4,069 research outputs found
Rational Construction of Stochastic Numerical Methods for Molecular Sampling
In this article, we focus on the sampling of the configurational
Gibbs-Boltzmann distribution, that is, the calculation of averages of functions
of the position coordinates of a molecular -body system modelled at constant
temperature. We show how a formal series expansion of the invariant measure of
a Langevin dynamics numerical method can be obtained in a straightforward way
using the Baker-Campbell-Hausdorff lemma. We then compare Langevin dynamics
integrators in terms of their invariant distributions and demonstrate a
superconvergence property (4th order accuracy where only 2nd order would be
expected) of one method in the high friction limit; this method, moreover, can
be reduced to a simple modification of the Euler-Maruyama method for Brownian
dynamics involving a non-Markovian (coloured noise) random process. In the
Brownian dynamics case, 2nd order accuracy of the invariant density is
achieved. All methods considered are efficient for molecular applications
(requiring one force evaluation per timestep) and of a simple form. In fully
resolved (long run) molecular dynamics simulations, for our favoured method, we
observe up to two orders of magnitude improvement in configurational sampling
accuracy for given stepsize with no evident reduction in the size of the
largest usable timestep compared to common alternative methods
Enabling scalable stochastic gradient-based inference for Gaussian processes by employing the Unbiased LInear System SolvEr (ULISSE)
In applications of Gaussian processes where quantification of uncertainty is
of primary interest, it is necessary to accurately characterize the posterior
distribution over covariance parameters. This paper proposes an adaptation of
the Stochastic Gradient Langevin Dynamics algorithm to draw samples from the
posterior distribution over covariance parameters with negligible bias and
without the need to compute the marginal likelihood. In Gaussian process
regression, this has the enormous advantage that stochastic gradients can be
computed by solving linear systems only. A novel unbiased linear systems solver
based on parallelizable covariance matrix-vector products is developed to
accelerate the unbiased estimation of gradients. The results demonstrate the
possibility to enable scalable and exact (in a Monte Carlo sense)
quantification of uncertainty in Gaussian processes without imposing any
special structure on the covariance or reducing the number of input vectors.Comment: 10 pages - paper accepted at ICML 201
A relative entropy rate method for path space sensitivity analysis of stationary complex stochastic dynamics
We propose a new sensitivity analysis methodology for complex stochastic
dynamics based on the Relative Entropy Rate. The method becomes computationally
feasible at the stationary regime of the process and involves the calculation
of suitable observables in path space for the Relative Entropy Rate and the
corresponding Fisher Information Matrix. The stationary regime is crucial for
stochastic dynamics and here allows us to address the sensitivity analysis of
complex systems, including examples of processes with complex landscapes that
exhibit metastability, non-reversible systems from a statistical mechanics
perspective, and high-dimensional, spatially distributed models. All these
systems exhibit, typically non-gaussian stationary probability distributions,
while in the case of high-dimensionality, histograms are impossible to
construct directly. Our proposed methods bypass these challenges relying on the
direct Monte Carlo simulation of rigorously derived observables for the
Relative Entropy Rate and Fisher Information in path space rather than on the
stationary probability distribution itself. We demonstrate the capabilities of
the proposed methodology by focusing here on two classes of problems: (a)
Langevin particle systems with either reversible (gradient) or non-reversible
(non-gradient) forcing, highlighting the ability of the method to carry out
sensitivity analysis in non-equilibrium systems; and, (b) spatially extended
Kinetic Monte Carlo models, showing that the method can handle high-dimensional
problems
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