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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}
}

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