Sampling and Change of Measure for Monte Carlo Integration on Simplices

Abstract

Simplices are the fundamental domain when integrating over convex polytopes. The aim of this work is to establish a novel framework of Monte Carlo integration over simplices, throughout from sampling to variance reduction. Namely, we develop a uniform sampling method on the standard simplex consisting of two independent procedures and construct theories on change of measure on each of the two independent elements in the developed sampling technique with a view towards variance reduction by importance sampling. We provide illustrative figures and numerical results to support our theoretical findings and demonstrate the strong potential of the developed framework for effective implementation and acceleration of Monte Carlo integration over simplices.

A Simple Algorithm for Converting Random Number Generator Outputs to Universal Distributions to Aid Teaching and Research in Modern Physical Chemistry

“…Educators could use this as a chance to describe other numerical integration methods, such as Riemann sums or Simpson's rule, or more sophisticated methods, such as Gaussian quadrature or Monte Carlo integration algorithms. 12,13 The grid density and number of bins could be tested by the students as an exercise, experimenting with the algorithm's behavior and convergence.…”