An Adaptive CGNR Algorithm for Solving Large Linear Systems

In this paper, an adaptive algorithm based on the normal equations for solving large nonsymmetric linear systems is presented. The new algorithm is a hybrid method combining polynomial preconditioning with the CGNR method. Residual polynomial is used in the preconditioning to estimate the eigenvalues of the s.p.d. matrix A ^T A, and the residual polynomial is generated from several steps of CGNR by recurrence. The algorithm is adaptive during its implementation. The robustness is maintained, and the iteration convergence is speeded up. A numerical test result is also reported.     

基于自适应参数校正策略求解SDP的Mehrotra型内点算法

最近,Salahi对线性规划提出了一个基于新的自适应参数校正策略的Mehrotra型预估-校正算法,该策略使其在不使用安全策略的情况下,证明了算法的多项式迭代复杂界.本文将这一算法推广到半定规划的情形.通过利用Zhang的对称化技术,得到了算法的多项式迭代复杂界,这与求解线性规划的相应算法有相同的迭代复杂性阶.     

Adaptive polynomial preconditioning for hermitian indefinite linear systems

This paper explores the use of polynomial preconditioned CG methods for hermitian indefinite linear systems,Ax=b. Polynomial preconditioning is attractive for several reasons. First, it is well-suited to vector and/or parallel architectures. It is also easy to employ, requiring only matrix-vector multiplication and vector addition. To obtain an optimum polynomial preconditioner we solve a minimax approximation problem. The preconditioning polynomial,C(), is optimum in that it minimizes a bound on the condition number of the preconditioned matrix,C(A)A. We also characterize the behavior of this minimax polynomial, which makes possible a thorough understanding of the associated CG methods. This characterization is also essential to the development of an adaptive procedure for dynamically determining the optimum polynomial preconditioner. Finally, we demonstrate the effectiveness of polynomial preconditioning in a variety of numerical experiments on a Cray X-MP/48. Our results suggest that high degree (20–50) polynomials are usually best.This research was supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Dept. of Energy, by Lawrence Livermore National Laboratory under contract W-7405-ENG-48.This research was supported in part by the Dept. of Energy and the National Science Foundation under grant DMS 8704169.This research was supported in part by U.S. Dept. of Energy grant DEFG02-87ER25026 and by National Science Foundation grant DMS 8703226.     

Large Matrix Computations on Vector Computers

Preconditionings have proved to be a powerful technique for accelerating the rate of convergence of an iterative method. This paper, which is concerned with the conjugate gradient algorithm for large matrix computations, investigates an approximate polynomial preconditioning strategy. The method is particularly attractive for implementation on vector computers.     

Optimal multigrid preconditioned semi‐monotonic augmented Lagrangians applied to the Stokes problem

We propose an optimal computational complexity algorithm for the solution of quadratic programming problems with equality constraints arising from partial differential equations. The algorithm combines a variant of the semi‐monotonic augmented Lagrangian (SMALE) method with adaptive precision control and a multigrid preconditioning for the Hessian of the cost function and for the inner product on the space of Lagrange variables. The update rule for penalty parameter acts as preconditioning of constraints. The optimality of the algorithm is theoretically proven and confirmed by numerical experiments for the two‐dimensional Stokes problem. Copyright © 2007 John Wiley & Sons, Ltd.     

On the orthogonality of residual polynomials\newline of minimax polynomial preconditioning

Summary. This paper studies polynomials used in polynomial preconditioning for solving linear systems of equations. Optimum preconditioning polynomials are obtained by solving some constrained minimax approximation problems. The resulting residual polynomials are referred to as the de Boor-Rice and Grcar polynomials. It will be shown in this paper that the de Boor-Rice and Grcar polynomials are orthogonal polynomials over several intervals. More specifically, each de Boor-Rice or Grcar polynomial belongs to an orthogonal family, but the orthogonal family varies with the polynomial. This orthogonality property is important, because it enables one to generate the minimax preconditioning polynomials by three-term recursive relations. Some results on the convergence properties of certain preconditioning polynomials are also presented. Received February 1, 1992/Revised version received July 7, 1993     

