Hamiltonian Monte Carlo

Summary

The Hamiltonian Monte Carlo algorithm (originally known as hybrid Monte Carlo) is a Markov chain Monte Carlo method for obtaining a sequence of random samples which converge to being distributed according to a target probability distribution for which direct sampling is difficult. This sequence can be used to estimate integrals with respect to the target distribution (expected values). Hamiltonian Monte Carlo corresponds to an instance of the Metropolis–Hastings algorithm, with a Hamiltonian dynamics evolution simulated using a time-reversible and volume-preserving numerical integrator (typically the leapfrog integrator) to propose a move to a new point in the state space. Compared to using a Gaussian random walk proposal distribution in the Metropolis–Hastings algorithm, Hamiltonian Monte Carlo reduces the correlation between successive sampled states by proposing moves to distant states which maintain a high probability of acceptance due to the approximate energy conserving properti

Official source

https://en.wikipedia.org/wiki/Hamiltonian_Monte_Carlo

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Quantum Monte Carlo calculations using new trialfunctions

Philippe Aeberhard

EPFL2007

A novel approach is presented for constructing polynomial chaos representations of scalar quantities of interest (QoI) that extends previously developed methods for adaptation in Homogeneous Chaos spaces. In this work, we develop a Bayesian formulation of the problem that characterizes the posterior distributions of the series coefficients and the adaptation rotation matrix acting on the Gaussian input variables. The adaptation matrix is thus construed as a new parameter of the map from input to QoI, estimated through Bayesian inference. For the computation of the coefficients' posterior distribution, we use a variational inference approach that approximates the posterior with a member of the same exponential family as the prior, such that it minimizes a Kuilback-Leibler criterion. On the other hand, the posterior distribution of the rotation matrix is explored by employing a Geodesic Monte Carlo sampling approach, consisting of a variation of the Hamiltonian Monte Carlo algorithm for embedded manifolds, in our case, the Stiefel manifold of orthonormal matrices. The performance of our method is demonstrated through a series of numerical examples, including the problem of multiphase flow in heterogeneous porous media.

ROYAL SOC2018

MULTILEVEL ENSEMBLE KALMAN FILTERING FOR SPATIO-TEMPORAL PROCESSES

Hakon Andreas Hoel, Fabio Nobile

This work concerns state-space models, in which the state-space is an infinite-dimensional spatial field, and the evolution is in continuous time, hence requiring approximation in space and time. The multilevel Monte Carlo (MLMC) sampling strategy is leveraged in the Monte Carlo step of the ensemble Kalman filter (EnKF), thereby yielding a multilevel ensemble Kalman filter (MLEnKF) for spatio-temporal models, which has provably superior as- ymptotic error/cost ratio. A practically relevant stochastic partial differential equation (SPDE) example is presented, and numerical experiments with this example support our theoretical findings.

2017