

A split-step second-order predictor-corrector method for space-fractional reaction-diffusion equations with nonhomogeneous boundary conditions is presented and analyzed for the stability and convergence.The matrix transfer technique is used for spatial discretization of the problem.The method is shown to be unconditionally stable and second-order convergent.Numerical experiments are performed to confirm the stability and secondorder convergence of the method.The split-step predictor-corrector method is also compared with an IMEX predictor-corrector method which is found to incur oscillatory behavior for some time steps.Our method is seen to produce reliable and oscillatioresults for any time step when implemented on numerical examples with nonsmooth initial data.We also present a priori reliability constraint for the IMEX predictor-corrector method to avoid unwanted oscillations and show its validity numerically.

关键词： Fractional Laplacian Space-fractional reaction diffusion equations Non-homogeneous boundary conditions Matrix transfer technique predictor-corrector method

A POLYNOMIAL predictor-corrector INTERIOR-POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING

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Acta Mathematica Scientia 2006年第2期26卷 265-270页

作者：余谦黄崇超江燕 Institute of Systems Engineering of Wuhan University Wuhan 430072 China School of Mathematics and Statistics Wuhan University Wuhan 430072China

This article presents a polynomial predictor-corrector interior-point algorithm for convex quadratic programming based on a modified predictor-corrector interior-point algorithm. In this algorithm, there is only one corrector step after each predictor step, where Step 2 is a predictor step and Step 4 is a corrector step in the algorithm. In the algorithm, the predictor step decreases the dual gap as much as possible in a wider neighborhood of the central path and the corrector step draws iteration points back to a narrower neighborhood and make a reduction for the dual gap. It is shown that the algorithm has O（√nL） iteration complexity which is the best result for convex quadratic programming so far.

关键词： Convex quadratic programming predictor-corrector interior-point algorithm

A hybrid method of fractional steps with predictor-corrector difference-pseudospectrum for numerical solution of the convection-dominated flow problems

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Science China Mathematics 1995年第7期38卷 838-852页

作者：王喜君张法高吴江航 Department of Mechanics Peking University. Beijing 100871 China Institute of Oceanology Chinese Academy of Sciences. Qingdao 266071 China

A fractional-step method of predictor-corrector difference-pseudospectrum with unconditional L2-stability and exponential convergence is presented. The stability and convergence of this method is strictly proved mathematically for a nonlinear convection-dominated flow. The error estimation is given and the superiority of this method is verified by numerical test.

关键词： convective domination fractional-step method predictor-corrector method pseudospectral method.

A predictor-corrector INTERIOR-POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING

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Numerical Mathematics A Journal of Chinese Universities(English Series) 2002年第1期11卷 52-62页

作者： Liang Ximing(梁昔明) Qian Jixin(钱积新) College of Information Science ＆ Engineering Central South University Changsha 410083 Institute of Systems Engineering National Lab of Industrial Control Technology Zhejiang University Hangzhou 310027

The simplified Newton method, at the expense of fast convergence, reduces the work required by Newton method by reusing the initial Jacobian matrix. The composite Newton method attempts to balance the trade-off between expense and fast convergence by composing one Newton step with one simplified Newton step. Recently, Mehrotra suggested a predictor-corrector variant of primal-dual interior point method for linear programming. It is currently the interiorpoint method of the choice for linear programming. In this work we propose a predictor-corrector interior-point algorithm for convex quadratic programming. It is proved that the algorithm is equivalent to a level-1 perturbed composite Newton method. Computations in the algorithm do not require that the initial primal and dual points be feasible. Numerical experiments are made.

关键词： convex quadratic programming, interior-point methods, predictor-corrector algorithms, numerical experiments.

