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International Conference on Mathematical Modeling and Scientific Computing
April 2-6, 2001
Middle East Technical University and Selcuk University
Ankara and Konya, Turkey

Organizers
F. Bornemann (Munich University of Tecnology, Germany), H. Bulgak (Selcuk University, Konya, Turkey), V. Ganzha (Munich University of Technology, Germany), B. Karasozen (METU, Ankara, Turkey), A. Sinan (Selcuk University, Konya, Turkey), C. Zenger (Munich University of Technology, Germany)

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Runge-Kutta type methods for ill-posed problems
by
Christine Boeckmann
Institute of Mathematics, Potsdam University, Germany

A key issue in many scientific problems is the solution of linear equations Tv=y\delta. The unknown v may either be the solution of the entire problem or some intermediate quantity, when solving a nonlinear problem by Newton-type methods. In pratice the data y are rarely given exactly; instead, due to measurement errors, modelling errors and such, there is some inherent noise \delta in the data y\delta. While T is typically a bounded operator between Hilbert spaces V and Y, the inverse operator is unbounded and the solution, in general, is not unique. Therefore, one typically searches the unique solution v=Tfy\delta in X of minimal norm. During the solution process one needs with respect to the ill-posedness a regularization method using a regularization parameter \gamma to balance approximation error and propagated data error. Besides the well-known Tikhonov regularization and various iterative Landweber as well as conjugate gradient type methods in continuous regularization theory the asymptotic regularization is an other possibility
x'(t)  +  T*T x(t)  =  T* y\delta ,    x(0)=0,    t in IR+0.
(\theequation)
The regularization in using (1) is in integrating this initial value problem not as far as possible (theoretically to +\infty), but only up to an abscissa 1/\gamma, where \gamma is the regularization parameter. In fact, (1) can be seen as a continuous analogue of an iterative regularization method: if we would solve (1) by the forward Euler method with stepsize \omega, then we get the common Landweber method. In this paper we propose a new generalization. We investigate in the solution of (1) by an arbitrary explicite s-stage Runge-Kutta method with the Butcher tableau (c, a, B) and the order p. The new Runge-Kutta type iteration methods
xk+1s=( s
å
i=0 
 (-1)i\frac1i!(\omegaT*T)i)xks + \omega( s-1
å
i=0 
 (-1)i\frac1(i+1)!(\omegaT*T)i) T* y\delta,  x0=0,
s=p, with the positive filter function
Fl = 1-( s
å
i=0 
 (-1)i \frac1i! (\omega\sigmal2)i)\frac1\gamma    with  \frac1\gamma in IN ,   0 < \sigmal <= ||T||
 and 0 < \omega < \frac1||T|| as well as the singular values \sigmal of T, e.g., if T is compact, are convergent regularization methods. Furthermore, if xf fulfills a common source condition with \tau > 0 the following error estimation for s=1, 3 holds
||x\delta, s1/\gamma -xf||V    <=   c \delta\frac2\tau2\tau+1,
where c > 0 is a constant independently of \delta.
An application to Atmospheric Physics in determining the aerosol size distribution is shown.

Date received: February 16, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagk-65.