# Hybrid Monte Carlo on Hilbert spaces

@article{Beskos2011HybridMC, title={Hybrid Monte Carlo on Hilbert spaces}, author={Alexandros Beskos and F. J. Pinski and Jes{\'u}s Mar{\'i}a Sanz-Serna and Andrew M. Stuart}, journal={Stochastic Processes and their Applications}, year={2011}, volume={121}, pages={2201-2230} }

## 121 Citations

### A Novel Hybrid Monte Carlo Algorithm for Sampling Path Space

- MathematicsEntropy
- 2021

The ingredients of the new algorithm include the definition of the mass operator, the equations for the Hamiltonian flow, the (approximate) numerical integration of the evolution equations, and finally, the Metropolis–Hastings acceptance rule, which constitute a robust method for sampling the target distribution in an almost dimension-free manner.

### A Function Space HMC Algorithm With Second Order Langevin Diffusion Limit

- Mathematics, Computer Science
- 2013

The main result of this paper states that the new algorithm, appropriately rescaled, converges weakly to a second order Langevin diffusion on Hilbert space; as a consequence the algorithm explores the approximate target measures on R^N in a number of steps which is independent of N.

### Mixing rates for Hamiltonian Monte Carlo algorithms in finite and infinite dimensions

- MathematicsStochastics and Partial Differential Equations: Analysis and Computations
- 2021

We establish the geometric ergodicity of the preconditioned Hamiltonian Monte Carlo (HMC) algorithm defined on an infinite-dimensional Hilbert space, as developed in [Beskos et al., Stochastic…

### Optimal tuning of the hybrid Monte Carlo algorithm

- Computer Science
- 2010

It is proved that, to obtain an O(1) acceptance probability as the dimension d of the state space tends to, the leapfrog step-size h should be scaled as h=l ×d−1/ 4, which means that in high dimensions, HMC requires O(d1/ 4 ) steps to traverse the statespace.

### Advanced Markov Chain Monte Carlo methods for sampling on diffusion pathspace

- Mathematics
- 2012

The need to calibrate increasingly complex statistical models requires a persistent effort for further advances on available, computationally intensive MonteCarlo methods. We study here an advanced…

### Optimal Scaling and Diffusion Limits for the Langevin Algorithm in High Dimensions

- Computer Science, Mathematics
- 2011

It is proved that, started in stationarity, a suitably interpolated and scaled version of the Markov chain corresponding to MALA converges to an infinite dimensional diffusion process.

### Spectral gaps for a Metropolis–Hastings algorithm in infinite dimensions

- Mathematics
- 2014

We study the problem of sampling high and infinite dimensional target measures arising in applications such as conditioned diffusions and inverse problems. We focus on those that arise from…

### Entropy-based adaptive Hamiltonian Monte Carlo

- Computer ScienceNeurIPS
- 2021

A gradient-based algorithm is developed that allows for the adaptation of the mass matrix of Hamiltonian Monte Carlo by encouraging the leapfrog integrator to have high acceptance rates while also exploring all dimensions jointly.

### Quantum-Inspired Hamiltonian Monte Carlo for Bayesian Sampling

- Computer ScienceArXiv
- 2019

The proposed Quantum-Inspired Hamiltonian Monte Carlo algorithm (QHMC) allows a particle to have a random mass matrix with a probability distribution rather than a fixed mass, and proves the convergence property of QHMC and shows why such a randommass can improve the performance when the authors sample a broad class of distributions.

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It is proved that, to obtain an O(1) acceptance probability as the dimension d of the state space tends to, the leapfrog step-size h should be scaled as h=l ×d−1/ 4, which means that in high dimensions, HMC requires O(d1/ 4 ) steps to traverse the statespace.

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A Markov chain Monte Carlo based analysis of a multilevel model for functional MRI data and its applications in environmental epidemiology, educational research, and fisheries science are studied.

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An asymptotic diffusion limit theorem is proved and it is shown that, as a function of dimension n, the complexity of the algorithm is O(n1/3), which compares favourably with the O- complexity of random walk Metropolis algorithms.