Research reports

Years: 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

Rate of convergence of regularization procedures and Finite Element approximations for the total variation flow

by X. Feng and M. von Oehsen and A. Prohl

(Report number 2003-12)

Abstract
Following the work of [12], in this paper we continue to carry out mathematical and numerical analysis of the gradient flow for the total variation functional, which has applications to image processing, geometric analysis and materials sciences. The main objectives of the paper are to study the long time behavior of the total variation flow, to analyze rate of convergence for regularization procedures and finite element approximations for the total variation gradient flow with $L^2$ initial data, and to derive explicit scaling laws which relate mesh parameters to the regularization parameter. We also provide numerical experiments which complement our theoretical results. In addition, we present an a priori (model) error estimate for the total variation image denoising model of Rudin-Osher-Fatemi [17] and for other related image denoising models. This (model) error estimate provides a theoretical explanation for the good performance of the total variation model in image denoising.

Keywords: Bounded variation, gradient flow, variational inequality, finite elements, a priori error analysis

@Techreport{FvP03_324,
  author = {X. Feng and M. von Oehsen and A. Prohl},
  title = {Rate of convergence of regularization procedures and Finite Element approximations for the total variation flow},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2003-12},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2003/2003-12.pdf },
  year = {2003}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).