Monte Carlo on Manifolds: Sampling Densities and Integrating Functions

Zappa, Emilio; Holmes-Cerfon, Miranda; Goodman, Joel W.

doi:10.1002/cpa.21783

Cited by 55 publications

(99 citation statements)

References 35 publications

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“…Goodman, Holmes-Cerfon and Zappa recently showed, by a clever geometric construction, how to construct reversible Metropolis random walks on submanifolds [36]. In fact, their construction can be interpreted as a one-step HMC scheme where the gradient of the potential energy in the proposal move is zero.…”

Section: Introductionmentioning

confidence: 99%

Hybrid Monte Carlo methods for sampling probability measures on submanifolds

2019

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Probability measures supported on submanifolds can be sampled by adding an extra momentum variable to the state of the system, and discretizing the associated Hamiltonian dynamics with some stochastic perturbation in the extra variable. In order to avoid biases in the invariant probability measures sampled by discretizations of these stochastically perturbed Hamiltonian dynamics, a Metropolis rejection procedure can be considered. The so-obtained scheme belongs to the class of generalized Hybrid Monte Carlo (GHMC) algorithms. We show here how to generalize to GHMC a procedure suggested by Goodman, Holmes-Cerfon and Zappa for Metropolis random walks on submanifolds, where a reverse projection check is performed to enforce the reversibility of the algorithm for any timesteps and hence avoid biases in the invariant measure. We also provide a full mathematical analysis of such procedures, as well as numerical experiments demonstrating the importance of the reverse projection check on simple toy examples. Mathematical setting and resultsWe first describe in Section 2.1 the target probability measures we are interested in sampling. The core of our analysis is presented in Section 2.2, where we show how to rigorously formalize the reversibility properties of discretizations of the Hamiltonian dynamics on a submanifold for possibly large timesteps. One crucial ingredient in this analysis is the Lagrange multiplier function, which associates to a state on the submanifold and another state close to the submanifold (obtained by an unconstrained step of the dynamics) a new state on the submanifold. Examples of such Lagrange multiplier functions are provided in Section 2.3. Once the discretization of the Hamiltonian part of the dynamics is clear, GHMC schemes follow in a straightforward way; see Section 2.4. Geometric settingLet us first introduce the measures we are interested in sampling, namely the target measure (2) below, as well as the probability measure (4) on an extended space, which admits (2) as a marginal. We only provide the most essential objects needed for our analysis. For more details and motivations for the definitions and results given here, we refer the reader to the standard reference textbooks [1,4], and also to [22, Chapter 3.3.2] for a self-contained presentation. Target measureWe denote by d the dimension of the ambient space, and consider a submanifold M ⊂ R d defined as the zero level set of a given smooth function ξ :For example, the function ξ encodes molecular constraints or reaction coordinates in molecular dynamics, or is a summary statistics in Approximate Bayesian Computations. Let M ∈ R d×d be a fixed symmetric positive definite matrix, which is interpreted as the mass tensor below. The ambient space R d is endowed with the scalar product q,q M = q T Mq. One could take for simplicity M = Id but it is sometimes useful to consider non identity mass matrices for numerical purposes [13]. The function ξ is assumed to be smooth on a neighborhood of M in R d and such thatis an invertible matrix for...

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Section: Introductionmentioning

confidence: 99%

Hybrid Monte Carlo methods for sampling probability measures on submanifolds

2019

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“…Other methods have considered how to sample probability densities directly on manifolds and have shown themselves to be significantly more efficient than using short-range forces to keep a process near a manifold (e.g. [13,64,92].) A third step, and perhaps the most challenging, will be to adapt the method to processes which are sticky on even lower-dimensional "corners".…”

Section: Discussionmentioning

confidence: 99%

Sticky Brownian Motion and Its Numerical Solution

Bou-Rabee¹,

Holmes-Cerfon²

2020

SIAM Rev.

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Sticky Brownian motion is the simplest example of a diffusion process that can spend finite time both in the interior of a domain and on its boundary. It arises in various applications such as in biology, materials science, and finance. This article spotlights the unusual behavior of sticky Brownian motions from the perspective of applied mathematics, and provides tools to efficiently simulate them. We show that a sticky Brownian motion arises naturally for a particle diffusing on R + with a strong, short-ranged potential energy near the origin. This is a limit that accurately models mesoscale particles, those with diameters ≈ 100nm-10µm, which form the building blocks for many common materials. We introduce a simple and intuitive sticky random walk to simulate sticky Brownian motion, that also gives insight into its unusual properties. In parameter regimes of practical interest, we show this sticky random walk is two to five orders of magnitude faster than alternative methods to simulate a sticky Brownian motion. We outline possible steps to extend this method towards simulating multi-dimensional sticky diffusions.

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“…We compute the automorphism group, the point group and the sticky symmetry group of loops and chains for N = 4 − 10, 15, 20. In order to test that our algorithm works, we randomly choose a point x in the manifold of configurations using the sampling algorithm in [25], and compute P x and T x . In this way, x will have no "a priori" symmetry axes.…”

Section: Examplesmentioning

confidence: 99%

“…For each graph α so obtained we check that it is connected, and that it is not isomorphic to a graph we have already seen. 7 If it passes both tests, then we compute the integral in (5) using the method described in [25] (with center of mass of the cluster fixed), which is essentially a form of thermodynamic integration on a manifold, and call the resulting quantity I α . Then we calculate the symmetry number σ α using the method described in this paper.…”

Section: Examplesmentioning

confidence: 99%

“…where β ∈ R is a real parameter and Z > 0 is a normalization constant. We sample on the manifold from the density ρ using the Markov Chain Monte Carlo algorithm described in [25] and a step size parameter σ, again usually comparable to tol. The parameter β is called the "inverse temperature" in simulated annealing or other sampling techniques [18], and it controls how peaked the density is near the minimum of U (x): large β means the density is strongly peaked near the minimum so sampling pushes points toward x 1 , β ≈ 0 means the density is relatively flat so the manifold is sampled nearly uniformly (similar to the first method but with a Metropolis step to ensure we sample the correct density), and β < 0 means the density is lowest at the optimum, so sampling should push us away from x 1 in general.…”

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confidence: 99%

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Calculating the Symmetry Number of Flexible Sphere Clusters

Zappa

Holmes-Cerfon

2019

J Nonlinear Sci

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We present a theoretical and computational framework to compute the symmetry number of a flexible sphere cluster in R 3 , using a definition of symmetry that arises naturally when calculating the equilibrium probability of a cluster of spheres in the sticky-sphere limit. We define the sticky symmetry group of the cluster as the set of permutations and inversions of the spheres which preserve adjacency and can be realized by continuous deformations of the cluster that do not change the set of contacts or cause particles to overlap. The symmetry number is the size of the sticky symmetry group. We introduce a numerical algorithm to compute the sticky symmetry group and symmetry number, and show it works well on several test cases. Furthermore we show that once the sticky symmetry group has been calculated for indistinguishable spheres, the symmetry group for partially distinguishable spheres (those with non-identical interactions) can be efficiently obtained without repeating the laborious parts of the computations. We use our algorithm to calculate the partition functions of every possible connected cluster of 6 identical sticky spheres, generating data that may be used to design interactions between spheres so they self-assemble into a desired structure.

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Monte Carlo on Manifolds: Sampling Densities and Integrating Functions

Cited by 55 publications

References 35 publications

Hybrid Monte Carlo methods for sampling probability measures on submanifolds

Hybrid Monte Carlo methods for sampling probability measures on submanifolds

Sticky Brownian Motion and Its Numerical Solution

Calculating the Symmetry Number of Flexible Sphere Clusters

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