Maximale Konvergenzordnung bei der numerischen Lösung von Anfangswertproblemen mit Splines

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.

Iterative continuous maximum-likelihood reconstruction method

Behandlung eines Goursatproblems mit einer verallgemeinerten Riemannschen Methode

In dieser Arbeit wird ein lineares Goursat problem in zwei Zeit- und einer Raumvariablen behandelt. Die Koeffizienten der betrachteten Differentialgleichung mussen hierbei nach allen Variablen beliebig oft differenzierbar sein und nebst all ihren partiellen Ableitungen bestimmten Wachstumsbeschrankungen genugen. Fur die Inhomogenitat und die Vorgaben werden gesonderte Voraussetzungen gestellt. Zuerst wird fur ein hinsichtlich der Anfangsbedingungen verallgemeinertes Goursatproblem die eindeutige Losbarkeit in der gleichen Funktionenklasse bewiesen, in der die Koeffizienten der Differentialgleichung liegen. Auf Grund dieses Ergebnisses gelingt es dann, mit Hilfe einer verallgemeinerten Riema…

On properties of the iterative maximum likelihood reconstruction method

In this paper, we continue our investigations6 on the iterative maximum likelihood reconstruction method applied to a special class of integral equations of the first kind, where one of the essential assumptions is the positivity of the kernel and the given right-hand side. Equations of this type often occur in connection with the determination of density functions from measured data. There are certain relations between the directed Kullback–Leibler divergence and the iterative maximum likelihood reconstruction method some of which were already observed by other authors. Using these relations, further properties of the iterative scheme are shown and, in particular, a new short and elementar…

On an iterative method for a class of integral equations of the first kind

In this paper, we investigate an iterative method which has been proposed [1] for the numerical solution of a special class of integral equations of the first kind, where one of the essential assumptions is the positivity of the kernel and the given right-hand side. Integral equations of this special type occur in experimental physics, astronomy, medical tomography and other fields where density functions cannot be measured directly, but are related to observable functions via integral equations. In order to take into account the non-negativity of density functions, the proposed iterative scheme was defined in such a way that only non-negative solutions can be approximated. The first part o…

Zur numerischen Lösung gewöhnlicher Differential-gleichungen mit Splines in einem Sonderfall

In an earlier paper [1] a general procedure has been presented to obtain polynomial spline approximations for the solution of the initial value problem for ordinary differential equations. In this paper the general procedure is described by an equivalent one step method. Furthermore two convergence theorems are proved for a special case which is not included in the general convergence or divergence theory given in [1].