求解大型线性方程组的一类非定常内外迭代法

1 引 言 求解大型线性方程组 Ax=b, A∈R~(?),det(A)≠0. x,b∈R~n (1.1)的内外迭代法,首先由Nichols于1973年提出。由于这类算法在求解大型问题。特别对由边值问题离散化得到的大型稀疏方程组求解,显示了优越性,而受到众多的关注。1991年,Lanzkron.Rose.Szvld等人进一步降其发展成为成套迭代法,为预条件组的近似及同步和     

局部紧空间上的Fuzzy测度

本文研究局部紧Ｈａｕｓｄｏｒｆ空间上Ｆｕｚｙ测度的一些特征;引进了Ｆｕｚｙ测度正则性的概念,并在一定条件下证明了正则Ｆｕｚｙ测度的几个定理。     

Intrinsic volumes and successive radii

Motivated by a problem of Teissier to bound the intrinsic volumes of a convex body in terms of the inradius and the circumradius of the body, we give upper and lower bounds for the intrinsic volumes of a convex body in terms of the elementary symmetric functions of the so-called successive inner and outer radii. These results improve on former bounds and, in particular, they also provide bounds for the elementary symmetric functions of the roots of Steiner polynomials in terms of the elementary symmetric functions of these radii.     

Lebesgue测度等测包及其应用

利用Lebesgue测度的等测内核和等测外包讨论内外测度的一系列性质,给出外测度有限可加的一个充要条件,该条件改进了原有外测度有限可加性相关问题的结果.     

Forms of the Pasch axiom in ordered geometry

We prove that, in the framework of ordered geometry, the inner form of the Pasch axiom ( IP ) does not imply its outer form ( OP ). We also show that OP can be properly split into IP and the weak Pasch axiom ( WP ) (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

关于具优势对称部分的不定线性代数方程组的分裂极小残量算法

１．引 言 考虑大型稀疏线性代数方程组 为利用系数矩阵的稀疏结构以尽可能减少存储空间和计算开销，Ｋｒｙｌｏｖ子空间迭代算法［１，１６，２３］及其预处理变型［６，８，１３，１８，１９］通常是求解（１）的有效而实用的方法．当系数矩阵对称正定时，共轭梯度法（ＣＧ（     

Measure and integration: the basic extension theorems

The present article returns to the new foundations of measure and integration due to the author. In this development the basic extension procedures lead from the so-called outer and inner premeasures to their unique maximal extensions. The initial version was for extended real valued set functions. In the sequel we want to achieve a major simplification, in that we develop the procedures—with no loss in the essentials—in the traditional frame of nonnegative set functions. The final section then will obtain an important extension theorem in the inner theories.     

A New and Self-Contained Proof of Borwein’s Norm Duality Theorem

Borwein’s norm duality theorem establishes the equality between the outer (inner) norm of a sublinear mapping and the inner (outer) norm of its adjoint mappings. In this note we provide an extended version of this theorem with a new and self-contained proof relying only on the Hahn-Banach theorem. We also give examples showing that the assumptions of the theorem cannot be relaxed. This author is supported by Grant BES-2003-0188 from FPI Program of MEC (Spain).     

Lebesgue积分几何意义的推广

证明了非负有界函数的Lebesgue上积分等于函数下方图形的Lebesgue外测度,其Lebesgue下积分等于函数下方图形的Lebesgue内测度,从而将积分的几何意义从可测情形推广到一般情形.     

On the stability of inner and outer approximations of a convex compact set by a ball

The finite-dimensional problems of outer and inner estimation of a convex compact set by a ball of some norm (circumscribed and inscribed ball problems) are considered. The stability of the solution with respect to the error in the specification of the estimated compact set is generally characterized. A new solution criterion for the outer estimation problem is obtained that relates the latter to the inner estimation problem for the lower Lebesgue set of the distance function to the most distant point of the estimated compact set. A quantitative estimate for the stability of the center of an inscribed ball is given under the additional assumption that the compact set is strongly convex. Assuming that the used norm is strongly quasi-convex, a quantitative stability estimate is obtained for the center of a circumscribed ball.     

Analytic approximation and differentiability of joint chance constraints

ABSTRACT

An inner–outer approximation approach was recently developed to solve single chance constrained optimization (SCCOPT) problems. In this paper, we extend this approach to address joint chance constrained optimization (JCCOPT) problems. Using an inner–outer approximation, two smooth parametric optimization problems are defined whose feasible sets converge to the feasible set of JCCOPT from inside and outside, respectively. Any optimal solution of the inner approximation problem is a priori feasible to the JCCOPT. As the approximation parameter tends to zero, a subsequence of the solutions of the inner and outer problems, respectively, converge asymptotically to an optimal solution of the JCCOPT. As a main result, the continuous differentiability of the probability function of a joint chance constraint is obtained by examining the uniform convergence of the gradients of the parametric approximations.     

