96,193 research outputs found
Some Results on the Complexity of Numerical Integration
This is a survey (21 pages, 124 references) written for the MCQMC 2014
conference in Leuven, April 2014. We start with the seminal paper of Bakhvalov
(1959) and end with new results on the curse of dimension and on the complexity
of oscillatory integrals. Some small errors of earlier versions are corrected
Lagrangian Reachabililty
We introduce LRT, a new Lagrangian-based ReachTube computation algorithm that
conservatively approximates the set of reachable states of a nonlinear
dynamical system. LRT makes use of the Cauchy-Green stretching factor (SF),
which is derived from an over-approximation of the gradient of the solution
flows. The SF measures the discrepancy between two states propagated by the
system solution from two initial states lying in a well-defined region, thereby
allowing LRT to compute a reachtube with a ball-overestimate in a metric where
the computed enclosure is as tight as possible. To evaluate its performance, we
implemented a prototype of LRT in C++/Matlab, and ran it on a set of
well-established benchmarks. Our results show that LRT compares very favorably
with respect to the CAPD and Flow* tools.Comment: Accepted to CAV 201
The adaptive patched cubature filter and its implementation
There are numerous contexts where one wishes to describe the state of a
randomly evolving system. Effective solutions combine models that quantify the
underlying uncertainty with available observational data to form scientifically
reasonable estimates for the uncertainty in the system state. Stochastic
differential equations are often used to mathematically model the underlying
system.
The Kusuoka-Lyons-Victoir (KLV) approach is a higher order particle method
for approximating the weak solution of a stochastic differential equation that
uses a weighted set of scenarios to approximate the evolving probability
distribution to a high order of accuracy. The algorithm can be performed by
integrating along a number of carefully selected bounded variation paths. The
iterated application of the KLV method has a tendency for the number of
particles to increase. This can be addressed and, together with local dynamic
recombination, which simplifies the support of discrete measure without harming
the accuracy of the approximation, the KLV method becomes eligible to solve the
filtering problem in contexts where one desires to maintain an accurate
description of the ever-evolving conditioned measure.
In addition to the alternate application of the KLV method and recombination,
we make use of the smooth nature of the likelihood function and high order
accuracy of the approximations to lead some of the particles immediately to the
next observation time and to build into the algorithm a form of automatic high
order adaptive importance sampling.Comment: to appear in Communications in Mathematical Sciences. arXiv admin
note: substantial text overlap with arXiv:1311.675
The Neural Particle Filter
The robust estimation of dynamically changing features, such as the position
of prey, is one of the hallmarks of perception. On an abstract, algorithmic
level, nonlinear Bayesian filtering, i.e. the estimation of temporally changing
signals based on the history of observations, provides a mathematical framework
for dynamic perception in real time. Since the general, nonlinear filtering
problem is analytically intractable, particle filters are considered among the
most powerful approaches to approximating the solution numerically. Yet, these
algorithms prevalently rely on importance weights, and thus it remains an
unresolved question how the brain could implement such an inference strategy
with a neuronal population. Here, we propose the Neural Particle Filter (NPF),
a weight-less particle filter that can be interpreted as the neuronal dynamics
of a recurrently connected neural network that receives feed-forward input from
sensory neurons and represents the posterior probability distribution in terms
of samples. Specifically, this algorithm bridges the gap between the
computational task of online state estimation and an implementation that allows
networks of neurons in the brain to perform nonlinear Bayesian filtering. The
model captures not only the properties of temporal and multisensory integration
according to Bayesian statistics, but also allows online learning with a
maximum likelihood approach. With an example from multisensory integration, we
demonstrate that the numerical performance of the model is adequate to account
for both filtering and identification problems. Due to the weightless approach,
our algorithm alleviates the 'curse of dimensionality' and thus outperforms
conventional, weighted particle filters in higher dimensions for a limited
number of particles
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