Renormalization Multigrid (RMG): 
               Statistically Optimal Renormalization Group Flow 
                 and Coarse-to-Fine Monte Carlo Acceleration 

                          Achi Brandt and Dorit Ron 
            Department of Applied Mathematics and Computer Science,
             Weizmann Institute of Science, Rehovot 76100, Israel 

                                   Abstract

New renormalization-group algorithms are developed with adaptive
representations of the renormalized action which automatically express only
significant interactions.  As the amount of statistics grows, more
interactions enter, thereby systematically reducing the truncation error.
This allows statistically optimal calculation of thermodynamic limits, in the
sense that it achieves accuracy e in just O(1/(e*e)) random number
generations.  There are practically no finite-size effects and the
renormalization transformation can be repeated arbitrarily many times.
Consequently, the desired fixed point is obtained and the correlation-length
critical exponent $\nu$ is extracted.In addition, we introduce a new
multiscale coarse-to-fine acceleration method, based on a multigrid-like
approach.  This general (non-cluster) algorithm generates independent
equilibrium configurations without slow down.  A particularly simple version
of it can be used at criticality.  The methods are of great generality; here
they are demonstrated on the 2D Ising model.

Key words.  Ising model, Renormalization Multigrid, P+ probabilities,
neighborhoods, criticalization, coarse-to-fine Monte Carlo acceleration,
Compatible Monte Carlo, Post Relaxation.