Beschränktheitsaussagen für sublineare und bisublineare Funktionale

title: Bemerkungen zum von Neumannschen Minimaxtheorem

title: 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.

title: Subproper splitting for rectangular matrices
abstract: In this paper iterative schemes for approximating a solution to a rectangular but consistent linear system Ax = b are studied. Let A ϵ Cm × nr. The splitting A = M − N is called subproper if R(A) ⊆ R(M) and R(A ∗ ) ⊆R(M ∗ ) . Consider the iteration x i = M † Nx i −1 + M † b . We characterize the convergence of this scheme to a solution of the linear system. When A ϵ Rm×nr, monotonicity and the concept of subproper regular splitting are used to determine a necessary and a sufficient condition for the scheme to converge to a solution.

title: Models of parallel chaotic iteration methods
abstract: We consider two models of parallel multisplitting chaotic iterations for solving large nonsingular systems of equations Ax = b . In the first model each processor can carry out an arbitrary number of local iterations before the next global approximation to the solution is formed. In the second model any processor can update the global approximation which resides in the central processor at any time. This model is a generalization of a sequential iterative scheme due to Ostrowski called the free steering group Jacobi iterative scheme and a chaotic relaxation point iterative scheme due to Chazan and Miranker. We show that when A is a monotone matrix and all the splittings are weak regular, both models lead to convergent schemes.

title: Discrete approximations to reachability cones of linear differential systems
abstract: In recent years the authors both individually and jointly with Professor Abraham Berman have been interested in the following cone reachability problem: Given the linear differential system x = Ax , where A is an n × n essentially nonnegative matrix, characterize the set of all initial points in R n such that the trajectories emanating from these points enter the nonnegative orthant and, due to the positive invariance of the nonnegative orthant under A , remain in the orthant thereafter. The authors now show that a discrete method of approximation to the differential equation such as the Cauchy-Euler scheme can be employed to test whether a point reaches the nonnegative orthant without the necessity of using very small time steps.

title: Algorithms for Computing Bases for the Perron Eigenspace with PrescribedNonnegativity and Combinatorial Properties
abstract: Let $P$ be an $n \times n$ nonnegative matrix. In this paper the authors introduce a method called the SCANBAS algorithm for computing a union of (Jordan) chains $\cal{C}$ corresponding to the Perron eigenvalue of $P$, such that $\cal{C}$ consists of nonnegative vectors only and such that at each height, $\cal{C}$ contains the maximal number of nonnegative vectors of that height possible in a height basis for the Perron eigenspace of $P$. It is further shown that $\cal{C}$ can be extended to a height basis for the Perron eigenspace of $P$. The chains are extracted from transform components of $P$ that are, in turn, polynomials in $P$. When the Perron eigenspace has a Jordan basis consisting of nonnegative vectors only, this algorithm computes such a basis. The paper concludes with various examples computed by the algorithm using MATLAB. The work here continues and deepens work on computing nonnegative bases for the Perron eigenspace from polynomials in the matrix already begun by Hartwig, Neumann, and Rose and by Neumann and Schneider.