Convergence rates for nonlinear ill-posed problems based on a variational inequalities approach
Bernd Hofmann ( TU Chemnitz, Germany)
MSI Computational Mathematics Seminar SeriesDATE: 2009-10-26
TIME: 11:00:00 - 12:00:00
LOCATION: JD G35
CONTACT: JavaScript must be enabled to display this email address.
ABSTRACT:
Twenty years ago {sc Engl, Kunisch} and {sc Neubauer} presented the fundamentals of a systematic theory for convergence rates in Tikhonov regularization $$|F(u)-v^delta|^2+alpha |u-u^*|^2 o min$$ of nonlinear ill-posed problems $F(u)=v$ with solutions $u^dagger$ and noisy data $u^delta$ in a Hilbert space setting. On the one hand, results are based on structural conditions concerning the nonlinearity of $F$, in principle Lipschitz continuity of a Frechet derivative $F^prime(u)$ in a neighbourhood of $u^dagger$. On the other hand, source conditions $u^dagger-u^*=F^prime(u^dagger)^* w $ concerning the solution smoothness with some additional smallness assumption on $|w|$ are required. In this talk, following the lines of [3], [4] and [5], we show that both nonlinearity and smoothness conditions can be expressed by variational inequalities in a unified manner characterizing the capability of yielding convergence rates. This approach essentially exploits the concept of variable Hilbert scales developed by {sc Hegland} in the seminal paper [2]. In this context, we also extend the ideas of approximate source conditions presented in [1] for linear ill-posed problems to the nonlinear case. To formulate the results in a Banach space setting we use Bregman distances for measuring the regularization error.
BIO:
http://www.tu-chemnitz.de/mathematik/inverse_probleme/hofmann/