Exponential behaviour of the Butkovi?-Zimmermann algorithm for solving two-sided linear systems in max-algebra

In [P. Butkovi?, K. Zimmermann, A strongly polynomial algorithm for solving two-sided linear systems in max-algebra, Discrete Applied Mathematics 154 (3) (2006) 437-446] an ingenious algorithm for solving systems of two-sided linear equations in max-algebra was given and claimed to be strongly polynomial. However, in this note we give a sequence of examples showing exponential behaviour of the algorithm. We conclude that the problem of finding a strongly polynomial algorithm is still open.     

棋盘游戏与有限域上多项式分解算法

将有限域F_2上多项式分解问题转化为一种对应的棋盘游戏,利用后者的性质设计了一个F_2上m+n-2次多项式f(x)分解为一个m-1次多项式与一个n-1次多项式的判断、分解算法,并对算法的复杂度进行了分析.算法的一个优势是,如果f(x)不能按要求分解,也可以找到一个与f(x)相近(这里指系数相异项较少)的多项式的分解.     

AN ADAPTIVE TRUST-REGION METHOD FOR GENERALIZED EIGENVALUES OF SYMMETRIC TENSORS

For symmetric tensors,computing generalized eigenvalues is equivalent to a homogenous polynomial optimization over the unit sphere.In this paper,we present an adaptive trustregion method for generalized eigenvalues of symmetric tensors.One of the features is that the trust-region radius is automatically updated by the adaptive technique to improve the algorithm performance.The other one is that a projection scheme is used to ensure the feasibility of all iteratives.Global convergence and local quadratic convergence of our algorithm are established,respectively.The preliminary numerical results show the efficiency of the proposed algorithm.     

Unconstrained and constrained global optimization of polynomial functions in one variable

In Floudas and Visweswaran (1990), a new global optimization algorithm (GOP) was proposed for solving constrained nonconvex problems involving quadratic and polynomial functions in the objective function and/or constraints. In this paper, the application of this algorithm to the special case of polynomial functions of one variable is discussed. The special nature of polynomial functions enables considerable simplification of the GOP algorithm. The primal problem is shown to reduce to a simple function evaluation, while the relaxed dual problem is equivalent to the simultaneous solution of two linear equations in two variables. In addition, the one-to-one correspondence between the x and y variables in the problem enables the iterative improvement of the bounds used in the relaxed dual problem. The simplified approach is illustrated through a simple example that shows the significant improvement in the underestimating function obtained from the application of the modified algorithm. The application of the algorithm to several unconstrained and constrained polynomial function problems is demonstrated.     

对一类overlay层组播路由问题的模型的改进

根据overlay层虚拟网图的特点，本给出了一类overlay层组播路由问题的数学模型的改进，及相应的一种启发式算法，即MMD算法，并分析了该算法的性质，证明了它是一个多项式时间算法。     

与有限域上置换多项式交换的多项式的算法

An efficient algorithm for determining polynomials that commute with apermutation polynomial over a finite field is given. The complexity of the algorithm isdiscussed and examples of applying the algorithm are also provided.     

1. Introduction

1.IntroductionLetpbeprime,nbeapositiveinteger,andFa=GF(P")betheGajoisfieldwithq=p"elements.Apolynomialf(x)overFiissaidtobeapermutationpolynomialiff(x),consideredasafunctionfromFitoFi,isabijection.Givenapermutationpolynomialf(x)overFi,weconsidertheproblemofdeterminingailpolynomials(orallpermutationpolynomials)g(x)overFisuchthatf(g(x))=g(f(x)),forallzinFi,i.e.,gcommuteswithfasfunctionsonFi.In[5],Wellscharacterizedthepolynomialsthatcommutewithatranslationf(x)=x a.In[4],Mullencharacterizedth…     

Accurate evaluation of a polynomial in Chebyshev form

This paper presents a compensated algorithm to accurately evaluate a polynomial expressed in Chebyshev basis of the first and second kind with floating-point coefficients. The principle is to apply error-free transformations to improve the traditional Clenshaw algorithm. The new algorithm is as accurate as the Clenshaw algorithm performed in twice the working precision. Forward error analysis and numerical experiments illustrate the accuracy and properties of the proposed algorithm.     

