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International Journal of Modern Physics C

PARALLEL ACCELERATION ALGORITHM FOR SPIN MODELS

Gyan V Bhanot, Stephen L. Adler

Published Aug 1, 1992• 5 citations
Publisher

Abstract

We describe and implement a multi-scale acceleration algorithm for spin models on a massively parallel supercomputer, the Connection Machine CM-200. Unlike usual cluster algorithms, our algorithm is completely parallelizable. The time to update all variables in a system of volume Ld scales as Ld log 2L. We prove this by computing the time for one lattice sweep for the 2-d XY model for our algorithm on lattices of size up to 2048×2048.

DOI
10.1142/s0129183192000397

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We describe and implement a multi-scale acceleration algorithm for spin models on a massively parallel supercomputer, the Connection Machine CM-200. Unlike usual cluster algorithms, our algorithm is completely parallelizable. The time to update all variables in a system of volume Ld scales as Ld log 2L. We prove this by computing the time for one lattice sweep for the 2-d XY model for our algorithm on lattices of size up to 2048×2048.
Parallel computers are ideally suited to the Monte Carlo simulation of spin models using the standard Metropolis algorithm, since it is regular and local. However local algorithms have the major drawback that near a phase transition the number of sweeps needed to generate a statistically independent configuration increases as the square of the lattice size. New algorithms have recently been developed which dramatically reduce this `critical slowing down' by updating clusters of spins at a time. The highly irregular and non-local nature of these algorithms means that they are much more difficult to parallelize efficiently. Here we introduce the new cluster algorithms, explain some sequential algorithms for identifying and labelling connected clusters of spins, and then outline some parallel algorithms which have been implemented on MIMD machines. 1. Introduction Computer simulations are extremely useful in the study of spin models in condensed matter physics. In these models the spins ...
A comparison between single-cluster and single-spin algorithms is made for the Ising model in 2 and 3 dimensions. We compare the amount of computer time needed to achieve a given level of statistical accuracy, rather than the speed in terms of site updates per second or the dynamical critical exponents. Our main result is that the cluster algorithms become more efficient when the system size, Ld, exceeds, L~70–300 for d=2 and l~80–200 for d=3. The exact value of the crossover is dependent upon the computer being used. The lower end of the crossover range is typical of workstations while the higher end is typical of vector computers. Hence, even for workstations, the system sizes needed for efficient use of the cluster algorithm is relatively large.
An implementation of the Lanczos algorithm for the exact diagonalization of the two dimensional Hubbard model on a 4×4 square lattice on the Connection Machine CM-2 system is described. The CM-2 is a massively parallel machine with distributed memory. The program is written in C/PARIS. This implementation minimizes memory usage by generating the matrix elements as needed instead of storing them. The Lanczos vectors are stored across the local memory of the processors. Using translational symmetry only, the dimension of the Hilbert space at half filling is more than 10 million. A speed of about 2.4 min per iteration is achieved on a 64K CM-2. This implementation is scalable. Running it on a bigger machine with more processors speeds up the process. The performance analysis of this implementation is shown and discuss its advantages and disadvantages are discussed.
One of the vital roles of computing is to solve large-scale combinatorial optimization problems in a short time. In recent years, methods have been proposed that map optimization problems to ones of searching for the ground state of an Ising model by using a stochastic process. Simulated annealing (SA) is a representative algorithm. However, it is inherently difficult to perform a parallel search. Here we propose an algorithm called momentum annealing (MA), which, unlike SA, updates all spins of fully connected Ising models simultaneously and can be implemented on GPUs that are widely used for scientific computing. MA running in parallel on GPUs is 250 times faster than SA running on a modern CPU at solving problems involving 100 000 spin Ising models.