Biased Metropolis Sampling for Rugged Free Energy Landscapes

Metropolis simulations of all-atom models of peptides (i.e. small proteins) are considered. Inspired by the funnel picture of Bryngelson and Wolyness, a transformation of the updating probabilities of the dihedral angles is defined, which uses probability densities from a higher temperature to improve the algorithmic performance at a lower temperature. The method is suitable for canonical as well as for generalized ensemble simulations. A simple approximation to the full transformation is tested at room temperature for Met-Enkephalin in vacuum. Integrated autocorrelation times are found to be reduced by factors close to two and a similar improvement due to generalized ensemble methods enters multiplicatively.Comment: Plenary talk at the Los Alamos conference, The Monte Carlo Method in Physical Sciences: Celebrating the 50th Anniversary of the Metropolis Algorithm, to appear in the proceedings, 11 pages, 4 figures, one table. Inconsistencies corrected and references adde

New Algorithm to Investigate Neural Networks

Random cost simulations were introduced as a method to investigate optimization problems in systems with conflicting constraints. Here I study the approach in connection with the training of a feed-forward multilayer perceptron, as used in high energy physics applications. It is suggested to use random cost simulations for generating a set of selected configurations. On each of those final minimization may then be performed by a standard algorithm. For the training example at hand many almost degenerate local minima are thus found. Some effort is spend to discuss whether they lead to equivalent classifications of the data.Comment: 16 pages and 8 figures. Typos in eqn.(1) and various misleading formulations eliminate

The lowest crossing in 2D critical percolation

We study the following problem for critical site percolation on the triangular lattice. Let A and B be sites on a horizontal line e separated by distance n. Consider, in the half-plane above e, the lowest occupied crossing R from the half-line left of A to the half-line right of B. We show that the probability that R has a site at distance smaller than m from AB is of order (log (n/m))^{-1}, uniformly in 1 <= m < n/2. Much of our analysis can be carried out for other two-dimensional lattices as well.Comment: 16 pages, Latex, 2 eps figures, special macros: percmac.tex. Submitted to Annals of Probabilit