High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs

We consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solutions to partial differential equations (PDEs) depending on a possibly large number of parameters which result from the application of generalised polynomial chaos discretisations to random and stochastic PDEs. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space is enriched at different stages of the computation. In this paper, we study an interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving. This construction is based on the standard principle of tensorisation of a one-dimensional interpolation scheme and sparsification. We derive bounds on the Lebesgue constants for this interpolation process in terms of their univariate counterpart. For a class of model elliptic parametric PDE’s, we have shown in Chkifa et al. (Modél. Math. Anal. Numér. 47(1):253–280, 2013) that certain polynomial approximations based on Taylor expansions converge in terms of the polynomial dimension with an algebraic rate that is robust with respect to the parametric dimension. We show that this rate is preserved when using our interpolation algorithm. We also propose a greedy algorithm for the adaptive selection of the polynomial spaces based on our interpolation scheme, and illustrate its performance both on scalar valued functions and on parametric elliptic PDE’s.     

Efficient numerical methods in non-uniform sampling theory

Summary. We present a new ``second generation" reconstruction algorithm for irregular sampling, i\@.e\@. for the problem of recovering a band-limited function from its non-uniformly sampled values. The efficient new method is a combination of the adaptive weights method which was developed by the two first named authors and the method of conjugate gradients for the solution of positive definite linear systems. The choice of "adaptive weights" can be seen as a simple but very efficient method of preconditioning. Further substantial acceleration is achieved by utilizing the Toeplitz-type structure of the system matrix. This new algorithm can handle problems of much larger dimension and condition number than have been accessible so far. Furthermore, if some gaps between samples are large, then the algorithm can still be used as a very efficient extrapolation method across the gaps. Received December 22, 1993     

Componentwise adaptation for high dimensional MCMC

Summary  We introduce a new adaptive MCMC algorithm, based on the traditional single component Metropolis-Hastings algorithm and on our earlier adaptive Metropolis algorithm (AM). In the new algorithm the adaption is performed component by component. The chain is no more Markovian, but it remains ergodic. The algorithm is demonstrated to work well in varying test cases up to 1000 dimensions.     

A successful algorithm for solving directed Hamiltonian path problems

In this paper we present a graph-theoretic polynomial algorithm which has positive probability of finding a Hamiltonian path in a given graph, if there is one; if the algorithm fails, it can be rerun with a randomly chosen starting solution, and there is again a positive probability it will find an answer. If there is no Hamiltonian path, the algorithm will always terminate with failure. We call this a Successful Algorithm because it has high (close to 1) empirical probability of success and it works in polynomial time. Some basic theoretical results concerning spanning arborescences of a graph are given. The concept of a ramification index is defined and it is shown that ramification index of a Hamiltonian path is zero. The algorithm starts with finding any spanning arborescence and by suitable pivots it endeavors to reduce the ramification index to zero. Probabilistic properties of the algorithm are discussed. Computational experience with graphs up to 30 000 nodes is included.     

满足路径约束的最优路算法

满足路径约束的最优路问题已被证明是NP-hard问题。本针对源点到宿点满足两个QoS(服务质量)度量的路由问题，给出一种保证时延的最小费用路由启发式算法。这个算法的优点是计算较简单、占用内存小、时间短。算法的复杂度是多项式的，表明算法是有效的。     

预条件广义极小残余新算法

研究Krylov子空间广义极小残余算法(GMRES(m))的基本理论，给出GMRES(m)算法透代求解所满足的代数方程组．深入探讨算法的收敛性与方程组系数矩阵的密切关系，提出一种改进GMRES(m)算法收敛性的新的预条件方法，并作出相关论证．