Adaptive solution of infinite linear systems by Krylov subspace methods

In this paper we consider the problem of approximating the solution of infinite linear systems, finitely expressed by a sparse coefficient matrix. We analyse an algorithm based on Krylov subspace methods embedded in an adaptive enlargement scheme. The management of the algorithm is not trivial, due to the irregular convergence behaviour frequently displayed by Krylov subspace methods for nonsymmetric systems. Numerical experiments, carried out on several test problems, indicate that the more robust methods, such as GMRES and QMR, embedded in the adaptive enlargement scheme, exhibit good performances.     

Comparison of Krylov subspace methods on the PageRank problem

PageRank algorithm plays a very important role in search engine technology and consists in the computation of the eigenvector corresponding to the eigenvalue one of a matrix whose size is now in the billions. The problem incorporates a parameter that determines the difficulty of the problem. In this paper, the effectiveness of stationary and nonstationary methods are compared on some portion of real web matrices for different choices of . We see that stationary methods are very reliable and more competitive when the problem is well conditioned, that is for small values of . However, for large values of the parameter the problem becomes more difficult and methods such as preconditioned BiCGStab or restarted preconditioned GMRES become competitive with stationary methods in terms of Mflops count as well as in number of iterations necessary to reach convergence.     

一类多元重复测量模型参数的似然比检验及其功效分析

在多元重复测量试验模型下,当受试对象观测矩阵的协方差矩阵∑为等方差等协方差结构时,给出了参数的似然比检验统计量.给出该检验在原假设下的渐近零分布和在备择假设下的渐近非零分布,并就检验的功效进行了分析.     

On RD‐rational Krylov approximations to the core‐functions of exponential integrators

The paper deals with the application of the restricted‐denominator rational Krylov method, recently discussed in (BIT 2004; 44 (3):595–615; SIAM J. Sci. Comput. 2005; 27 :1438–1457), to the computation of the action of the so‐called φ‐functions, which play a fundamental role in several modern exponential integrators. The analysis here presented is devoted in particular to the construction of error estimates of easy practical use. Copyright © 2007 John Wiley & Sons, Ltd.     

BiCGstab(l) and other hybrid Bi-CG methods

It is well-known that Bi-CG can be adapted so that the operations withA ^T can be avoided, and hybrid methods can be constructed in which it is attempted to further improve the convergence behaviour. Examples of this are CGS, Bi-CGSTAB, and the more general BiCGstab(l) method. In this paper it is shown that BiCGstab(l) can be implemented in different ways. Each of the suggested approaches has its own advantages and disadvantages. Our implementations allow for combinations of Bi-CG with arbitrary polynomial methods. The choice for a specific implementation can also be made for reasons of numerical stability. This aspect receives much attention. Various effects have been illustrated by numerical examples.     

AVERAGE σ-K WIDTH OF CLASS OF Lp（R^n） IN Lq（R^n）

ＡＶＥＲＡＧＥ　σ－Ｋ　ＷＩＤＴＨ　ＯＦ　ＣＬＡＳＳ　ＯＦ　Ｌ_ｐ（Ｒ~ｎ）ＩＮ　Ｌ_ｑ（Ｒ~ｎ）ＡＶＥＲＡＧＥσ－ＫＷＩＤＴＨＯＦＣＬＡＳＳＯＦＬ_ｐ（Ｒ~ｎ）ＩＮＬ_ｑ（Ｒ~ｎ）￥ＬＩＵＹＯＮＧＰＩＮＧＡｂｓｔｒａｃｔ：Ｔｈｅａｖｅｒａｇｅσ－Ｋｗｉｄｔ...     

Multigrid preconditioned Krylov subspace methods for three-dimensional numerical solutions of the incompressible Navier–Stokes equations

We consider numerical methods for the incompressible Reynolds averaged Navier–Stokes equations discretized by finite difference techniques on non-staggered grids in body-fitted coordinates. A segregated approach is used to solve the pressure–velocity coupling problem. Several iterative pressure linear solvers including Krylov subspace and multigrid methods and their combination have been developed to compare the efficiency of each method and to design a robust solver. Three-dimensional numerical experiments carried out on scalar and vector machines and performed on different fluid flow problems show that a combination of multigrid and Krylov subspace methods is a robust and efficient pressure solver. This revised version was published online in June 2006 with corrections to the Cover Date.     

不变子空间问题的一个等价条件

本文得到了自反Banach空间X上有界线性算子全体所成的Banach代数的子代数的不变子空间问题的一个新的等价条件。     

A note on the efficiency of the conjugate gradient method for a class of time‐dependent problems

We discuss the efficiency of the conjugate gradient (CG) method for solving a sequence of linear systems; Auⁿ⁺¹ = uⁿ, where A is assumed to be sparse, symmetric, and positive definite. We show that under certain conditions the Krylov subspace, which is generated when solving the first linear system Au¹ = u⁰, contains the solutions {uⁿ} for subsequent time steps. The solutions of these equations can therefore be computed by a straightforward projection of the right‐hand side onto the already computed Krylov subspace. Our theoretical considerations are illustrated by numerical experiments that compare this method with the order‐optimal scheme obtained by applying the multigrid method as a preconditioner for the CG‐method at each time step. Copyright © 2007 John Wiley & Sons, Ltd.