Simulating Hamiltonian dynamics

By Ernst Hairer, Gerhard Wanner. 2006

993 citations recorded by the source corpus.

Subfields: Probabilistic & Bayesian Methods, Numerical Methods & Scientific Computing

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Citations in the corpus

Handbook of Markov Chain Monte Carlo (2011, neutral): “We can then simulate a Markov chain in which each iteration resamples the momentum and then does a Metropolis update with a proposal found using Hamiltonian dynamics.”
The No-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo (2011, methodology): “The blue arrow is the vector from the positions associated with the leftmost to the rightmost leaf nodes in the rightmost height-3 subtree, and the magenta arrow is the (normalized) momentum vector at the ﬁnal state in the trajectory.”
Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions (2009, neutral): “See [LR04] for a broad introduction to symplectic numerical methods and their conservation properties.”
Riemannian Manifold Hamiltonian Monte Carlo (2009, methodology): “Whilst this step of the integrator is defined iteratively it should be stressed that the overall symplectic integrator that is pre ented in this paper is, computationally, an enormous improvement to fully implicit methods such as the Generalised Leapfrog, see e.g. Leimkuhler and Reich (2004).”
A survey of uncertainty in deep neural networks (2021, methodology): “The so-called leapfrog integrator is used as a solver [161].”
On the convergence of Hamiltonian Monte Carlo (2017, background): “These properties are discussed in the literature [8, 12] and we just brieﬂy sketch the main ideas with admittedly loose notations and deﬁnitions.”
A General Metric for Riemannian Manifold Hamiltonian Monte Carlo (2012, background): “Nominally, the O ( N3 ) computational burden of RMHMC is significantly worse than the O(N) burden of EMHMC.”
Optimal tuning of the hybrid Monte Carlo algorithm (2010, neutral): “may be shown to be volume-preserving and time-reversible (see [19,25,41]), but it does not exactly conserve energy.”
The Geometric Foundations of Hamiltonian Monte Carlo (2014, background): “…optimal step size, however, requires a deeper understanding of the interaction between the geometry of the exact Hamiltonian ﬂow and that of the symplectic integrator developed in backwards error analysis (Leimkuhler and Reich, 2004; Hairer, Lubich and Wanner, 2006; Izaguirre and Hampton, 2004).”
Discrete mechanics and optimal control for constrained systems (2010, background): “On the other hand, these evolution equations have a highly complicated structure, causing the derivation of their temporal discrete form to be expensive and therefore, in most cases, not recommended [11, 12].”
Hybrid Monte Carlo on Lefschetz thimbles — A study of the residual sign problem (2013, methodology): “5), we employ the second order constraint-preserving symmetric integrator[38]: it is assumed first that zn and wn satisfy the constraints z = z[e, t′(n)], (3.”
A Conceptual Introduction to Hamiltonian Monte Carlo (2017, methodology): “Fortunately, we can use the geometry of phase space itself to construct an extremely powerful family of numerical solvers, known as symplectic integrators (Leimkuhler and Reich, 2004; Hairer, Lubich and Wanner, 2006), that are robust to phenomena like drift and enable high-performance implementation”