Absolute Monte Carlo estimation of integrals and partition functions

Abstract

Owing to their favorable scaling with dimensionality, Monte Carlo (MC) methods have become the tool of choice for numerical integration across the quantitative sciences. Almost invariably, efficient MC integration schemes are strictly designed to compute ratios of integrals, their efficiency being intimately tied to the degree of overlap between the given integrands. Consequently, substantial user insight is required prior to the use of such methods, either to mitigate the oft-encountered lack of overlap in ratio computations, or to find closely related integrands of known quadrature in absolute integral estimation. Here a simple physical idea--measuring the volume of a container by filling it up with an ideal gas--is exploited to design a new class of MC integration schemes that can yield efficient, absolute integral estimates for a broad class of integrands with simple transition matrices as input. The methods are particularly useful in cases where existing (importance sampling) strategies are most demanding, namely when the integrands are concentrated in relatively small and unknown regions of configuration space (e.g. physical systems in ordered/low-temperature phases). Examples ranging from a volume with infinite support to the partition function of the 2D Ising model are provided to illustrate the application and scope of the methods.