Speed-up of the Monte Carlo method by using a physical model of the Dempster-Shafer theory

Abstract

By using the Monte Carlo method, we can obtain the minimum value of a function V(r) that is generally associated with the potential energy. In this paper we present a method that makes it possible to speed up the classical Monte Carlo method. The new method is based on the observation that the Bolzmann transition probability and the concept of local thermodynamical equilibrium give rise to an initial state of maximum entropy, which is subsequently modified by using the information on the internal structure of the system. The classical thermodynamic model does not take into account any structures inside the system, and therefore in many cases does not accurately model the system itself. In an attempt to take into account the internal structure of the system, we propose a physical model of the belief measure as defined in the Dempster-Shafer theory. The recent discovery by Resconi of an algorithm to calculate the probability distribution that has previously been developed by Harmanec and Klir, and which is consistent with the belief measure, opens the way to utilizing the Bolzmann distribution not only with a uniform distribution of probability, but instead with an arbitrary distribution of probability to guide the Monte Carlo iterative method to obtain the global minimum value of the potential energy. Starting from local thermodynamic equilibrium (i.e., local symmetry), the algorithm computes a new distribution over subsystems, resulting in a nonuniform distribution and in symmetry breaking. In the general case one can start with a different initial distribution induced by other local symmetries, corresponding to specific differential equations (e.g., the Fokker-Planck equation) and calculate from this the global distribution corresponding to the breaking of local symmetry. © 1998 John Wiley & Sons, Inc.