Multidimensional integration through Markovian sampling under steered function morphing: a physical guise from statistical mechanics

TL;DR

This paper introduces a novel computational method for multidimensional integration using Markovian sampling and a physical analogy from statistical mechanics, specifically leveraging Jarzynski's equality, to improve efficiency and accuracy.

Contribution

The paper develops a new integration technique combining stochastic thermodynamics principles with Markovian sampling, extending Jarzynski's equality to multidimensional integrals with sign-changing functions.

Findings

Demonstrates improved computational efficiency and accuracy.

Provides a formulation for integrands with zeros and sign changes.

Shows similarity to Annealed Importance Sampling in a physical context.

Abstract

We present a computational strategy for the evaluation of multidimensional integrals on hyper-rectangles based on Markovian stochastic exploration of the integration domain while the integrand is being morphed by starting from an initial appropriate profile. Thanks to an abstract reformulation of Jarzynski's equality applied in stochastic thermodynamics to evaluate the free-energy profiles along selected reaction coordinates via non-equilibrium transformations, it is possible to cast the original integral into the exponential average of the distribution of the pseudo-work (that we may term "computational work") involved in doing the function morphing, which is straightforwardly solved. Several tests illustrate the basic implementation of the idea, and show its performance in terms of computational time, accuracy and precision. The formulation for integrand functions with zeros and possible sign changes is also presented. It will be stressed that our usage of Jarzynski's equality shares similarities with a practice already known in statistics as Annealed Importance Sampling (AIS), when applied to computation of the normalizing constants of distributions. In a sense, here we dress the AIS with its "physical" counterpart borrowed from statistical mechanics.