251 research outputs found
Adaptive Thermostats for Noisy Gradient Systems
We study numerical methods for sampling probability measures in high
dimension where the underlying model is only approximately identified with a
gradient system. Extended stochastic dynamical methods are discussed which have
application to multiscale models, nonequilibrium molecular dynamics, and
Bayesian sampling techniques arising in emerging machine learning applications.
In addition to providing a more comprehensive discussion of the foundations of
these methods, we propose a new numerical method for the adaptive
Langevin/stochastic gradient Nos\'{e}--Hoover thermostat that achieves a
dramatic improvement in numerical efficiency over the most popular stochastic
gradient methods reported in the literature. We also demonstrate that the newly
established method inherits a superconvergence property (fourth order
convergence to the invariant measure for configurational quantities) recently
demonstrated in the setting of Langevin dynamics. Our findings are verified by
numerical experiments
Geometric Generalisations of SHAKE and RATTLE
A geometric analysis of the Shake and Rattle methods for constrained
Hamiltonian problems is carried out. The study reveals the underlying
differential geometric foundation of the two methods, and the exact relation
between them. In addition, the geometric insight naturally generalises Shake
and Rattle to allow for a strictly larger class of constrained Hamiltonian
systems than in the classical setting.
In order for Shake and Rattle to be well defined, two basic assumptions are
needed. First, a nondegeneracy assumption, which is a condition on the
Hamiltonian, i.e., on the dynamics of the system. Second, a coisotropy
assumption, which is a condition on the geometry of the constrained phase
space. Non-trivial examples of systems fulfilling, and failing to fulfill,
these assumptions are given
Efficient stochastic thermostatting of path integral molecular dynamics
The path integral molecular dynamics (PIMD) method provides a convenient way
to compute the quantum mechanical structural and thermodynamic properties of
condensed phase systems at the expense of introducing an additional set of
high-frequency normal modes on top of the physical vibrations of the system.
Efficiently sampling such a wide range of frequencies provides a considerable
thermostatting challenge. Here we introduce a simple stochastic path integral
Langevin equation (PILE) thermostat which exploits an analytic knowledge of the
free path integral normal mode frequencies. We also apply a recently-developed
colored-noise thermostat based on a generalized Langevin equation (GLE), which
automatically achieves a similar, frequency-optimized sampling. The sampling
efficiencies of these thermostats are compared with that of the more
conventional Nos\'e-Hoover chain (NHC) thermostat for a number of physically
relevant properties of the liquid water and hydrogen-in-palladium systems. In
nearly every case, the new PILE thermostat is found to perform just as well as
the NHC thermostat while allowing for a computationally more efficient
implementation. The GLE thermostat also proves to be very robust delivering a
near-optimum sampling efficiency in all of the cases considered. We suspect
that these simple stochastic thermostats will therefore find useful application
in many future PIMD simulations.Comment: Accepted for publication on JC
Bulgac-Kusnezov-Nos\'e-Hoover thermostats
In this paper we formulate Bulgac-Kusnezov constant temperature dynamics in
phase space by means of non-Hamiltonian brackets. Two generalized versions of
the dynamics are similarly defined: one where the Bulgac-Kusnezov demons are
globally controlled by means of a single additional Nos\'e variable, and
another where each demon is coupled to an independent Nos\'e-Hoover thermostat.
Numerically stable and efficient measure-preserving time-reversible algorithms
are derived in a systematic way for each case. The chaotic properties of the
different phase space flows are numerically illustrated through the
paradigmatic example of the one-dimensional harmonic oscillator. It is found
that, while the simple Bulgac-Kusnezov thermostat is apparently not ergodic,
both of the Nos\'e-Hoover controlled dynamics sample the canonical distribution
correctly
Strongly hyperbolic Hamiltonian systems in numerical relativity: Formulation and symplectic integration
We consider two strongly hyperbolic Hamiltonian formulations of general
relativity and their numerical integration with a free and a partially
constrained symplectic integrator. In those formulations we use hyperbolic
drivers for the shift and in one case also for the densitized lapse. A system
where the densitized lapse is an external field allows to enforce the momentum
constraints in a holonomically constrained Hamiltonian system and to turn the
Hamilton constraint function from a weak to a strong invariant.
