Abstract:

The integration to steady state of many initial value ODEs and PDEs
using the forward Euler method can alternatively be considered as
gradient descent for an associated minimization problem. Greedy
algorithms such as steepest descent for determining the step size are as
slow to reach steady state as is forward Euler integration with the
best uniform step size. But other, much faster methods using bolder
step size selection exist. Various alternatives are investigated from
both theoretical and practical points of view.

The steepest descent method is also known for the regularizing or
smoothing effect that the first few steps have for certain inverse
problems, amounting to a finite time regularization. We further
investigate the retention of this property using the faster gradient
descent variants in the context of two or three applications:
denoising and deblurring of images, and shape optimization involving
data inversion of elliptic PDEs. When the combination of
regularization and accuracy demands more than a dozen or so steepest
descent steps, the alternatives offer an advantage, even though
(indeed because) the absolute stability limit of forward Euler is
carefully yet severely violated.

(Joint work with Kees van den Doel, Hui Huang and Benar Svaiter)