一种基于LVI求解二次规划问题的数值算法

给出并研究了一种数值算法(简称94LVI算法),用于求解带等式和双端约束的二次规划问题. 这类带约束的二次规划问题首先被转换为线性变分不等式问题,该问题等价于分段线性投影等式.接着使用94LVI算法求解上述分段线性投影等式,从而得到QP问题的最优解. 进一步给出了94LVI算法的全局收敛性证明. 94LVI算法与经典有效集算法的对比实验结果证实了给出的94LVI算法在求解二次规划问题上的高效性与优越性.     

Fixed points of C2 maps

S.N. Chow and J.A. Yorke have proposed in abstract terms an algorithm for computing fixed points of C² maps that is globally convergent with probability one. A numerical implementation of that algorithm is presented here, where careful attention has been paid to computational efficiency, accuracy, and robustness. Convergence proofs for the numerical algorithm require differential geometry, and are given elsewhere. FORTRAN subroutines are given and explained in detail, and some typical numerical results are presented. It is shown how to modify the subroutines to compute zeros and handle some large sparse problems.     

A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions

In this paper, we propose a new numerical algorithm for solving linear and non linear fractional differential equations based on our newly constructed integer order and fractional order generalized hat functions operational matrices of integration. The linear and nonlinear fractional order differential equations are transformed into a system of algebraic equations by these matrices and these algebraic equations are solved through known computational methods. Further some numerical examples are given to illustrate and establish the accuracy and reliability of the proposed algorithm. The results obtained, using the scheme presented here, are in full agreement with the analytical solutions and numerical results presented elsewhere.     

API stepsize control for the numerical solution of ordinary differential equations

A control-theoretic approach is used to design a new automatic stepsize control algorithm for the numerical integration of ODE's. The new control algorithm is more robust at little extra expense. Its improved performance is particularly evident when the stepsize is limited by numerical stability. Comparative numerical tests are presented.     

一类单边约束问题的数值方法

本文分别基于原始变分形式与对偶混合变分形式，对一类单边约束问题进行了数值求解，提出了求解离散对偶混合变分问题的Uzawa型算法，并用数值例子验证了算法的有效性．     

AN EFFECT ITERATION ALGORITHM FOR NUMERICAL SOLUTION OF DISCRETE HAMILTON-JACOBIBELLMAN EQUATIONS

§1Introduction ConsidertheHamilton-Jacobi-Bellmanequation max1≤v≤m[A(v)u(x)-f(v)(x)]=0,x∈Ω(1.1)withtheboundarycondition u(x)=0,x∈Ω(1.2)whereΩisabounded,smoothdomaininEuclideanspaceRd,d∈N;f(v)(x)aregiven functionsfromC2(Ω);A(v)aresecond-orderuniformlyellipticoperatorsoftheform A(v)=-d i,j=1a(v)ij2xixj+di=1b(v)ixi+c(v).(1.3)Intheaboveexpression(1.3)therearecoefficientsa(v)ij,b(v)i,c(v)∈C2(Ω)satisfying,forall1≤v≤m,a(v)ij(x)=a(v)ji(x),1≤i,j≤d,c(v)≥c0≥0,x∈Ω,a…     

On One method for the numerical solution of a system of parabolic equations

An algorithm for the numerical solution of a system of two parabolic equations of a special form has been presented, and its stability has been proved. Estimates of the numerical solutions obtained have been made. Conditions guaranteeing the applicability of the sweep method are determined, and the time step at which the iterative method has the inner convergence has been estimated. An example of the application of the algorithm in soil science has been presented.     

An algorithm for global optimization of Lipschitz continuous functions

An algorithm is presented which locates the global minimum or maximum of a function satisfying a Lipschitz condition. The algorithm uses lower bound functions defined on a partitioned domain to generate a sequence of lower bounds for the global minimum. Convergence is proved, and some numerical results are presented.     

An alternative to the Bathe algorithm

This paper presents a new composite sub-steps algorithm for solving reliable numerical responses in structural dynamics. The newly developed algorithm is a two sub-steps, second-order accurate and unconditionally stable implicit algorithm with the same numerical properties as the Bathe algorithm. The detailed analysis of the stability and numerical accuracy is presented for the new algorithm, which shows that its numerical characteristics are identical to those of the Bathe algorithm. Hence, the new sub-steps scheme could be considered as an alternative to the Bathe algorithm. Meanwhile, the new algorithm possesses the following properties: (a) it produces the same accurate solutions as the Bathe algorithm for solving linear and nonlinear problems; (b) it does not involve any artificial parameters and additional variables, such as the Lagrange multipliers; (c) The identical effective stiffness matrices can be obtained inside two sub-steps; (d) it is a self-starting algorithm. Some numerical experiments are given to show the superiority of the new algorithm and the Bathe algorithm over the dissipative CH-α algorithm and the non-dissipative trapezoidal rule.     

