### 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 language | English (US) |
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Journal | Communications on Pure and Applied Mathematics |

DOIs | |

State | Accepted/In press - Jan 1 2018 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*. https://doi.org/10.1002/cpa.21783

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

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Monte Carlo on Manifolds

T2 - Sampling Densities and Integrating Functions

AU - Zappa, Emilio

AU - Holmes-Cerfon, Miranda

AU - Goodman, Jonathan

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85052405750&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85052405750&partnerID=8YFLogxK

U2 - 10.1002/cpa.21783

DO - 10.1002/cpa.21783

M3 - Article

AN - SCOPUS:85052405750

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -