Various scienti c applications compute iterations on a huge set of input data. Examples include parameter sweep, which processes the same model through hundreds or thousands of input data, scienti c visualization and numeric methods to solve hyperbolic equations. The state of the art for executing such applications is to model them as scienti c work ows and to run them in HPC infrastructure. Th...

A wide range of free boundary problems occurring in engineering and industry can be rewritten as a minimization problem for a strictly convex, piecewise smooth but non{diierentiable energy functional. The fast solution of related discretized problems is a very delicate question, because usual Newton techniques cannot be applied. We propose a new approach based on convex minimization and constra...

Chaotic iterations have been introduced on the one hand by Chazan, Miranker [6] and Miellou [10] in a numerical analysis context, and on the other hand by Robert [12] and Pellegrin [11] in the discrete dynamical systems framework. In both cases, the objective was to derive conditions of convergence of such iterations to a fixed state. In this paper, a new point of view is presented, the goal he...

Important matrix-valued functions f(A) are, e.g., the inverse A−1, the square root √ A, the sign function and the exponent. Their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices A and f(A) (often f(A) possesses this structure only approximately). However, intermediate matrices arising during the evaluatio...

Consider the map $S$ which sends a planar polygon $P$ to new $S(P)$ whose vertices are intersection points of second nearest sides $P$. This is inverse famous pentagram map. In this paper we investigate dynamics $S$. Namely, address question whether convex stays under iterations Computer experiments suggest that almost never happens. We prove indeed set polygons remain has measure zero, and mor...

Abstract: We establish a relation between the length T of the integration window of a linear differential equation x′+Ax = b and a spectral parameter s∗. This parameter is determined by comparing the exact solution x(T ) at the end of the integration window to the solution of a linear system obtained from the Laplace transform of the differential equation by freezing the system matrix. We propo...

Sierpiński graphs S(n, 3) are the graphs of the Tower of Hanoi with n disks, while Sierpiński gasket graphs Sn are the graphs naturally defined by the finite number of iterations that lead to the Sierpiński gasket. An explicit labeling of the vertices of Sn is introduced. It is proved that Sn is uniquely 3-colorable, that S(n, 3) is uniquely 3-edgecolorable, and that χ′(Sn) = 4, thus answering ...

The solution of nonsymmetric systems of linear equations continues to be a diicult problem. A main algorithm for solving nonsymmetric problems is restarted GMRES. The algorithm is based on restarting full GMRES every s iterations, for some integer s>0. This paper considers the impact of the restart frequency s on the convergence and work requirements of the method. It is shown that a good choic...

Let C be a convex subset of a complete generalized convex metric space X, and S and T be two self mappings on C. In this paper it is shown that if the sequence of modified Ishikawa iterations with errors in the sense of Xu [19] associated with S and T converges, then its limit point is the common fixed point of S and T . This result extends and generalizes the corresponding results of Niampally...

While the forward trajectory of a point in discrete dynamical system is always unique, general can have infinitely many backward trajectories. The union limit points all trajectories through $x$ was called by M.~Hero "special $\alpha$-limit" ($s\alpha$-limit for short) $x$. In this article we show that there hierarchy $s\alpha$-limits under iterations S-unimodal map: size $s\alpha$-limit increa...