67 research outputs found
Efficient Monte Carlo sampling by parallel marginalization
Markov chain Monte Carlo sampling methods often suffer from long correlation
times. Consequently, these methods must be run for many steps to generate an
independent sample. In this paper a method is proposed to overcome this
difficulty. The method utilizes information from rapidly equilibrating coarse
Markov chains that sample marginal distributions of the full system. This is
accomplished through exchanges between the full chain and the auxiliary coarse
chains. Results of numerical tests on the bridge sampling and
filtering/smoothing problems for a stochastic differential equation are
presented.Comment: 7 figures, 2 figures, PNAS .cls and .sty files, submitted to PNA
The Brownian fan
We provide a mathematical study of the modified Diffusion Monte Carlo (DMC)
algorithm introduced in the companion article \cite{DMC}. DMC is a simulation
technique that uses branching particle systems to represent expectations
associated with Feynman-Kac formulae. We provide a detailed heuristic
explanation of why, in cases in which a stochastic integral appears in the
Feynman-Kac formula (e.g. in rare event simulation, continuous time filtering,
and other settings), the new algorithm is expected to converge in a suitable
sense to a limiting process as the time interval between branching steps goes
to 0. The situation studied here stands in stark contrast to the "na\"ive"
generalisation of the DMC algorithm which would lead to an exponential
explosion of the number of particles, thus precluding the existence of any
finite limiting object. Convergence is shown rigorously in the simplest
possible situation of a random walk, biased by a linear potential. The
resulting limiting object, which we call the "Brownian fan", is a very natural
new mathematical object of independent interest.Comment: 53 pages, 2 figures. Formerly 2nd part of arXiv:1207.286
Fast randomized iteration: diffusion Monte Carlo through the lens of numerical linear algebra
We review the basic outline of the highly successful diffusion Monte Carlo
technique commonly used in contexts ranging from electronic structure
calculations to rare event simulation and data assimilation, and propose a new
class of randomized iterative algorithms based on similar principles to address
a variety of common tasks in numerical linear algebra. From the point of view
of numerical linear algebra, the main novelty of the Fast Randomized Iteration
schemes described in this article is that they work in either linear or
constant cost per iteration (and in total, under appropriate conditions) and
are rather versatile: we will show how they apply to solution of linear
systems, eigenvalue problems, and matrix exponentiation, in dimensions far
beyond the present limits of numerical linear algebra. While traditional
iterative methods in numerical linear algebra were created in part to deal with
instances where a matrix (of size ) is too big to store, the
algorithms that we propose are effective even in instances where the solution
vector itself (of size ) may be too big to store or manipulate.
In fact, our work is motivated by recent DMC based quantum Monte Carlo schemes
that have been applied to matrices as large as . We
provide basic convergence results, discuss the dependence of these results on
the dimension of the system, and demonstrate dramatic cost savings on a range
of test problems.Comment: 44 pages, 7 figure
Sharp entrywise perturbation bounds for Markov chains
For many Markov chains of practical interest, the invariant distribution is
extremely sensitive to perturbations of some entries of the transition matrix,
but insensitive to others; we give an example of such a chain, motivated by a
problem in computational statistical physics. We have derived perturbation
bounds on the relative error of the invariant distribution that reveal these
variations in sensitivity.
Our bounds are sharp, we do not impose any structural assumptions on the
transition matrix or on the perturbation, and computing the bounds has the same
complexity as computing the invariant distribution or computing other bounds in
the literature. Moreover, our bounds have a simple interpretation in terms of
hitting times, which can be used to draw intuitive but rigorous conclusions
about the sensitivity of a chain to various types of perturbations
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