Points of increase of the Brownian sheet

Summary. It is well-known that Brownian motion has no points of increase. We show that an analogous statement for the Brownian sheet is false. More precisely, for the standard Brownian sheet in the positive quadrant, we prove that there exist monotone curves along which the sheet has a point of increase. Received: 7 December 1994 / In revised form: 6 August 1996     

Regions of exact convergence of the USSOR method applied on generalized consistently ordered matrices

Summary. Using the theory of nonnegative matrices and regular splittings, exact convergence and divergence domains of the Unsymmetric Successive Overrelaxation (USSOR) method, as it pertains to the class of Generalized Consistently Ordered (GCO) matrices, are determined. Our recently derived upper bounds, for the convergence of the USSOR method, re also used as effective tools. Received October 17, 1993 / Revised version received December 19, 1994     

Probabilistic aspects of finite-fuel,reflected follower problems

The issue is that of following the path of a Brownian particle by a process of bounded total variation and subject to a reflecting barrier at the origin, in such a way as to minimize expected total cost over a finite horizon. We establish the existence of optimal processes and the dynamic programming equations for this question, and show (by purely probabilistic arguments) its relation to an appropriatefamily of optimal stopping problems with absorption at the origin.Work carried out during a visit by the second author at the University Pierre et Marie Curie (Paris VI), and at INRIA (Institut National de Recherche en Informatique et en Automatique). The hospitality of these institutions is gratefully acknowledged.Research supported in part by the U.S. Air Force Office of Scientific Research, under grant AFOSR-86-0203.     

A new algorithm for the huber estimator in linear models

This paper considers algorithms for solving the linear robust regression problem by minimizing the Huber function. In the computational methods for this problem used so far, the scale estimate is adjusted separately. The new algorithm, based on Newton's method, treats both the scale and the location parameters as independent variables. The special form of the Hessian allows for an efficient updating scheme.     

Some convergence results for asynchronous algorithms

Resumé Nous présentons dans cet article des résultats de convergence des algorithmes asynchrones basés essentiellement sur la notion classique de contraction.Nous généralisons, en particulier, tous les résultats de convergence de ces algorithmes qui font l'hypothèse de contraction en norme vectorielle qui récemment a été très souvant utilisée.Par ailleurs, l'hypothèse de contraction en norme vectorielle peut se trouver difficile, voire impossible à vérifier pour certains problèmes qui peuvent être cependant abordés dans le cadre de la contraction classique que nous adoptons.

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Some convergence results for asynchronous algorithms

Summary In this paper we present convergence results for the asynchronous algorithms based essentially on the notion of classical contraction.We generalize, in particular, all convergence results for those algorithms which are based on the vectorial norm hypothesis, in wide spread use recently.Certain problems, for which the vectorial norm hypothesis can be difficult or even impossible to verify, can nontheless be tackled within the scope of the classical contraction that we adopte.

     

Unicity of best one-sidedL 1-approximations

Summary This paper deals with the problem of uniqueness in one-sidedL ₁-approximation. The chief purpose is to characterize finite dimensional subspacesG of the space of continuous or differentiable functions which have a unique best one-sidedL ₁-approximation. In addition, we study a related problem in moment theory. These considerations have an important application to the uniqueness of quadrature formulae of highest possible degree of precision.     

On finite element subspaces over quadrilateral and hexahedral meshes for incompressible viscous flow problems

Summary Optimal rates of convergence for the approximate solution of the stationary Stokes equations are obtained for finite element schemes which use piecewise constants to approximate the pressure and piecewise linear or piecewise bilinear or trilinear polynomials to approximate the velocity over fairly general quadrilateral or hexahedral meshes.This research was supported by NSF Grant MC80-16532     

Über die Einzugsbereiche der Nullstellen von Polynomen beim Newton-Verfahren

Summary The region of attraction of a zero of a polynomial consists of those points from which the Newton method may be started when this zero is to be computed. The regions of attraction are approximately characterized by studying a continuous analog of the Newton method.

     

Maximale Konvergenzordnung bei der numerischen Lösung von Anfangswertproblemen mit Splines

Summary In [10] a general procedureV is presented to obtain spline approximations by collocation for the solutions of initial value problems for first order ordinary differential equations. In this paper the attainable order of convergence with respect to the maximum norm is characterized in dependence of the parameters involved inV; in particular the appropriate choice of the collocation points is considered.

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Maximale Konvergenzordnung bei der numerischen Lösung von Anfangswertproblemen mit Splines

Zusammenfassung In [10] ist ein allgemeines VerfahrenV beschrieben, das die Lösungen von Anfangswertproblemen bei gewöhnlichen Differentialgleichungen erster Ordnung durch Splines approximiert. Die Konstruktion der Splines erfolgt hierbei mittels Kollokation. In dieser Arbeit wird die maximal erreichbare Konvergenzordnung vonV bezüglich der Maximumnorm in Abhängigkeit aller Parameter vonV charakterisiert, insbesondere wird auf die geeignete Wahl der Kollokationsknoten eingegangen.

     

Über die Effektivität einiger nichtlinearer Verfahren bei der numerischen Behandlung steifer Differentialgleichungssysteme

Summary The stability and accuracy of some explicit nonlinear methods for the numerical integration of stiff systems of ordinary differential equations are investigated. It is shown, that in the general case they can produce the essential error. The special class of stiff systems is singled out, for which these methods are highly efficient. Some numerical results are also presented.