2,951 research outputs found
Simulation of diffusions by means of importance sampling paradigm
The aim of this paper is to introduce a new Monte Carlo method based on
importance sampling techniques for the simulation of stochastic differential
equations. The main idea is to combine random walk on squares or rectangles
methods with importance sampling techniques. The first interest of this
approach is that the weights can be easily computed from the density of the
one-dimensional Brownian motion. Compared to the Euler scheme this method
allows one to obtain a more accurate approximation of diffusions when one has
to consider complex boundary conditions. The method provides also an
interesting alternative to performing variance reduction techniques and
simulating rare events.Comment: Published in at http://dx.doi.org/10.1214/09-AAP659 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Hybrid PDE solver for data-driven problems and modern branching
The numerical solution of large-scale PDEs, such as those occurring in
data-driven applications, unavoidably require powerful parallel computers and
tailored parallel algorithms to make the best possible use of them. In fact,
considerations about the parallelization and scalability of realistic problems
are often critical enough to warrant acknowledgement in the modelling phase.
The purpose of this paper is to spread awareness of the Probabilistic Domain
Decomposition (PDD) method, a fresh approach to the parallelization of PDEs
with excellent scalability properties. The idea exploits the stochastic
representation of the PDE and its approximation via Monte Carlo in combination
with deterministic high-performance PDE solvers. We describe the ingredients of
PDD and its applicability in the scope of data science. In particular, we
highlight recent advances in stochastic representations for nonlinear PDEs
using branching diffusions, which have significantly broadened the scope of
PDD.
We envision this work as a dictionary giving large-scale PDE practitioners
references on the very latest algorithms and techniques of a non-standard, yet
highly parallelizable, methodology at the interface of deterministic and
probabilistic numerical methods. We close this work with an invitation to the
fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European
Journal of Applied Mathematics (EJAM
Asymptotic Exit Location Distributions in the Stochastic Exit Problem
Consider a two-dimensional continuous-time dynamical system, with an
attracting fixed point . If the deterministic dynamics are perturbed by
white noise (random perturbations) of strength , the system state
will eventually leave the domain of attraction of . We analyse the
case when, as , the exit location on the boundary
is increasingly concentrated near a saddle point of the
deterministic dynamics. We show that the asymptotic form of the exit location
distribution on is generically non-Gaussian and asymmetric,
and classify the possible limiting distributions. A key role is played by a
parameter , equal to the ratio of the stable
and unstable eigenvalues of the linearized deterministic flow at . If
then the exit location distribution is generically asymptotic as
to a Weibull distribution with shape parameter , on the
length scale near . If it is generically
asymptotic to a distribution on the length scale, whose
moments we compute. The asymmetry of the asymptotic exit location distribution
is attributable to the generic presence of a `classically forbidden' region: a
wedge-shaped subset of with as vertex, which is reached from ,
in the limit, only via `bent' (non-smooth) fluctuational paths
that first pass through the vicinity of . We deduce from the presence of
this forbidden region that the classical Eyring formula for the
small- exponential asymptotics of the mean first exit time is
generically inapplicable.Comment: This is a 72-page Postscript file, about 600K in length. Hardcopy
requests to [email protected] or [email protected]
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