Solving large-scale semidefinite programs in parallel

We describe an approach to the parallel and distributed solution of large-scale, block structured semidefinite programs using the spectral bundle method. Various elements of this approach (such as data distribution, an implicitly restarted Lanczos method tailored to handle block diagonal structure, a mixed polyhedral-semidefinite subdifferential model, and other aspects related to parallelism) are combined in an implementation called LAMBDA, which delivers faster solution times than previously possible, and acceptable parallel scalability on sufficiently large problems. This work was supported in part by NSF grants DMS-0215373 and DMS-0238008.     

Combining Complementary Ligands into one Framework for the Construction of a Ferromagnetically Coupled [MnIII12] Wheel

Phenolic oxime and diethanolamine moieties have been combined into one organic framework, resulting in the formation of a novel ligand type that can be employed to construct a rare and unusual dodecametallic Mn wheel, within which nearest neighbours are coupled ferromagnetically.     

A matrix rational Lanczos method for model reduction in large‐scale first‐ and second‐order dynamical systems

In the present paper, we describe an adaptive modified rational global Lanczos algorithm for model‐order reduction problems using multipoint moment matching‐based methods. The major problem of these methods is the selection of some interpolation points. We first propose a modified rational global Lanczos process and then we derive Lanczos‐like equations for the global case. Next, we propose adaptive techniques for choosing the interpolation points. Second‐order dynamical systems are also considered in this paper, and the adaptive modified rational global Lanczos algorithm is applied to an equivalent state space model. Finally, some numerical examples will be given.     

A note on simultaneous preconditioning and symmetrization of non‐symmetric linear systems

Motivated by the theory of self‐duality that provides a variational formulation and resolution for non‐self‐adjoint partial differential equations (Ann. Inst. Henri Poincaré (C) Anal Non Linéaire 2007; 24 :171–205; Selfdual Partial Differential Systems and Their Variational Principles. Springer: New York, 2008), we propose new templates for solving large non‐symmetric linear systems. The method consists of combining a new scheme that simultaneously preconditions and symmetrizes the problem, with various well‐known iterative methods for solving linear and symmetric problems. The approach seems to be efficient when dealing with certain ill‐conditioned, and highly non‐symmetric systems. The numerical and theoretical results are provided to show the efficiency of our approach. Copyright © 2010 John Wiley & Sons, Ltd.     

On the Vector ε-Algorithm for Solving Linear Systems of Equations

The four vector extrapolation methods, minimal polynomial extrapolation, reduced rank extrapolation, modified minimal polynomial extrapolation and the topological epsilon algorithm, when applied to linearly generated vector sequences are Krylov subspace methods and it is known that they are equivalent to some well-known conjugate gradient type methods. However, the vector -algorithm is an extrapolation method, older than the four extrapolation methods above, and no similar results are known for it. In this paper, a determinantal formula for the vector -algorithm is given. Then it is shown that, when applied to a linearly generated vector sequence, the algorithm is also a Krylov subspace method and for a class of matrices the method is equivalent to a preconditioned Lanczos method. A new determinantal formula for the CGS is given, and an algebraic comparison between the vector -algorithm for linear systems and CGS is also given.     

On Generating Discrete Orthogonal Bivariate Polynomials

In this paper we present an algorithm for recursively generating orthogonal bivariate polynomials on a discrete set S ². For this purpose we employ commuting pairs of real symmetric matrices H, K ^n×n to obtain, in a certain sense, a two dimensional Hermitian Lanczos method. The resulting algorithm relies on a recurrence having a slowly growing length. Practical implementation issues an applications are considered. The method can be generalized to compute orthogonal polynomials depending on an arbitrary number of variables.     

Ordered and Periodic Chaos of the Bounded One-Dimensional Multibarrier Potential

Numerical analysis indicates that there exists an unexpected new ordered chaos for the bounded one-dimensional multibarrier potential. For certain values of the number of barriers, repeated identical forms (periods) of the wavepackets result upon passing through the multibarrier potential.     

Computing Eigenelements of Real Symmetric Matrices via Optimization

In certain circumstances, it is advantageous to use an optimization approach in order to solve the generalized eigenproblem, Ax = Bx, where A and B are real symmetric matrices and B is positive definite. In particular, this is the case when the matrices A and B are very large the computational cost, prohibitive, of solving, with high accuracy, systems of equations involving these matrices. Usually, the optimization approach involves optimizing the Rayleigh quotient.We first propose alternative objective functions to solve the (generalized) eigenproblem via (unconstrained) optimization, and we describe the variational properties of these functions.We then introduce some optimization algorithms (based on one of these formulations) designed to compute the largest eigenpair. According to preliminary numerical experiments, this work could lead the way to practical methods for computing the largest eigenpair of a (very) large symmetric matrix (pair).     

计算最小奇异组的一个精化调和 Lanczos双对角化方法

在很多实际应用中需要计算大规模矩阵的若干个最小奇异组.调和投影方法是计算内部特征对的常用方法,其原理可用于求解大规模奇异值分解问题.本文证明了,当投影空间足够好时,该方法得到的近似奇异值收敛,但近似奇异向量可能收敛很慢甚至不收敛.根据第二作者近年来提出的精化投影方法的原理,本文提出一种精化的调和Lanczos双对角化方法,证明了它的收敛性.然后将该方法与Sorensen提出的隐式重新启动技术相结合,开发出隐式重新启动的调和Lanczos双对角化算法(IRHLB)和隐式重新启动的精化调和Lanczos双对角化算法(IRRHLB).位移的合理选取是算法成功的关键之一,本文对精化算法提出了一种新的位移策略,称之为"精化调和位移".理论分析表明,精化调和位移比IRHLB中所用的调和位移要好,且可以廉价可靠地计算出来.数值实验表明,IRRHLB比IRHLB要显著优越,而且比目前常用的隐式重新启动的Lanczos双对角化方法(IRLB)和精化算法IRRLB更有效.