Reversible and non-reversible Markov chain Monte Carlo algorithms for reservoir simulation problems

Dobson, Paul; Fursov, I.; Lord, Gabriel J.

doi:10.1007/s10596-020-09947-4

Cited by 9 publications

(7 citation statements)

References 16 publications

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“…Some comments on the above definition (a) If the gradient flow part of ( 34) is quadratic, i.e. if the evolution is given by 14 We recall that if F is convex and continuously differentiable then (x − y)(dF(x) − dF(y)) ⩾ 0 for every x, y.…”

Section: Pre-genericmentioning

confidence: 99%

Non-reversible processes: GENERIC, hypocoercivity and fluctuations

Duong

2023

Nonlinearity

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We consider two approaches to study non-reversible Markov processes, namely the hypocoercivity theory and general equations for non-equilibrium reversible–irreversible coupling; the basic idea behind both of them is to split the process into a reversible component and a non-reversible one, and then quantify the way in which they interact. We compare such theories and provide explicit formulas to pass from one formulation to the other; as a bi-product we give a simple proof of the link between reversibility of the dynamics and gradient flow structure of the associated Fokker–Planck equation. We do this both for linear Markov processes and for a class of nonlinear Markov process as well. We then characterise the structure of the large deviation functional of generalised-reversible processes; this is a class of non-reversible processes of large relevance in applications. Finally, we show how our results apply to two classes of Markov processes, namely non-reversible diffusion processes and a class of piecewise deterministic Markov processes (PDMPs), which have recently attracted the attention of the statistical sampling community. In particular, for the PDMPs we consider we prove entropy decay.

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Section: Pre-genericmentioning

confidence: 99%

Non-reversible processes: GENERIC, hypocoercivity and fluctuations

Duong

2023

Nonlinearity

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“…Reflected Langevin dynamics have also been used in the context of optimisation, see [67], in the case of a bounded smooth domain. Finally we mention that reflections can also be used with Hamiltonian dynamics [15,23], although we note here that a Metropolis-Hastings step has been included to ensure the algorithm has the desired invariant distribution. Since all of these algorithms are for bounded domains they are not applicable to our setting and extra care is required as the domain is not smooth.…”

Section: Related Workmentioning

confidence: 99%

Efficient Bayesian computation for low-photon imaging problems

Melidonis¹,

Dobson²,

Altmann³

et al. 2022

Preprint

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This paper studies a new and highly efficient Markov chain Monte Carlo (MCMC) methodology to perform Bayesian inference in low-photon imaging problems, with particular attention to situations involving observation noise processes that deviate significantly from Gaussian noise, such as binomial, geometric and low-intensity Poisson noise. These problems are challenging for many reasons. From an inferential viewpoint, low-photon numbers lead to severe identifiability issues, poor stability and high uncertainty about the solution. Moreover, low-photon models often exhibit poor regularity properties that make efficient Bayesian computation difficult; e.g., hard non-negativity constraints, non-smooth priors, and log-likelihood terms with exploding gradients. More precisely, the lack of suitable regularity properties hinders the use of state-of-the-art Monte Carlo methods based on numerical approximations of the Langevin stochastic differential equation (SDE), as both the SDE and its numerical approximations behave poorly. We address this difficulty by proposing an MCMC methodology based on a reflected and regularised Langevin SDE, which is shown to be well-posed and exponentially ergodic under mild and easily verifiable conditions. This then allows us to derive four reflected proximal Langevin MCMC algorithms to perform Bayesian computation in lowphoton imaging problems. The proposed approach is demonstrated with a range of experiments related to image deblurring, denoising, and inpainting under binomial, geometric and Poisson noise.

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“…A classical example which belongs to this setup is the so-called Andersen thermostat (see [32,) which, from a sampling perspective, is the Hybrid Monte Carlo algorithm (see [44] and [14,42] for related Hamiltonian samplers). The Andersen thermostat can be understood as follows: suppose we want to sample from the measure µ defined in (2), or, more precisely, from its multidimensional version,…”

Section: Examples: Diffusion Processes and Pdmpsmentioning

confidence: 99%

Non-reversible processes: GENERIC, Hypocoercivity and fluctuations

Duong¹

2021

Preprint

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We consider two approaches to study non-reversible Markov processes, namely the Hypocoercivity Theory (HT) and GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling); the basic idea behind both of them is to split the process into a reversible component and a non-reversible one, and then quantify the way in which they interact. We compare such theories and provide explicit formulas to pass from one formulation to the other; as a bi-product we give a simple proof of the link between reversibility of the dynamics and gradient flow structure of the associated Fokker-Planck equation. We do this both for linear Markov processes and for a class of nonlinear Markov process as well. We then characterize the structure of the Large deviation functional of generalised-reversible processes; this is a class of non-reversible processes of large relevance in applications. Finally, we show how our results apply to two classes of Markov processes, namely non-reversible diffusion processes and a class of Piecewise Deterministic Markov Processes (PDMPs), which have recently attracted the attention of the statistical sampling community. In particular, for the PDMPs we consider we prove entropy decay.

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Reversible and non-reversible Markov chain Monte Carlo algorithms for reservoir simulation problems

Cited by 9 publications

References 16 publications

Non-reversible processes: GENERIC, hypocoercivity and fluctuations

Non-reversible processes: GENERIC, hypocoercivity and fluctuations

Efficient Bayesian computation for low-photon imaging problems

Non-reversible processes: GENERIC, Hypocoercivity and fluctuations

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