Artificial time integration |
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Authors: | UM Ascher H Huang K van den Doel |
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Institution: | (1) Department of Computer Science, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada;(2) Institute of Applied Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada |
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Abstract: | Many recent algorithmic approaches involve the construction of a differential equation model for computational purposes, typically
by introducing an artificial time variable. The actual computational model involves a discretization of the now time-dependent differential system, usually
employing forward Euler. The resulting dynamics of such an algorithm is then a discrete dynamics, and it is expected to be
“close enough” to the dynamics of the continuous system (which is typically easier to analyze) provided that small – hence
many – time steps, or iterations, are taken. Indeed, recent papers in inverse problems and image processing routinely report
results requiring thousands of iterations to converge. This makes one wonder if and how the computational modeling process
can be improved to better reflect the actual properties sought.
In this article we elaborate on several problem instances that illustrate the above observations. Algorithms may often lend
themselves to a dual interpretation, in terms of a simply discretized differential equation with artificial time and in terms
of a simple optimization algorithm; such a dual interpretation can be advantageous. We show how a broader computational modeling
approach may possibly lead to algorithms with improved efficiency.
AMS subject classification (2000) 65L05, 65M32, 65N21, 65N22, 65D18 |
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Keywords: | artificial time discrete dynamics continuous dynamics geometric integration level set shape optimization surface mesh smoothing |
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