Stable Explicit p-Laplacian Flows Based on Nonlinear Eigenvalue Analysis

Ido Cohen, Adi Falik, Guy Gilboa

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

6 Scopus citations

Abstract

Implementation of nonlinear flows by explicit schemes can be very convenient, due to their simplicity and low-computational cost per time step. A well known drawback is the small time step bound, referred to as the CFL condition, which ensures a stable flow. For p-Laplacian flows, with (Formula Presented), explicit schemes without gradient regularization require, in principle, a time step approaching zero. However, numerical implementations show explicit flows with small time-steps are well behaved. We can now explain and quantify this phenomenon. In this paper we examine explicit p-Laplacian flows by analyzing the evolution of nonlinear eigenfunctions, with respect to the p-Laplacian operator. For these cases analytic solutions can be formulated, allowing for a comprehensive analysis. A generalized CFL condition is presented, relating the time step to the inverse of the nonlinear eigenvalue. Moreover, we show that the flow converges and formulate a bound on the error of the discrete scheme. Finally, we examine general initial conditions and propose a dynamic time-step bound, which is based on a nonlinear Rayleigh quotient.

Original languageEnglish
Title of host publicationScale Space and Variational Methods in Computer Vision - 7th International Conference, SSVM 2019, Proceedings
EditorsJan Lellmann, Jan Modersitzki, Martin Burger
Pages315-327
Number of pages13
DOIs
StatePublished - 2019
Externally publishedYes
Event7th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2019 - Hofgeismar, Germany
Duration: 30 Jun 20194 Jul 2019

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume11603 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference7th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2019
Country/TerritoryGermany
CityHofgeismar
Period30/06/194/07/19

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