Differentiating densities on smooth manifolds

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 smoo...

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Bibliographic Details
Published in:arXiv.org 2021-03
Main Authors: Sliwiak, Adam A, Wang, Qiqi
Format: Article
Language:English
Subjects:
Computing costs
Density
Derivatives
Differential equations
Manifolds
Mathematical analysis
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2103.07380