Rehabilitation of the Gauss-Jordan algorithm

Summary In this paper a Gauss-Jordan algorithm with column interchanges is presented and analysed. We show that, in contrast with Gaussian elimination, the Gauss-Jordan algorithm has essentially differing properties when using column interchanges instead of row interchanges for improving the numerical stability. For solutions obtained by Gauss-Jordan with column interchanges, a more satisfactory bound for the residual norm can be given. The analysis gives theoretical evidence that the algorithm yields numerical solutions as good as those obtained by Gaussian elimination and that, in most practical situations, the residuals are equally small. This is confirmed by numerical experiments. Moreover, timing experiments on a Cyber 205 vector computer show that the algorithm presented has good vectorisation properties.     

求解非线性互补问题的一类光滑Broyden-like方法

非线性互补问题可以等价地转换为光滑方程组来求解. 基于一种新的非单调线搜索准则, 提出了求解非线性互补问题等价光滑方程组的一类新的非单调光滑 Broyden-like 算法.在适当的假设条件下, 证明了该算法的全局收敛性与局部超线性收敛性. 数值实验表明所提出的算法是有效的.     

一类带约束min-max-min问题的区间算法

构造了求解一类带不等式约束的min-max-min问题的区间算法,其中目标函数和约束函数都是一阶连续可微函数,证明了方法的收敛性,给出了数值算例.该方法可以同时求出问题的最优值和全部全局最优解,是有效和可靠的.     

自适应迎风格式求解奇异摄动两点边值问题的高精度算法(二)

0引言 考虑与文[1]相同的奇异摄动两点边值问题的数值解法: Tu(x):=-εu″(x)-p(x)u′(x)=f(x),x∈(0,1); (1) u(0)=0,u(1)=1. (2) 其中ε是一个常数,0<ε≤1,f∈C2[0,1].假定P∈C3[0,1]且存在常数β和-β使得0<β≤p(x)≤-β,|p′(x)|≤-β,(V)x∈[0,1] (3) 成立.     

Steepest-descent failure analysis

An extensive failure analysis of the steepest-descent optimization algorithm has been made. Each of the ways in which the algorithm can fail is discussed in terms of both the mathematical and numerical manifestations of a failure and the information which each type of failure provides about the formulation of the physical problem. Numerical tests for each of the various types of failure are described; several faulty problem formulations are presented, each of which illustrates a particular type of failure. A table is presented in which all failure modes are summarized and the corresponding numerical tests are exhibited.     

Estimates for numerical approximations of rank one convex envelopes

Summary. We present a convergence analysis of an algorithm for the numerical computation of the rank-one convex envelope of a function . A rate of convergence for the scheme is established, and numerical experiments are presented to illustrate the analytical results and applications of the algorithm. Received February 1, 1999 / Published online April 20, 2000     

A new PEC algorithm for the numerical solution of ordinary differential equations

Significant time reduction in obtaining numerical solutions of ordinary differential equations for which function evaluations are time consuming can be obtained with PEC methods as compared to PECE methods. In this report we present two PEC methods: a fourth-order algorithm for which stability characteristics and numerical examples are presented, and a second-order algorithm which is just mentioned. It is believed that PEC methods represent a useful addition to the library of solution techniques.     

Numerical simulation of the process of nonequilibrium counterflow capillary imbibition

The convergence of the Galerkin method of solving the Cauchy–Dirichlet problem for the Barenbratt–Gilman equation is studied. On the basis of theoretical results, a numerical algorithm for this problem is developed. Results of a numerical experiment are presented.     

A globally convergent algorithm for computing fixed points of C2 maps

Chow, Mallet-Paret, and Yorke have recently proposed in abstract terms an algorithm for computing Brouwer fixed points of C² maps. A numerical implementation of that algorithm is presented here. Careful attention has been paid to computational efficiency, accuracy, and robustness. Some convergence theorems and results of numerical tests are also included.     

Multiscale computational method for nonstationary integrated heat transfer problem in periodic porous materials

This article discusses multiscale analysis and numerical algorithm for the nonstationary integrated heat transfer problem with rapidly oscillating coefficients. The multiscale asymptotic expansion of the solution for this kind of problems is presented first. Then, error estimates of the multiscale approximate solution are derived, and a numerical algorithm based on the multiscale method for temperature field is introduced. Finally, using some numerical models, we verify the validity and relevancy of the proposed algorithm. The numerical results show that the algorithm is effective to predict the heat transfer performance of porous materials, and support the convergence theorem reported in this article. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 510–530, 2016     

Super-fast validated solution of linear systems

Validated solution of a problem means to compute error bounds for a solution in finite precision. This includes the proof of existence of a solution. The computed error bounds are to be correct including all possible effects of rounding errors. The fastest known validation algorithm for the solution of a system of linear equations requires twice the computing time of a standard (purely) numerical algorithm. In this paper we present a super-fast validation algorithm for linear systems with symmetric positive definite matrix. This means that the entire computing time for the validation algorithm including computation of an approximated solution is the same as for a standard numerical algorithm. Numerical results are presented.