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Sehr geehrte Fakultätsmitglieder,<br>
<br>
anbei ein Vortrag zu den Vortragsankündigungen der KW 40.<br>
<br>
<b>Prof. Peter Deuflhard</b> (Freien Universität Berlin, Konrad Zuse
Zentrum für Informationstechnik Berlin, Beijing University of
Technology)<br>
Donnerstag, 2. Oktober 2014, 14:00-15:30, SR 10<br>
<b>Titel: "Affine covariant versus affine contravariant Gauss-Newton
methods"</b><br>
<br>
Abstract:<br>
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. <br>
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. <br>
Affine contravariant GN methods, also called residual based GN
methods, can also be consistently derived. These methods converge
locally only for ``small residual'' <br>
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. <br>
<br>
mit freundlichen Grüßen<br>
<br>
Martina Fellner<br>
<pre class="moz-signature" cols="72">--
Martina Fellner
Sekretariat
Fakultät für Mathematik
Oskar-Morgenstern-Platz 1 - 10.140
1090 Wien
Tel.: 0043 (1) 4277 50602
Fax.: 0043 (1) 4277 9506</pre>
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