 
Summary: Artificial time integration
U. Ascher
H. Huang
K. van den Doel
November 11, 2006
Abstract
Many recent algorithmic approaches involve the construction of a differen
tial equation model for computational purposes, typically by introducing an
artificial time variable. The actual computational model involves a discretiza
tion of the now timedependent differential system, usually employing forward
Euler. The resulting dynamics of such an algorithm is then a discrete dynam
ics, 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 interpre
tation, in terms of a simply discretized differential equation with artificial time
