  We describe a reverse integration approach for the exploration of
low-dimensional effective potential landscapes. Coarse reverse integration
initialized on a ring of coarse states enables efficient "navigation" on the
landscape terrain: escape from local effective potential wells, detection of
saddle points, and identification of significant transition paths between
wells. We consider several distinct ring evolution modes: backward stepping in
time, solution arc--length, and effective potential. The performance of these
approaches is illustrated for a deterministic problem where the energy
landscape is known explicitly. Reverse ring integration is then applied to
"noisy" problems where the ring integration routine serves as an outer
"wrapper" around a forward-in-time inner simulator. Three versions of such
inner simulators are considered: a system of stochastic differential equations,
a Gillespie--type stochastic simulator, and a molecular dynamics simulator. In
these "equation-free" computational illustrations, estimation techniques are
applied to the results of short bursts of "inner" simulation to obtain the
unavailable (in closed form) quantities (local drift and diffusion coefficient
estimates) required for reverse ring integration; this naturally leads to
approximations of the effective landscape.

Chemical Physics