title: Comparative Monte Carlo efficiency by Monte Carlo analysis.
abstract: We propose a modified power method for computing the subdominant eigenvalue λ{2} of a matrix or continuous operator. While useful both deterministically and stochastically, we focus on defining simple Monte Carlo methods for its application. The methods presented use random walkers of mixed signs to represent the subdominant eigenfunction. Accordingly, the methods must cancel these signs properly in order to sample this eigenfunction faithfully. We present a simple procedure to solve this sign problem and then test our Monte Carlo methods by computing λ{2} of various Markov chain transition matrices. As |λ{2}| of this matrix controls the rate at which Monte Carlo sampling relaxes to a stationary condition, its computation also enabled us to compare efficiencies of several Monte Carlo algorithms as applied to two quite different types of problems. We first computed λ{2} for several one- and two-dimensional Ising models, which have a discrete phase space, and compared the relative efficiencies of the Metropolis and heat-bath algorithms as functions of temperature and applied magnetic field. Next, we computed λ{2} for a model of an interacting gas trapped by a harmonic potential, which has a mutidimensional continuous phase space, and studied the efficiency of the Metropolis algorithm as a function of temperature and the maximum allowable step size Δ. Based on the λ{2} criterion, we found for the Ising models that small lattices appear to give an adequate picture of comparative efficiency and that the heat-bath algorithm is more efficient than the Metropolis algorithm only at low temperatures where both algorithms are inefficient. For the harmonic trap problem, we found that the traditional rule of thumb of adjusting Δ so that the Metropolis acceptance rate is around 50% is often suboptimal. In general, as a function of temperature or Δ , λ{2} for this model displayed trends defining optimal efficiency that the acceptance ratio does not. The cases studied also suggested that Monte Carlo simulations for a continuum model are likely more efficient than those for a discretized version of the model.
authors.affiliation: Chemistry Department, Columbia University, New York, New York 10027, USA.
authors.forename: B M
authors.initials: BM
authors.lastname: Rubenstein
authors.forename: J E
authors.initials: JE
authors.lastname: Gubernatis
authors.forename: J D
authors.initials: JD
authors.lastname: Doll