Differentiating densities on smooth manifolds

TL;DR

This paper develops formulas for computing the density gradient function on smooth manifolds, improving Monte Carlo integration of oscillatory functions in computational science.

Contribution

It provides analytical formulas for the density gradient on smooth manifolds, addressing a key challenge in integrating oscillatory functions over complex domains.

Findings

Derived explicit formulas for the density gradient on manifolds.

Demonstrated the importance of the density gradient in Monte Carlo integration.

Showcased numerical examples involving oscillatory integrands.

Abstract

Lebesgue integration of derivatives of strongly-oscillatory functions is a recurring challenge in computational science and engineering. Integration by parts is an effective remedy for huge computational costs associated with Monte Carlo integration schemes. In case of Lebesgue integrals over a smooth manifold, however, integration by parts gives rise to a derivative of the density implied by charts describing the domain manifold. This paper focuses on the computation of that derivative, which we call the density gradient function, on general smooth manifolds. We analytically derive formulas for the density gradient and present examples of manifolds determined by popular differential equation-driven systems. We highlight the significance of the density gradient by demonstrating a numerical example of Monte Carlo integration involving oscillatory integrands.