55,060 research outputs found

    Importance Sampling Variance Reduction for the Fokker-Planck Rarefied Gas Particle Method

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    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

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    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

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    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 kk-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., 22-set packing) problem is the design of an \emph{adaptive} algorithm that queries only a constant number of edges per vertex and achieves a (1−ϵ)(1-\epsilon) fraction of the omniscient optimal solution, for an arbitrarily small ϵ>0\epsilon>0. 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 (0.5−ϵ)(0.5 - \epsilon) fraction of the omniscient optimum. We also extend both our results to stochastic kk-set packing by designing an adaptive algorithm that achieves a (2k−ϵ)(\frac{2}{k} - \epsilon) fraction of the omniscient optimal solution, again with only O(1)O(1) queries per element. This guarantee is close to the best known polynomial-time approximation ratio of 3k+1−ϵ\frac{3}{k+1} -\epsilon for the \emph{deterministic} kk-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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