Modified Efficient Families of Two and Three-Step predictor-corrector Iterative Methods for Solving Nonlinear Equations

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Applied Mathematics 2010年第3期1卷 153-158页

作者： Sanjeev Kumar Vinay Kanwar Sukhjit Singh 不详

In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting near the root. The main advantage of our methods is that they perform better and moreover, have the same efficiency indices as that of existing multipoint iterative methods. Furthermore, the convergence analysis of the new methods is discussed and several examples are given to illustrate their efficiency.

关键词： Nonlinear Equations Iterative Methods Multipoint Iterative Methods Newton’s Method Traub-Ostrowski’s Method predictor-corrector Methods Order of Convergence

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一类凸规划的多项式预估校正内点法

高等学校计算数学学报 2005年第3期27卷 5-13页

作者：余谦黄崇超王先甲武汉大学数学与统计学院武汉大学系统工程研究所武汉大学系统工程研究所武汉 430072

1引言1990年由Mehrotra对线性规划问题提出了一个称为预估校正的方法,并在1992年给出了其数值算法.1993年Mizuno,Todd和Y.Ye.给出了改进的预估校正内点法,使得一个预估步后只跟一个校正步.1994年F.A.Potra给出了不可行预估校正内点法,... 详细信息

1引言1990年由Mehrotra对线性规划问题提出了一个称为预估校正的方法,并在1992年给出了其数值算法.1993年Mizuno,Todd和Y.Ye.给出了改进的预估校正内点法,使得一个预估步后只跟一个校正步.1994年F.A.Potra给出了不可行预估校正内点法,使得可以从一个不可行的初始点开始算法的迭代,并证明了其为二次收敛.实际上预估校正内点法等价于应用简化一步牛顿法与一步牛顿法的合成方法求解该

关键词： convex programming predictor-corrector interior-point algorithm.

Simulation of Time-Dependent Schrodinger Equation in the Position and Momentum Domains

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American Journal of Computational Mathematics 2015年第3期5卷 291-303页

作者： Michael Jennings Chicago IL USA

The paper outlines the development of a new, spectral method of simulating the Schrödinger equation in the momentum domain. The kinetic energy operator is approximated in the momentum domain by exploiting the derivative property of the Fourier transform. These results are compared to a position-domain simulation generated by a fourth-order, centered-space, finite-difference formula. The time derivative is approximated by a four-step predictor-corrector in both domains. Free-particle and square-well simulations of the time-dependent Schrödinger equation are run in both domains to demonstrate agreement between the new, spectral methods and established, finite-difference methods. The spectral methods are shown to be accurate and precise.

关键词： Schrodinger Equation predictor-corrector Fourier Transform Momentum Domain

Improving Wilson-θ and Newmark-β Methods for Quasi-Periodic Solutions of Nonlinear Dynamical Systems

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Journal of Applied Mathematics and Physics 2018年第8期6卷 1625-1635页

作者： G. Liu Z. R. Lv Y. M. Chen Department of Applied Mechanics and Engineering School of Aeronautics and Astronautics Sun Yat-sen University Guangzhou China

Quasi-periodic responses can appear in a wide variety of nonlinear dynamical systems. To the best of our knowledge, it has been a tough job for years to solve quasi-periodic solutions, even by numerical algorithms. Here in this paper, we will present effective and accurate algorithms for quasi-periodic solutions by improving Wilson-θ and Newmark-β methods, respectively. In both the two methods, routinely, the considered equations are rearranged in the form of incremental equilibrium equations with the coefficient matrixes being updated in each time step. In this study, the two methods are improved via a predictor-corrector algorithm without updating the coefficient matrixes, in which the predicted solution at one time point can be corrected to the true one at the next. Numerical examples show that, both the improved Wilson-θ and Newmark-β methods can provide much more accurate quasi-periodic solutions with a smaller amount of computational resources. With a simple way to adjust the convergence of the iterations, the improved methods can even solve some quasi-periodic systems effectively, for which the original methods cease to be valid.

关键词： Wilson-θ Method Newmark-β Method Quasi-Periodic Solution predictor-corrector Algorithm