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 talk 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.