[Vortraege] Vortrag KW 40

[Martina Fellner]

Sehr geehrte Fakultätsmitglieder,

anbei ein Vortrag zu den Vortragsankündigungen der KW 40.

*Prof. Peter Deuflhard* (Freien Universität Berlin, Konrad Zuse Zentrum 
für Informationstechnik Berlin, Beijing University of Technology)
Donnerstag, 2. Oktober 2014, 14:00-15:30, SR 10
*Titel: "Affine covariant versus affine contravariant Gauss-Newton methods"*

Abstract:
Two versions of Gauss-Newton methods for the numeral solution of 
nonlinear least squares problems (NLSPs) are described synoptically in 
terms of both affine invariance structure and adaptive algorithms. To 
begin with, the two affine invariance classes are exemplified at Newton 
methods for nonlinear systems of equations.
Affine covariant Gauss-Newton methods, also called error oriented GN 
methods, are in common use and quite successful for NLSPs in 
differential equations. The are constructed on the conceptual background 
of a local and global Gauss-Newton path. These methods converge locally 
only for a class of "adequate'' NLSPs wherein the so-called 
incompatibility factor must be less than 1 in a sufficiently large 
neighborhood of the solution. A convenient numerical estimation of the 
incompatibility factor is possible. In order to achieve some kind of 
global convergence, an adaptive trust region strategy has been 
constructed. The treatment of rank-deficient cases yields  additional 
insight of the nature of the inverse problem at hand. An illustrative 
example from systems biology will be given.
Affine contravariant GN methods, also called residual based GN methods, 
can also be consistently derived. These methods converge locally only 
for ``small residual''
problems, which are algebraically equivalent to the geometrically 
derived convergence condition of  P.-A. Wedin. A convenient numerical 
estimation of this small residual factor is available. As above, in 
order to achieve some kind of global convergence, an adaptive trust 
region strategy can be derived, which is, however, different from the 
one for the affine covariant case. Finally, a possible extension to 
mollifier methods for inverse Radon problems due to Louis/Maas will be 
sketched. Here, a contravariant approach may possibly be preferable.

mit freundlichen Grüßen

Martina Fellner

-- 
Martina Fellner
Sekretariat
Fakultät für Mathematik
Oskar-Morgenstern-Platz 1 - 10.140
1090 Wien
Tel.: 0043 (1) 4277 50602
Fax.: 0043 (1) 4277 9506

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