Upper integral and its geometric meaning

The Hahn definition of the integral is recalled, the requirement of measurability of the integrand omitted. Both the upper and lower integrals comply with this definition and so does any measurable function between them. The outer product measure of the hypograph of a nonnegative bounded nonmeasurable function is equal to the upper integral which is equal to one of the Fan integrals. The outer measure of the graph of a bounded nonmeasurable function is equal to the difference between the upper and lower integrals. A norm for not necessarily measurable functions is defined with the upper integral. The linear space with this norm is complete. The convergence in this space implies the convergence in outer measure. The distance as an outer measure of the symmetric difference of two sets gives us a complete metric space of classes of subsets.      

Carathéodory outer measure on IF-sets

In this paper an outer measure on IF-sets is studied as a mapping to the set of all compact subintervals of the unit interval. We characterize the properties of the outer measure by the help of the properties of functions given by the edges of the intervals. Then there are defined measurable elements and there is proved that the family of measurable elements is a lattice. Finally the outer measure induced by a measure is constructed. This paper was supported by Grant VEGA 1/2002/05.     

Mean projection and section radii of convex bodies

We introduce new series of mean outer and inner radii, which are defined as the outer (respectively, inner) radius of, either the projection of the convex body onto an i-dimensional subspace, or the i-dimensional section, 1 ≤ i ≤ n, averaged over the Grassmannian manifold, and with respect to the Haar probability measure. We study some properties of these new functionals, establishing inequalities among them, as well as their relation with other measures as the volume or the quermassintegrals.     

Inexact inner–outer Golub–Kahan bidiagonalization method: A relaxation strategy

We study an inexact inner–outer generalized Golub–Kahan algorithm for the solution of saddle-point problems with a two-times-two block structure. In each outer iteration, an inner system has to be solved which in theory has to be done exactly. Whenever the system is getting large, an inner exact solver is, however, no longer efficient or even feasible and iterative methods must be used. We focus this article on a numerical study showing the influence of the accuracy of an inner iterative solution on the accuracy of the solution of the block system. Emphasis is further given on reducing the computational cost, which is defined as the total number of inner iterations. We develop relaxation techniques intended to dynamically change the inner tolerance for each outer iteration to further minimize the total number of inner iterations. We illustrate our findings on a Stokes problem and validate them on a mixed formulation of the Poisson problem.     

各向异性相态场模型的Gibbs—Thompson关系

本文利用奇异摄动的内、外解匹配方法,分析了各向异性时在相态场模型的边界层上表面张菌、法向速度、平均曲率和各向异性函数的影响,得到了各向异性时的Ｇｉｂｂｓ－Ｔｈｏｍｐｓｏｎ关系,以及边界层所满足的方程。     

On the Inner and Outer Norms of Sublinear Mappings

In this short note we show that the outer norm of a sublinear mapping F, acting between Banach spaces X and Y and with dom F = X, is finite only if F is single-valued. This implies in particular that for a sublinear multivalued mapping the inner and the outer norms cannot be finite simultaneously.      

Identification of non-minimum phase transfer function using higher-order spectrum

The present paper treats the identification of parametric nonminimum phase transfer function. We propose a method of identification based on the inner outer factorization of stable transfer function. It consists of identifying the outer and inner parts of a transfer function separately. The outer part is identified by the use of the second-order spectral estimate from the observed linear process, while the inner part is identified by the use of a higher-order cumulant spectral estimate from the observed process. Respective parameter estimators are determined in the light of asymptotic efficiency. In order to estimate the order of the inner part of a transfer function, a criterion is proposed. It is introduced based on the same principle as in the case of Akaike's AIC.     

Measure theory based on lattices and transfinite recursion

In this methodological study we develop the foundations of measure theory using lattices as prime structures instead of rings. Topological as well as abstract regularity is incorporated into this approach from the outset. The use of inner and outer measures is replaced by transfinite constructions. Basic extension steps are transfinitely iterated to yield generalizations of Carathéodory’s theorem which are optimal with respect to inner and outer approximations. Received: 5 May 2008     

Inner iterations in the shift-invert residual Arnoldi method and the Jacobi-Davidson method

We establish a general convergence theory of the Shift-Invert Residual Arnoldi(SIRA)method for computing a simple eigenvalue nearest to a given targetσand the associated eigenvector.In SIRA,a subspace expansion vector at each step is obtained by solving a certain inner linear system.We prove that the inexact SIRA method mimics the exact SIRA well,i.e.,the former uses almost the same outer iterations to achieve the convergence as the latter does if all the inner linear systems are iteratively solved with low or modest accuracy during outer iterations.Based on the theory,we design practical stopping criteria for inner solves.Our analysis is on one step expansion of subspace and the approach applies to the Jacobi-Davidson(JD)method with the fixed targetσas well,and a similar general convergence theory is obtained for it.Numerical experiments confirm our theory and demonstrate that the inexact SIRA and JD are similarly effective and are considerably superior to the inexact SIA.