55,060 research outputs found
Importance Sampling Variance Reduction for the Fokker-Planck Rarefied Gas Particle Method
Models and methods that are able to accurately and efficiently predict the
flows of low-speed rarefied gases are in high demand, due to the increasing
ability to manufacture devices at micro and nano scales. One such model and
method is a Fokker-Planck approximation to the Boltzmann equation, which can be
solved numerically by a stochastic particle method. The stochastic nature of
this method leads to noisy estimates of the thermodynamic quantities one wishes
to sample when the signal is small in comparison to the thermal velocity of the
gas. Recently, Gorji et al have proposed a method which is able to greatly
reduce the variance of the estimators, by creating a correlated stochastic
process which acts as a control variate for the noisy estimates. However, there
are potential difficulties involved when the geometry of the problem is
complex, as the method requires the density to be solved for independently.
Importance sampling is a variance reduction technique that has already been
shown to successfully reduce the noise in direct simulation Monte Carlo
calculations. In this paper we propose an importance sampling method for the
Fokker-Planck stochastic particle scheme. The method requires minimal change to
the original algorithm, and dramatically reduces the variance of the estimates.
We test the importance sampling scheme on a homogeneous relaxation, planar
Couette flow and a lid-driven-cavity flow, and find that our method is able to
greatly reduce the noise of estimated quantities. Significantly, we find that
as the characteristic speed of the flow decreases, the variance of the noisy
estimators becomes independent of the characteristic speed
Spatially Adaptive Stochastic Multigrid Methods for Fluid-Structure Systems with Thermal Fluctuations
In microscopic mechanical systems interactions between elastic structures are
often mediated by the hydrodynamics of a solvent fluid. At microscopic scales
the elastic structures are also subject to thermal fluctuations. Stochastic
numerical methods are developed based on multigrid which allow for the
efficient computation of both the hydrodynamic interactions in the presence of
walls and the thermal fluctuations. The presented stochastic multigrid approach
provides efficient real-space numerical methods for generating the required
stochastic driving fields with long-range correlations consistent with
statistical mechanics. The presented approach also allows for the use of
spatially adaptive meshes in resolving the hydrodynamic interactions. Numerical
results are presented which show the methods perform in practice with a
computational complexity of O(N log(N))
Ignorance is Almost Bliss: Near-Optimal Stochastic Matching With Few Queries
The stochastic matching problem deals with finding a maximum matching in a
graph whose edges are unknown but can be accessed via queries. This is a
special case of stochastic -set packing, where the problem is to find a
maximum packing of sets, each of which exists with some probability. In this
paper, we provide edge and set query algorithms for these two problems,
respectively, that provably achieve some fraction of the omniscient optimal
solution.
Our main theoretical result for the stochastic matching (i.e., -set
packing) problem is the design of an \emph{adaptive} algorithm that queries
only a constant number of edges per vertex and achieves a
fraction of the omniscient optimal solution, for an arbitrarily small
. Moreover, this adaptive algorithm performs the queries in only a
constant number of rounds. We complement this result with a \emph{non-adaptive}
(i.e., one round of queries) algorithm that achieves a
fraction of the omniscient optimum. We also extend both our results to
stochastic -set packing by designing an adaptive algorithm that achieves a
fraction of the omniscient optimal solution, again
with only queries per element. This guarantee is close to the best known
polynomial-time approximation ratio of for the
\emph{deterministic} -set packing problem [Furer and Yu, 2013]
We empirically explore the application of (adaptations of) these algorithms
to the kidney exchange problem, where patients with end-stage renal failure
swap willing but incompatible donors. We show on both generated data and on
real data from the first 169 match runs of the UNOS nationwide kidney exchange
that even a very small number of non-adaptive edge queries per vertex results
in large gains in expected successful matches
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