121595993
Continuity of sublinear operators on F-spaces
In this note, some of the fundamental theorems concerning the automatic continuity of positive linear functionals and positive linear operators (see [9,14,21]) as well as certain uniform boundedness type theorems (see [5,13,15,19]) will be extended to derive various continuity properties of (even discontinuous) sublinear operators from some metrizable topological vector space to some ordered topological vector space.
123207765
On the Strassen disintegration theorem
122751787
Varianten zum Konvergenzsatz von Simons und Anwendungen in der Choquettheorie
120206552
Beschränktheitsaussagen für sublineare und bisublineare Funktionale
118524160
Automatic Continuity of Linear Operators
The present survey is essentially based on some recent automatic continuity results which have been obtained by Ernst Albrecht and the author in the joint papers [1;2;3]. In contrast to the excellent account of automatic continuity theory in Sinclair's lecture notes [29] and in Dales' recent survey article [9], the emphasis of the present exposition lies on the automatic continuity problem for linear operators acting between spaces of functions and distributions. The continuity of such operators will be derived from certain conditions which naturally arise in the theory of linear systems, namely from time-invariance and causality on the one side and from dissipativity on the other. The automatic continuity of all time-invariant and causal linear operators turns out to be just one achievement of a far-reaching new theory on automatic continuity which also applies to the problem of continuity for algebrahomomorphisms, derivations, intertwining operators, and local operators. Some central features of this theory will be sketched; in particular, some extensions and variants of the classical principle of uniform boundedness will be discussed.
122589104
Bemerkungen zum von Neumannschen Minimaxtheorem
120219316
On bounds for the convergence of the SSOR method for H-matrices
In [1] G. Alefeld and R S. Varga show that if A is an H-matrix, then where and JA are, respectively, the SSOR and the Jacobi iteration matrices associated with A. We investigate the possibility for[M0001] when Ais an H-matrix and show that unless does not hold. Comparison is made with the sharpness of a similar bound on the ω's for which the SOR iteration matrix associated with A converges.
122204875
Theorems of Stein-Rosenberg Type. III. The Singular Case*
Abstract In the theory of iterative methods, the classical Stein-Rosenberg theorem can be viewed as giving the simultaneous convergence (or divergence) of the extrapolated Jacobi (JOR) matrix J ω and the successive overrelaxation (SOR) matrix , in the case when the Jacobi matrix J 1 is nonnegative. As has been established recently by Buoni and Varga, necessary and sufficient conditions for the simultaneous convergence (or divergence) of J ω and have been established which do not depend on the assumption that J 1 is nonnegative. Our aim here is to extend these results to the singular case, using the notion of semiconvergence. In particular, for a real singular matrix A with nonpositive off-diagonal entries, we find conditions (Theorem 3.4) which imply that J ω and simultaneously semiconverge for all ω in the real interval [0,1).
2662369
CONVERGENCE OF INFINITE PRODUCTS OF MATRICES AND INNER{OUTER ITERATION SCHEMES
We develop conditions under which a product Q1=0 Ti of matrices chosen from a possibly innite set of matricesS =fTjjj2 Jg converges. We obtain the following conditions which are sucient for the convergence of the product: There exists a vector norm such that all matrices in S are nonexpansive with respect to this norm and there exists a subsequencefikg 1=0 of the sequence of the nonnegative integers such that the corresponding sequence of operators Tik 1=0 converges to an operator which is paracontracting with respect to this norm. We deduce the continuity of the limit of the product of matrices as a function of the sequencesfikg1=0 . But more importantly, we apply our results to the question of the convergence of inner{outer iteration schemes for solving singular consistent linear systems of equations, where the outer splitting is regular and the inner splitting is weak regular.
121609264
A combined direct-iterative approach for solving large scale singular and rectangular consistent systems of linear equations
Abstract We construct a certain iterative scheme for solving large scale consistent systems of linear equations Ax = b , (*) where A is a complex m X n matrix of rank r , m ⩾ n , and where A is assumed reasonably well conditioned. The iterative method is obtained through a careful exploitation of an LU -decomposition of A , and, disregarding roundoff errors, it converges to a solution to ( * ), though not necessarily the minimal l 2 -norm one, from any starting vector in r iterations. Moreover, once the LU -decomposition of A is complete, only about r 3 /2 arithmetic operations (multiplications and divisions) are needed to execute the r iterations.
122950167
Monotonic sequences and rates of convergence of asynchronized iterative methods
Abstract In a recent paper B. Vemmer and the authors investigated the effect of varying the number of processors on the rate of convergence of the asynchronized parallel block Jacobi method associated with monotone matrices. It was found that, under certain simplifying assumptions, increasing the number of processors in relation to the number of blocks (or, what comes to the same in more general settings, the number of iteration operators) slows down the convergence. One interpretation for these results given in that paper was that increasing the number of processors means that when the current global approximation is updated by a local approximation from one of the processors, that local approximation was computed from a “much” earlier global approximation received from the host node. Hence the slowdown in the rate of convergence. The principal purpose of this paper is to remove some of the simplifying assumptions that were made in the above-mentioned paper and to prove that many of the results there hold under much more general conditions. Our present assumptions do not yield a fixed iteration matrix which models the process as was the case previously. This means that different tools have to be developed to establish results comparing the rate of convergence of two assynchronized processes.