475,893 research outputs found
A probabilistic interpretation of the parametrix method
In this article, we introduce the parametrix technique in order to construct
fundamental solutions as a general method based on semigroups and their
generators. This leads to a probabilistic interpretation of the parametrix
method that is amenable to Monte Carlo simulation. We consider the explicit
examples of continuous diffusions and jump driven stochastic differential
equations with H\"{o}lder continuous coefficients.Comment: Published at http://dx.doi.org/10.1214/14-AAP1068 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Counting connected hypergraphs via the probabilistic method
In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number
of connected graphs on with edges, whenever and the nullity
tend to infinity. Asymptotic formulae for the number of connected
-uniform hypergraphs on with edges and so nullity
were proved by Karo\'nski and \L uczak for the case ,
and Behrisch, Coja-Oghlan and Kang for . Here we prove such a
formula for any fixed, and any satisfying and
as . This leaves open only the (much simpler) case
, which we will consider in future work. ( arXiv:1511.04739 )
Our approach is probabilistic. Let denote the random -uniform
hypergraph on in which each edge is present independently with
probability . Let and be the numbers of vertices and edges in
the largest component of . We prove a local limit theorem giving an
asymptotic formula for the probability that and take any given pair
of values within the `typical' range, for any in the supercritical
regime, i.e., when where
and ; our enumerative result then follows
easily.
Taking as a starting point the recent joint central limit theorem for
and , we use smoothing techniques to show that `nearby' pairs of values
arise with about the same probability, leading to the local limit theorem.
Behrisch et al used similar ideas in a very different way, that does not seem
to work in our setting.
Independently, Sato and Wormald have recently proved the special case ,
with an additional restriction on . They use complementary, more enumerative
methods, which seem to have a more limited scope, but to give additional
information when they do work.Comment: Expanded; asymptotics clarified - no significant mathematical
changes. 67 pages (including appendix
Rigid abelian groups and the probabilistic method
The construction of torsion-free abelian groups with prescribed endomorphism
rings starting with Corner's seminal work is a well-studied subject in the
theory of abelian groups. Usually these construction work by adding elements
from a (topological) completion in order to get rid of (kill) unwanted
homomorphisms. The critical part is to actually prove that every unwanted
homomorphism can be killed by adding a suitable element. We will demonstrate
that some of those constructions can be significantly simplified by choosing
the elements at random. As a result, the endomorphism ring will be almost
surely prescribed, i.e., with probability one.Comment: 12 pages, submitted to the special volume of Contemporary Mathematics
for the proceedings of the conference Group and Model Theory, 201
Probabilistic Linear Solvers: A Unifying View
Several recent works have developed a new, probabilistic interpretation for
numerical algorithms solving linear systems in which the solution is inferred
in a Bayesian framework, either directly or by inferring the unknown action of
the matrix inverse. These approaches have typically focused on replicating the
behavior of the conjugate gradient method as a prototypical iterative method.
In this work surprisingly general conditions for equivalence of these disparate
methods are presented. We also describe connections between probabilistic
linear solvers and projection methods for linear systems, providing a
probabilistic interpretation of a far more general class of iterative methods.
In particular, this provides such an interpretation of the generalised minimum
residual method. A probabilistic view of preconditioning is also introduced.
These developments unify the literature on probabilistic linear solvers, and
provide foundational connections to the literature on iterative solvers for
linear systems
Global convergence rate analysis of unconstrained optimization methods based on probabilistic models
We present global convergence rates for a line-search method which is based
on random first-order models and directions whose quality is ensured only with
certain probability. We show that in terms of the order of the accuracy, the
evaluation complexity of such a method is the same as its counterparts that use
deterministic accurate models; the use of probabilistic models only increases
the complexity by a constant, which depends on the probability of the models
being good. We particularize and improve these results in the convex and
strongly convex case.
We also analyze a probabilistic cubic regularization variant that allows
approximate probabilistic second-order models and show improved complexity
bounds compared to probabilistic first-order methods; again, as a function of
the accuracy, the probabilistic cubic regularization bounds are of the same
(optimal) order as for the deterministic case
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