Monte Carlo on Manifolds: Sampling Densities and Integrating Functions

Emilio Zappa, Miranda Holmes-Cerfon, Jonathan Goodman

Research output: Contribution to journalArticle

Abstract

We describe and analyze some Monte Carlo methods for manifolds in euclidean space defined by equality and inequality constraints. First, we give an MCMC sampler for probability distributions defined by unnormalized densities on such manifolds. The sampler uses a specific orthogonal projection to the surface that requires only information about the tangent space to the manifold, obtainable from first derivatives of the constraint functions, hence avoiding the need for curvature information or second derivatives. Second, we use the sampler to develop a multistage algorithm to compute integrals over such manifolds. We provide single-run error estimates that avoid the need for multiple independent runs. Computational experiments on various test problems show that the algorithms and error estimates work in practice. The method is applied to compute the entropies of different sticky hard sphere systems. These predict the temperature or interaction energy at which loops of hard sticky spheres become preferable to chains.

Original languageEnglish (US)
JournalCommunications on Pure and Applied Mathematics
DOIs
StateAccepted/In press - Jan 1 2018

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Sampling
Derivatives
Hard Spheres
Probability distributions
Error Estimates
Monte Carlo methods
Entropy
Tangent Space
Orthogonal Projection
Equality Constraints
Second derivative
Inequality Constraints
Markov Chain Monte Carlo
Computational Experiments
Monte Carlo method
Test Problems
Euclidean space
Probability Distribution
Experiments
Curvature

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Monte Carlo on Manifolds : Sampling Densities and Integrating Functions. / Zappa, Emilio; Holmes-Cerfon, Miranda; Goodman, Jonathan.

In: Communications on Pure and Applied Mathematics, 01.01.2018.

Research output: Contribution to journalArticle

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