Open Access
Simulating Hamiltonian Dynamics
TLDR
Geometric integrators as mentioned in this paper are time-stepping methods, designed such that they exactly satisfy conservation laws, symmetries or symplectic properties of a system of differential equations.Abstract:
Geometric integrators are time-stepping methods, designed such that they exactly satisfy conservation laws, symmetries or symplectic properties of a system of differential equations. In this book the authors outline the principles of geometric integration and demonstrate how they can be applied to provide efficient numerical methods for simulating conservative models. Beginning from basic principles and continuing with discussions regarding the advantageous properties of such schemes, the book introduces methods for the N-body problem, systems with holonomic constraints, and rigid bodies. More advanced topics treated include high-order and variable stepsize methods, schemes for treating problems involving multiple time-scales, and applications to molecular dynamics and partial differential equations. The emphasis is on providing a unified theoretical framework as well as a practical guide for users. The inclusion of examples, background material and exercises enhance the usefulness of the book for self-instruction or as a text for a graduate course on the subject.read more
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BookDOI
MCMC using Hamiltonian dynamics
TL;DR: In this paper, the authors discuss theoretical and practical aspects of Hamiltonian Monte Carlo, and present some of its variations, including using windows of states for deciding on acceptance or rejection, computing trajectories using fast approximations, tempering during the course of a trajectory to handle isolated modes, and short-cut methods that prevent useless trajectories from taking much computation time.
Journal Article
The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo
Matthew D. Homan,Andrew Gelman +1 more
TL;DR: The No-U-Turn Sampler (NUTS), an extension to HMC that eliminates the need to set a number of steps L, and derives a method for adapting the step size parameter {\epsilon} on the fly based on primal-dual averaging.
Book ChapterDOI
MCMC Using Hamiltonian Dynamics
TL;DR: This volume focuses on perfect sampling or exact sampling algorithms, so named because such algorithms use Markov chains and yet obtain genuine i.i.d. draws—hence perfect or exact—from their limiting distributions within a finite numbers of iterations.
Journal ArticleDOI
The Magnus expansion and some of its applications
TL;DR: Magnusson expansion as discussed by the authors provides a power series expansion for the corresponding exponent and is sometimes referred to as Time-Dependent Exponential Perturbation Theory (TEPT).
Posted Content
A Conceptual Introduction to Hamiltonian Monte Carlo
TL;DR: This review provides a comprehensive conceptual account of these theoretical foundations of Hamiltonian Monte Carlo, focusing on developing a principled intuition behind the method and its optimal implementations rather of any exhaustive rigor.