10,208 research outputs found
Metastability in stochastic dynamics of disordered mean-field models
We study a class of Markov chains that describe reversible stochastic
dynamics of a large class of disordered mean field models at low temperatures.
Our main purpose is to give a precise relation between the metastable time
scales in the problem to the properties of the rate functions of the
corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin
theory in this case, showing that any transition can be decomposed, with
probability exponentially close to one, into a deterministic sequence of
``admissible transitions''. For these admissible transitions we give upper and
lower bounds on the expected transition times that differ only by a constant.
The distribution rescaled transition times are shown to converge to the
exponential distribution. We exemplify our results in the context of the random
field Curie-Weiss model.Comment: 73pp, AMSTE
Gradient and Passive Circuit Structure in a Class of Non-linear Dynamics on a Graph
We consider a class of non-linear dynamics on a graph that contains and
generalizes various models from network systems and control and study
convergence to uniform agreement states using gradient methods. In particular,
under the assumption of detailed balance, we provide a method to formulate the
governing ODE system in gradient descent form of sum-separable energy
functions, which thus represent a class of Lyapunov functions; this class
coincides with Csisz\'{a}r's information divergences. Our approach bases on a
transformation of the original problem to a mass-preserving transport problem
and it reflects a little-noticed general structure result for passive network
synthesis obtained by B.D.O. Anderson and P.J. Moylan in 1975. The proposed
gradient formulation extends known gradient results in dynamical systems
obtained recently by M. Erbar and J. Maas in the context of porous medium
equations. Furthermore, we exhibit a novel relationship between inhomogeneous
Markov chains and passive non-linear circuits through gradient systems, and
show that passivity of resistor elements is equivalent to strict convexity of
sum-separable stored energy. Eventually, we discuss our results at the
intersection of Markov chains and network systems under sinusoidal coupling
Joint state-parameter estimation of a nonlinear stochastic energy balance model from sparse noisy data
While nonlinear stochastic partial differential equations arise naturally in
spatiotemporal modeling, inference for such systems often faces two major
challenges: sparse noisy data and ill-posedness of the inverse problem of
parameter estimation. To overcome the challenges, we introduce a strongly
regularized posterior by normalizing the likelihood and by imposing physical
constraints through priors of the parameters and states. We investigate joint
parameter-state estimation by the regularized posterior in a physically
motivated nonlinear stochastic energy balance model (SEBM) for paleoclimate
reconstruction. The high-dimensional posterior is sampled by a particle Gibbs
sampler that combines MCMC with an optimal particle filter exploiting the
structure of the SEBM. In tests using either Gaussian or uniform priors based
on the physical range of parameters, the regularized posteriors overcome the
ill-posedness and lead to samples within physical ranges, quantifying the
uncertainty in estimation. Due to the ill-posedness and the regularization, the
posterior of parameters presents a relatively large uncertainty, and
consequently, the maximum of the posterior, which is the minimizer in a
variational approach, can have a large variation. In contrast, the posterior of
states generally concentrates near the truth, substantially filtering out
observation noise and reducing uncertainty in the unconstrained SEBM
Individual risk in mean-field control models for decentralized control, with application to automated demand response
Flexibility of energy consumption can be harnessed for the purposes of
ancillary services in a large power grid. In prior work by the authors a
randomized control architecture is introduced for individual loads for this
purpose. In examples it is shown that the control architecture can be designed
so that control of the loads is easy at the grid level: Tracking of a balancing
authority reference signal is possible, while ensuring that the quality of
service (QoS) for each load is acceptable on average. The analysis was based on
a mean field limit (as the number of loads approaches infinity), combined with
an LTI-system approximation of the aggregate nonlinear model. This paper
examines in depth the issue of individual risk in these systems. The main
contributions of the paper are of two kinds:
Risk is modeled and quantified:
(i) The average performance is not an adequate measure of success. It is
found empirically that a histogram of QoS is approximately Gaussian, and
consequently each load will eventually receive poor service.
(ii) The variance can be estimated from a refinement of the LTI model that
includes a white-noise disturbance; variance is a function of the randomized
policy, as well as the power spectral density of the reference signal.
Additional local control can eliminate risk:
(iii) The histogram of QoS is truncated through this local control, so that
strict bounds on service quality are guaranteed.
(iv) This has insignificant impact on the grid-level performance, beyond a
modest reduction in capacity of ancillary service.Comment: Publication without appendix to appear in the 53rd IEEE Conf. on
Decision and Control, December, 201
Concentration of measure and mixing for Markov chains
We consider Markovian models on graphs with local dynamics. We show that,
under suitable conditions, such Markov chains exhibit both rapid convergence to
equilibrium and strong concentration of measure in the stationary distribution.
We illustrate our results with applications to some known chains from computer
science and statistical mechanics.Comment: 28 page
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