Some observations on the l2 convergence of the additive Schwarz preconditioned GMRES method

Additive Schwarz preconditioned GMRES is a powerful method for solving large sparse linear systems of equations on parallel computers. The algorithm is often implemented in the Euclidean norm, or the discrete l² norm, however, the optimal convergence result is available only in the energy norm (or the equivalent Sobolev H¹ norm). Very little progress has been made in the theoretical understanding of the l² behaviour of this very successful algorithm. To add to the difficulty in developing a full l² theory, in this note, we construct explicit examples and show that the optimal convergence of additive Schwarz preconditioned GMRES in l² cannot be obtained using the existing GMRES theory. More precisely speaking, we show that the symmetric part of the preconditioned matrix, which plays a role in the Eisenstat–Elman–Schultz theory, has at least one negative eigenvalue, and we show that the condition number of the best possible eigenmatrix that diagonalizes the preconditioned matrix, key to the Saad–Schultz theory, is bounded from both above and below by constants multiplied by h^?1/2. Here h is the finite element mesh size. The results presented in this paper are mostly negative, but we believe that the techniques used in our proofs may have wide applications in the further development of the l² convergence theory and in other areas of domain decomposition methods. Copyright © 2002 John Wiley & Sons, Ltd.     

Some Properties of GMRES in Hilbert Spaces

Our purpose in this work is to explore the properties of GMRES in Hilbert spaces. We extend to the infinite dimensional context some main results that are known to hold in the finite dimensional case. A key assumption for these extensions is that the involved linear operator is an algebraic operator.     

Some results about GMRES in the singular case

In this paper, we study the Generalized Minimal Residual (GMRES) method for solving singular linear systems, particularly when the necessary and sufficient condition to obtain a Krylov solution is not satisfied. Thanks to some new results which may be applied in exact arithmetic or in finite precision, we analyze the convergence of GMRES and restarted GMRES. These formulas can also be used in the case when the systems are nonsingular. In particular, it allows us to understand what is often referred to as stagnation of the residual norm of GMRES. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some observations on cohomologicallyp-ample bundles

Sunto Usando un recente risultato di Deligne e Illusie si prova un teorema di annullamento per la coomologia di un fibrato coomologicamente p-ampio. Si studiano le proprietà dei gruppi di coomologia dei fibrati vettoriali nel caso in cui il morfismo di Frobenius si estenda a un sollevamento della varietà su W₂(k), anello dei vettori di Witt di lunghezza 2 del campo. Si studiano, infine, a titolo di esempio, i fibrati coomologicamente p-ampi su P^d.

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Partially supported by the MRST 40% and 60% funds.     

Some observations on HC-128

In this paper, we study HC-128 in detail from cryptanalytic point of view. First, we use linear approximation of the addition modulo 2 ⁿ of three n-bit integers to identify linear approximations of g ₁, g ₂, the feedback functions of HC-128. This, in turn, shows that the process of keystream output generation of HC-128 can be well approximated by linear functions. In this direction, we show that the ??least significant bit?? based distinguisher (presented by the designer himself) of HC-128 works for the complete 32-bit word. Using the above linear approximations of g ₁, g ₂, we present a new distinguisher for HC-128 which is slightly weaker than Wu??s distinguisher. Finally, in the line of Dunkelman??s observation, we also study how HC-128 keystream words leak secret state information of the cipher due to the properties of the functions h ₁, h ₂ and present improved results.     

Some observations on the oberwolfach problem

Some results concerning decompositions of K_n, K_n - F(where F denotes a 1-factor) and complements of a family of special cubic graphs into 2-factors of the same type are given. In particular, if 2d is a divisor of n, it is shown that K_n - F can be decomposed into 2-factors each of whose components is a cycle of length 2d.     

Some observations on local least squares

In previous work we introduced a construction to produce biorthogonal multiresolutions from given subdivisions. The approach involved estimating the solution to a least squares problem by means of a number of smaller least squares approximations on local portions of the data. In this work we use a result by Dahlquist, et al. on the method of averages to make observational comparisons between this local least squares estimation and full least squares approximation. We have explored examples in two problem domains: data reduction and data approximation. We observe that, particularly for design matrices with a repetitive pattern of column entries, the least squares solution is often well estimated by local least squares, that the estimation rapidly improves with the size of the local least squares problems, and that the quality of the estimate is largely independent of the size of the full problem. In memory of Germund Dahlquist (1925–2005).AMS subject classification (2000) 93E24     

Some observations on compact indestructible spaces

Inspired by a recent work of Dias and Tall, we show that a compact indestructible space is sequentially compact. We also prove that a Lindelöf _T2$T_2$ indestructible space has the finite derived set property and a compact _T2$T_2$ indestructible space is pseudoradial. Finally, we observe that under CH a compact weakly Whyburn space of countable tightness is indestructible.     

加权正则函数的一些性质

正则函数是Clifford分析中的一类重要函数,加权正则函数是正则函数的进一步发展,也是一类重要的函数,因此具有一定的研究意义.在正则函数的研究基础上,并利用加权正则函数自身的性质,讨论了加权正则函数的平均值定理,最大模原理,Weierstrass定理以及一些其它推论.     

A simpler GMRES

The generalized minimal residual (GMRES) method is widely used for solving very large, nonsymmetric linear systems, particularly those that arise through discretization of continuous mathematical models in science and engineering. By shifting the Arnoldi process to begin with Ar₀ instead of r₀, we obtain simpler Gram–Schmidt and Householder implementations of the GMRES method that do not require upper Hessenberg factorization. The Gram–Schmidt implementation also maintains the residual vector at each iteration, which allows cheaper restarts of GMRES(m) and may otherwise be useful.     

Some characterizations of weighted Hardy spaces

In this paper, we will characterize the weighted Hardy spaces by the intrinsic square functions including the Lusin area function, Littlewood-Paley G-function and -function.     

Some weighted Gagliardo-Nirenberg inequalities and applications

We obtain conditions on the measure so that the -norm of a function is controlled by the -norms of the function and its gradient. Applications to eigenvalues of the Schrödinger operator and to other inequalites are also given.

     

Some observations on practising successful operational research

Much has been written about the decline of Operational Research. In the 1960s it had already been proclaimed dead and the decreasing number of OR departments in industrial enterprises seems to support this thesis. The question seems to be: ‘Why is OR apparently disappearing at a time when the major obstacles to OR applications in the past, namely lack of computer-readable data, lack of software and hardware have been overcome?’ I report on 30 years of successful external OR activities and offer some observations which I have made while being active at a university and in OR companies in Germany.     

Some observations on left absolutely flat monoids

Communicated by J.M. Howie     

Some observations on repeated spring balance weighing designs

Summary Dey [3] has suggested a spring balance weighing design in preference to “repeated designs”, and later, Kulshreshtha and Dey [5] have suggested yet one more weighing design which, they say, would be preferred to “repeated designs” and to those suggested in [3], provided one is interested in estimating the weights of some of the objects with increased precision at the cost of precision for others. It has been shown here that, while the above findings may be true in some situations, one might, in a given problem, prefer “repeated designs” to those suggested in [3] and [5]. NSF Grant No. GP-28312 and GP-36562.     

Some observations on fuzzy relations over fuzzy subsets

This paper considers fuzzy relations defined over fuzzy subsets and settles some open problems regarding the distributivity and transitivity of such relations