Complete Interval Arithmetic and its Implementation

A Complete Interval Arithmetic and its Implementation is discussed

A Mathematical Basis for an Interval Arithmetic Standard

Basic concepts for an interval arithmetic standard are discussed in the paper. Interval arithmetic deals with closed and connected sets of real numbers. Unlike floating-point arithmetic it is free of exceptions. A complete set of formulas to approximate real interval arithmetic on the computer is displayed in section 3 of the paper. The essential comparison relations and lattice operations are discussed in section 6. Evaluation of functions for interval arguments is studied in section 7. The desirability of variable length interval arithmetic is also discussed in the paper. The requirement to adapt the digital computer to the needs of interval arithmetic is as old as interval arithmetic. An obvious, simple possible solution is shown in section 8

Convergence of an adaptive hp finite element strategy in higher space-dimensions

We develop a general convergence analysis for a class of inexact Newton-type regularizations for stably solving nonlinear ill-posed problems. Each of the methods under consideration consists of two components: the outer Newton iteration and an inner regularization scheme which, applied to the linearized system, provides the update. In this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including Landweber and steepest decent iterations, iterated Tikhonov method, and method of conjugate gradients

A geometric data structure for parallel finite elements and the application to multigrid methods with block smoothing

We present a parallel data structure which is directly linked to geometric quantities of an underlying mesh and which is well adapted to the requirements of a general finite element realization. In addition, we define an abstract linear algebra model which supports multigrid methods (extending our previous work in Comp. Vis. Sci. 1 (1997) 27-40). Finally, we apply the parallel multigrid preconditioner to several configurations in linear elasticity and we compute the condition number numerically for different smoothers, resulting in a quantitative evaluation of parallel multigrid performance

Up-to-date Interval Arithmetic From Closed Intervals to Connected Sets of Real Numbers

We consider biperiodic integral equations of the second kind with weakly singular kernels such as they arise in boundary integral equation methods. The equations are solved numerically using a collocation scheme based on trigonometric polynomials. The weak singularity is removed by a local change to polar coordinates. The resulting operators have smooth kernels and are discretized using the tensor product composite trapezodial rule. We prove stability and convergence of the scheme under suitable parameter choices, achieving algebraic convergence of any order under appropriate regularity assumptions. The method can be applied to typical boundary value problems such as potential and scattering problems both for bounded obstacles and for periodic surfaces. We present numerical results demonstrating that the expected convergence rates can be observed in practice

Begegnungen eines Mathematikers mit Informatik

Es handelt sich hier um die nachträgliche Niederschrift eines Vortrages, den der Autor während eines Festkolloquiums gehalten hat, das die Fakultät für Mathematik und das Institut für Angewandte und Numerische Mathematik anlässlich seines 75sten Geburtstages am 27. Juni 2008 veranstaltet haben. Der Vortrag skizziert zunächst kurz die Entwicklung der Rechengeschwindigkeit während der letzten zwanzig Jahre. Es wird dann auf die Entwicklung von Rechenanlagen in Deutschland und Europa in den 50ger und 60ger Jahren eingegangen. Schließlich wird anhand von 4 Fallbeispielen gezeigt, dass sich aus der Tatsache, dass es heute in Europa keine eigenständige Rechnerentwicklung mehr gibt, schwerwiegende Nachteile für zukunftsweisende Softwareentwicklungen ergeben