These schemes are tested in a perturbed Minkowski and the Schwarzschild
space-time. In those examples we find advantages of the strongly hyperbolic
formulations over the ADM system presented in [arXiv:0807.0734]. Furthermore we
observe stabilizing effects of the partially constrained evolution in
Schwarzschild space-time as long as the momentum constraints are enforced.Comment: This version clarifies some points concerning the interpretation of
the result
Variational modelling of wave-structure interactions with an offshore wind-turbine mast
We consider the development of a mathematical model of water waves interacting with the mast of an offshore wind turbine. A variational approach is used for which the starting point is an action functional describing a dual system comprising a potential-flow fluid, a solid structure modelled with nonlinear elasticity, and the coupling between them. We develop a linearized model of the fluid–structure or wave–mast coupling, which is a linearization of the variational principle for the fully coupled nonlinear model. Our numerical results for the linear case indicate that our variational approach yields a stable numerical discretization of a fully coupled model of water waves and an elastic beam. The energy exchange between the subsystems is seen to be in balance, yielding a total energy that shows only small and bounded oscillations amplitude of which tends to zero with the second-order convergence as the timestep approaches zero. Similar second-order convergence is observed for spatial mesh refinement. The linearized model so far developed can be extended to a nonlinear regime
Stochastic-dynamical thermostats for constraints and stiff restraints
A broad array of canonical sampling methods are available for molecular simulation based on stochastic-dynamical perturbation of Newtonian dynamics, including Langevin dynamics, Stochastic Velo- city Rescaling, and methods that combine Nosé-Hoover dynamics with stochastic perturbation. In this article we discuss several stochastic-dynamical thermostats in the setting of simulating systems with holonomic constraints. The approaches described are easily implemented and facilitate the recovery of correct canonical averages with minimal disturbance of the underlying dynamics. For the purpose of illustrating our results, we examine the numerical application of these methods to a simple atomic chain, where a Fixman term is required to correct the thermodynamic ensemble
CFD investigation of a complete floating offshore wind turbine
This chapter presents numerical computations for floating offshore wind turbines for a machine of 10-MW rated power. The rotors were computed using the Helicopter Multi-Block flow solver of the University of Glasgow that solves the Navier-Stokes equations in integral form using the arbitrary Lagrangian-Eulerian formulation for time-dependent domains with moving boundaries. Hydrodynamic loads on the support platform were computed using the Smoothed Particle Hydrodynamics method. This method is mesh-free, and represents the fluid by a set of discrete particles. The motion of the floating offshore wind turbine is computed using a Multi-Body Dynamic Model of rigid bodies and frictionless joints. Mooring cables are modelled as a set of springs and dampers. All solvers were validated separately before coupling, and the loosely coupled algorithm used is described in detail alongside the obtained results
Sampling constrained probability distributions using Spherical Augmentation
Statistical models with constrained probability distributions are abundant in
machine learning. Some examples include regression models with norm constraints
(e.g., Lasso), probit, many copula models, and latent Dirichlet allocation
(LDA). Bayesian inference involving probability distributions confined to
constrained domains could be quite challenging for commonly used sampling
algorithms. In this paper, we propose a novel augmentation technique that
handles a wide range of constraints by mapping the constrained domain to a
sphere in the augmented space. By moving freely on the surface of this sphere,
sampling algorithms handle constraints implicitly and generate proposals that
remain within boundaries when mapped back to the original space. Our proposed
method, called {Spherical Augmentation}, provides a mathematically natural and
computationally efficient framework for sampling from constrained probability
distributions. We show the advantages of our method over state-of-the-art
sampling algorithms, such as exact Hamiltonian Monte Carlo, using several
examples including truncated Gaussian distributions, Bayesian Lasso, Bayesian
bridge regression, reconstruction of quantized stationary Gaussian process, and
LDA for topic modeling.Comment: 41 pages, 13 figure
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