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Monte carlo algorithm for the double exchange model optimized for parallel computations

Nobuo Furukawa, Yukitoshi Motome, Hisaho Nakata

Published Nov 15, 2012• 16 citations
OpenAlex

Abstract

A new algorithm for Monte Carlo calculation of the double exchange model is studied. The algorithm is commonly applicable to wide classes of strongly correlated electron systems which involve itinerant electrons coupled with thermodynamically fluctuating fields. Using moment expansions of the density of states with Chebyshev polynomials, the algorithm provides an efficient calculation on large size clusters, especially on parallel computers. Benchmark calculations are performed on Beowulf-type cluster systems with over 100 CPUs in parallel. PACS code: 02.70.Lq 71.10.Fd 75.10.-b 75.70.Pa

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A new algorithm for Monte Carlo calculation of the double exchange model is studied. The algorithm is commonly applicable to wide classes of strongly correlated electron systems which involve itinerant electrons coupled with thermodynamically fluctuating fields. Using moment expansions of the density of states with Chebyshev polynomials, the algorithm provides an efficient calculation on large size clusters, especially on parallel computers. Benchmark calculations are performed on Beowulf-type cluster systems with over 100 CPUs in parallel. PACS code: 02.70.Lq 71.10.Fd 75.10.-b 75.70.Pa
A new Monte Carlo method is proposed for fermion systems interacting with classical degrees of freedom. To obtain a weight for each Monte Carlo sample with a fixed configuration of classical variables, the moment expansion of the density of states by Chebyshev polynomials is applied instead of the direct diagonalization of the fermion Hamiltonian. This reduces a cpu time to scale as O(Ndim2log⁡Ndim)O(N_{\rm dim}^{2} \log N_{\rm dim})O(Ndim2​logNdim​) compared to O(Ndim3)O(N_{\rm dim}^{3})O(Ndim3​) for the diagonalization in the conventional technique; NdimN_{\rm dim}Ndim​ is the dimension of the Hamiltonian. Another advantage of this method is that parallel computation with high efficiency is possible. These significantly save total cpu times of Monte Carlo calculations because the calculation of a Monte Carlo weight is the bottleneck part. The method is applied to the double-exchange model as an example. The benchmark results show that it is possible to make a systematic investigation using a system-size scaling even in three dimensions within a realistic cpu timescale.
Models of fermions interacting with classical degrees of freedom are applied to a large variety of systems in condensed matter physics. For this class of models, Weiße [Phys. Rev. Lett. 102 , 150604 (2009)] has recently proposed a very efficient numerical method, called O( N ) Green-Function-Based Monte Carlo (GFMC) method, where a kernel polynomial expansion technique is used to avoid the full numerical diagonalization of the fermion Hamiltonian matrix of size N , which usually costs O( N 3 ) computational complexity. Motivated by this background, in this paper we apply the GFMC method to the double exchange model in three spatial dimensions. We mainly focus on the implementation of GFMC method using both MPI on a CPU-based cluster and Nvidia's Compute Unified Device Architecture (CUDA) programming techniques on a GPU-based (Graphics Processing Unit based) cluster. The time complexity of the algorithm and the parallel implementation details on the clusters are discussed. We also show the performance scaling for increasing Hamiltonian matrix size and increasing number of nodes, respectively. The performance evaluation indicates that for a 32 3 Hamiltonian a single GPU shows higher performance equivalent to more than 30 CPU cores parallelized using MPI.
The exponential growth in computer power over the past few decades has been a huge boon to computational chemistry, physics, biology, and materials science. Now, a standard workstation or Linux cluster can calculate semi-quantitative properties of moderately sized systems. The next step in computational science is developing better algorithms which allow quantitative calculations of a system's properties. A relatively new class of algorithms, known collectively as Quantum Monte Carlo (QMC), has the potential to quantitatively calculate the properties of molecular systems. Furthermore, QMC scales as O(N3)O(N^3)O(N3) or better. This makes possible very high-level calculations on systems that are too large to be examined using standard high-level methods. This thesis develops (1) an efficient algorithm for determining "on-the-fly" the statistical error in serially correlated data, (2) a manager-worker parallelization algorithm for QMC that allows calculations to run on heterogeneous parallel computers and computational grids, (3) a robust algorithm for optimizing Jastrow functions which have singularities for some parameter values, and (4) a proof-of-concept demonstrating that it is possible to find transferable parameter sets for large classes of compounds.
An improved algorithm is proposed for Monte Carlo methods to study fermion systems interacting with adiabatical fields. To obtain a weight for each Monte Carlo sample with a fixed configuration of adiabatical fields, a series expansion using Chebyshev polynomials is applied. By introducing truncations of matrix operations in a systematic and controlled way, it is shown that the cpu time is reduced from O(N^3) to O(N) where N is the system size. Benchmark results show that the implementation of the algorithm makes it possible to perform systematic investigations of critical phenomena using system-size scalings even for an electronic model in three dimensions, within a realistic cpu timescale.
Coupled cluster theory is a vital cornerstone of electronic structure theory and is being applied to ever-larger systems. Stochastic approaches to quantum chemistry have grown in importance and offer compelling advantages over traditional deterministic algorithms in terms of computational demands, theoretical flexibility or lower scaling with system size. We present a highly parallelizable algorithm of the coupled cluster Monte Carlo method involving sampling of clusters of excitors over multiple time steps. The behaviour of the algorithm is investigated on the uniform electron gas and the water dimer at CCSD, CCSDT and CCSDTQ levels. We also describe two improvements to the original sampling algorithm, full non-composite and multi-spawn sampling. A stochastic approach to coupled cluster results in an efficient and scalable implementation at arbitrary truncation levels in the coupled cluster expansion.
Coupled cluster theory is a vital cornerstone of electronic structure theory and is being applied to ever-larger systems. Stochastic approaches to quantum chemistry have grown in importance and offer compelling advantages over traditional deterministic algorithms in terms of computational demands, theoretical flexibility, or lower scaling with system size. We present a highly parallelizable algorithm of the coupled cluster Monte Carlo method involving sampling of clusters of excitors over multiple time steps. The behavior of the algorithm is investigated on the uniform electron gas and the water dimer at coupled-cluster levels including up to quadruple excitations. We also describe two improvements to the original sampling algorithm, full non-composite, and multi-spawn sampling. A stochastic approach to coupled cluster results in an efficient and scalable implementation at arbitrary truncation levels in the coupled cluster expansion.