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Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Finite population-size effects in projection Monte Carlo methods

Nicolas J. Cerf, Olivier Martin

Published Apr 1, 1995• 22 citations
Publisher

Abstract

Projection (Green's function and diffusion) MOnte Carlo techniques sample a wave function by a stochastic iterative procedure. These method converge to a stationary distribution which is biased, i.e., differs from the exact ground state wave function. This bias occurs because of the use of population control procedures. We demonstrate that these biased Monte Carlo algorithms lead to a modified effective mass which is equal to the desired mass only in the limit of an infinite population of walkers. In general, the bias for the energy scales as 1/N for a population of walkers of size N and is proportional to the expectation value of the kinetic energy. Finally, we consider various strategies to reduce this bias.

DOI
10.1103/physreve.51.3679

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Projection (Green's function and diffusion) MOnte Carlo techniques sample a wave function by a stochastic iterative procedure. These method converge to a stationary distribution which is biased, i.e., differs from the exact ground state wave function. This bias occurs because of the use of population control procedures. We demonstrate that these biased Monte Carlo algorithms lead to a modified effective mass which is equal to the desired mass only in the limit of an infinite population of walkers. In general, the bias for the energy scales as 1/N for a population of walkers of size N and is proportional to the expectation value of the kinetic energy. Finally, we consider various strategies to reduce this bias.
The size of the population of random walkers required to obtain converged estimates in diffusion Monte Carlo (DMC) increases dramatically with system size. We illustrate this by comparing ground state energies of small clusters of parahydrogen (up to 48 molecules) computed by DMC and path integral ground state (PIGS) techniques. We contend that the bias associated with a finite population of walkers is the most likely cause of quantitative numerical discrepancies between PIGS and DMC energy estimates reported in the literature, for this few-body Bose system. We discuss the viability of DMC as a general-purpose ground state technique, and argue that PIGS, and even finite temperature methods, enjoy more favorable scaling, and are therefore a superior option for systems of large size.
We explore the existence of a residual population bias in Diffusion Monte Carlo algorithms. We develop a model problem, which mimics the single-particle, one dimensional simple harmonic oscillator of unitary ground state energy, to study these biases. We then apply an importance sampling algorithms with a population of a single random walker to calculate the ground state energy and wavefunction of our modeled system in cases of both perfect and imperfect importance sampling. In the former case, we perfectly calculate both the ground state energy and corresponding wavefunction of our system, thus validating our calculations and assumed answer. In the latter case, we show the existence of a residual population bias for populations of small numbers of random walkers. We demonstrate the convergence of our small population calculations and show that we cannot remove this bias through a number of single walker methods. Finally, we demonstrate the overall decay of this residual population bias as the population of random walkers expands, allowing our calculations to converge to the true answer. Key Words: Diffusion Monte Carlo, Population Dynamics, Random Walk
In this work we present a detailed study of the fermion Monte Carlo algorithm (FMC), a recently proposed stochastic method for calculating fermionic ground-state energies. A proof that the FMC method is an exact method is given. In this work the stability of the method is related to the difference between the lowest (bosonic-type) eigenvalue of the FMC diffusion operator and the exact Fermi energy. It is shown that within a FMC framework the lowest eigenvalue of the new diffusion operator is no longer the bosonic ground-state eigenvalue as in standard exact diffusion Monte Carlo (DMC) schemes but a modified value which is strictly greater. Accordingly, FMC can be viewed as an exact DMC method built from a correlated diffusion process having a reduced Bose–Fermi gap. As a consequence, the FMC method is more stable than any transient method (or nodal release-type approaches). It is shown that the most recent ingredient of the FMC approach (Kalos M H and Pederiva F 2000 Phys. Rev. Lett. 85 3547), namely the introduction of non-symmetric guiding functions, does not necessarily improve the stability of the algorithm. We argue that the stability observed with such guiding functions is in general a finite-size population effect disappearing for a very large population of walkers. The counterpart of this stability is a control population error which is different in nature from the standard diffusion Monte Carlo algorithm and which is at the origin of an uncontrolled approximation in FMC. We illustrate the various ideas presented in this work with calculations performed on a very simple model having only nine states but a full ‘sign problem’. Already for this toy model it is clearly seen that FMC calculations are inherently uncontrolled.
Population control is an essential component of any projector Monte Carlo algorithm. This control mechanism usually introduces a bias in the sampled quantities that is inversely proportional to the population size. In this paper, we investigate the population control bias in the full configuration interaction quantum Monte Carlo method. We identify the precise origin of this bias and quantify it in general. We show that it has different effects on different estimators and that the shift estimator is particularly susceptible. We derive a reweighting technique, similar to the one used in diffusion Monte Carlo, for correcting this bias and apply it to the shift estimator. We also show that by using importance sampling, the bias can be reduced substantially. We demonstrate the necessity and the effectiveness of applying these techniques for sign-problem-free systems where this bias is especially notable. Specifically, we show results for large one-dimensional Hubbard models and the two-dimensional Heisenberg model, where corrected FCIQMC results are comparable to the other high-accuracy results.
Projection methods such as Green's function and diffusion Monte Carlo are commonly used to calculate the leading eigenvalue and eigenvector of operators or large matrices. They thereby give access to ground state properties of quantum systems, and finite temperature properties of classical statistical mechanical systems having a transfer matrix. The basis of these approaches is a stochastic application of the power method in which a "projection" operator is applied iteratively. For the systematic errors to be small, the number of iterations must be large; however, in that limit, the statistical errors grow tremendously. We present an analytical study of the main variance reduction methods used for dealing with this problem. In particular, we discuss the consequences of guiding, replication, and population control on statistical and systematic errors.
We investigate a systematic statistical bias found in full configuration quantum Monte Carlo (FCIQMC) that originates from controlling a walker population with a fluctuating shift parameter. This bias can become the dominant error when the sign problem is absent, e.g., in bosonic systems. FCIQMC is a powerful statistical method for obtaining information about the ground state of a sparse and abstract matrix. We show that, when the sign problem is absent, the shift estimator has the nice property of providing an upper bound for the exact ground-state energy and all projected energy estimators, while a variational estimator is still an upper bound to the exact energy with substantially reduced bias. A scalar model of the general FCIQMC population dynamics leads to an exactly solvable It\^o stochastic differential equation. It provides further insights into the nature of the bias and gives accurate analytical predictions for delayed cross-covariance and autocovariance functions of the shift energy estimator and the walker number. The model provides a toehold on finding a cure for the population control bias. We provide evidence for nonuniversal power-law scaling of the population control bias with walker number in the Bose-Hubbard model for various estimators of the ground-state energy based on the shift or on projected energies. For the specific case of the noninteracting Bose-Hubbard Hamiltonian we obtain a full analytical prediction for the bias of the shift energy estimator.