Zero-Variance Principle for Monte Carlo Algorithms

Abstract

We present a general approach to greatly increase at little cost the efficiency of Monte Carlo algorithms. To each observable to be computed we associate a renormalized observable (improved estimator) having the same average but a different variance. By writing down the zero-variance condition a fundamental equation determining the optimal choice for the renormalized observable is derived (zerovariance principle for each observable separately). We show, with several examples including classical and quantum Monte Carlo calculations, that the method can be very powerful.

Annihilation rate in positronic systems by quantum Monte Carlo: e+LiH as test case

“…Always starting from Eq. 2, one could also exploit the approach proposed by Assaraf and Caffarel [15] to compute the expectation value needed to obtain nuclear forces by means of the HellmanFeynmann theorem. They showed that a judicious choice of a renormalized operator, whose mean value is equal to the original one, can reduce the infinite variance to a finite value [16].…”

Geometry optimization in quantum Monte Carlo with solution mapping: Application to formaldehyde

“…16,17 In this approach, terms that lead to large variance are canceled, making it possible to evaluate the average force with a relatively small variance. Recently Casalegno et al 18 calculated forces by introducing correction terms to the Hellmann-Feynman force as proposed by Pulay.…”

Transition matrix Monte Carlo method

“…With exception of quasi Monte Carlo [2] for numerical integration, as long as we use a probabilistic approach, it does not appear possible to overcome this barrier. Most of the work to improve the efficiency of Monte Carlo method has been via variance reduction [3,4], which reduces the value of coefficient in front of the 1/ √ t law. Next, while the traditional Monte Carlo method is good for computing expectation values such as the internal energy and its derivatives, it is more difficult to compute the free energy or entropy [5].…”