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SUMMARY:Efficient and Stable Schemes for 2D Forward-and-Backward Diffusion
  - Joachim Weickert (Universität des Saarlandes )
DTSTART:20170904T085000Z
DTEND:20170904T094000Z
UID:TALK77601@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:<span>Co-author: Martin Welk		(UMIT Hall\, Austria)        <br
 ></span><span><br>Image enhancement with forward-and-backward (FAB) diffus
 ion is numerically very challenging due to its negative diffusivities. As 
 a remedy\, we first extend the explicit nonstandard scheme by Welk et al. 
 (2009) from the 1D scenario to the practically relevant two-dimensional se
 tting. We prove that under a fairly severe time step restriction\, this 2D
  scheme preserves a maximum--minimum principle. Moreover\, we find an inte
 resting Lyapunov sequence which guarantees convergence to a flat steady st
 ate. Since a global application of the time step size restriction leads to
  very slow algorithms and is more restrictive than necessary for most pixe
 ls\, we introduce a much more efficient scheme with locally adapted time s
 tep sizes. It applies diffusive interactions of adjacent pixel  pairs in a
  randomized order and adapts the time step size locally. These space-varia
 nt time steps are synchronized at sync times which are  determined by stab
 ility properties of the explicit forward diffusion scheme. Experiments sho
 w that our novel two-pixel scheme allows to compute FAB diffusion with gua
 ranteed stability in the maximum norm at a speed that can be three orders 
 of magnitude larger than its explicit counterpart with a global time step 
 size.&nbsp\;</span>
LOCATION:Seminar Room 1\, Newton Institute
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