2,570 research outputs found
Optimal prediction for moment models: Crescendo diffusion and reordered equations
A direct numerical solution of the radiative transfer equation or any kinetic
equation is typically expensive, since the radiative intensity depends on time,
space and direction. An expansion in the direction variables yields an
equivalent system of infinitely many moments. A fundamental problem is how to
truncate the system. Various closures have been presented in the literature. We
want to study moment closure generally within the framework of optimal
prediction, a strategy to approximate the mean solution of a large system by a
smaller system, for radiation moment systems. We apply this strategy to
radiative transfer and show that several closures can be re-derived within this
framework, e.g. , diffusion, and diffusion correction closures. In
addition, the formalism gives rise to new parabolic systems, the reordered
equations, that are similar to the simplified equations.
Furthermore, we propose a modification to existing closures. Although simple
and with no extra cost, this newly derived crescendo diffusion yields better
approximations in numerical tests.Comment: Revised version: 17 pages, 6 figures, presented at Workshop on Moment
Methods in Kinetic Gas Theory, ETH Zurich, 2008 2 figures added, minor
correction
Small-Noise Analysis and Symmetrization of Implicit Monte Carlo Samplers
Implicit samplers are algorithms for producing independent, weighted samples from multivariate probability distributions. These are often applied in Bayesian data assimilation algorithms. We use Laplace asymptotic expansions to analyze two implicit samplers in the small noise regime. Our analysis suggests a symmetrization of the algorithms that leads to improved implicit sampling schemes at a relatively small additional cost. Computational experiments confirm the theory and show that symmetrization is effective for small noise sampling problems.© 2016 Wiley Periodicals, Inc
A discrete approach to stochastic parametrization and dimensional reduction in nonlinear dynamics
Many physical systems are described by nonlinear differential equations that
are too complicated to solve in full. A natural way to proceed is to divide the
variables into those that are of direct interest and those that are not,
formulate solvable approximate equations for the variables of greater interest,
and use data and statistical methods to account for the impact of the other
variables. In the present paper the problem is considered in a fully
discrete-time setting, which simplifies both the analysis of the data and the
numerical algorithms. The resulting time series are identified by a NARMAX
(nonlinear autoregression moving average with exogenous input) representation
familiar from engineering practice. The connections with the Mori-Zwanzig
formalism of statistical physics are discussed, as well as an application to
the Lorenz 96 system.Comment: 12 page, includes 2 figure
A comparative study of two stochastic mode reduction methods
We present a comparative study of two methods for the reduction of the
dimensionality of a system of ordinary differential equations that exhibits
time-scale separation. Both methods lead to a reduced system of stochastic
differential equations. The novel feature of these methods is that they allow
the use, in the reduced system, of higher order terms in the resolved
variables. The first method, proposed by Majda, Timofeyev and Vanden-Eijnden,
is based on an asymptotic strategy developed by Kurtz. The second method is a
short-memory approximation of the Mori-Zwanzig projection formalism of
irreversible statistical mechanics, as proposed by Chorin, Hald and Kupferman.
We present conditions under which the reduced models arising from the two
methods should have similar predictive ability. We apply the two methods to
test cases that satisfy these conditions. The form of the reduced models and
the numerical simulations show that the two methods have similar predictive
ability as expected.Comment: 35 pages, 6 figures. Under review in Physica
Optimal prediction in molecular dynamics
Optimal prediction approximates the average solution of a large system of
ordinary differential equations by a smaller system. We present how optimal
prediction can be applied to a typical problem in the field of molecular
dynamics, in order to reduce the number of particles to be tracked in the
computations. We consider a model problem, which describes a surface coating
process, and show how asymptotic methods can be employed to approximate the
high dimensional conditional expectations, which arise in optimal prediction.
The thus derived smaller system is compared to the original system in terms of
statistical quantities, such as diffusion constants. The comparison is carried
out by Monte-Carlo simulations, and it is shown under which conditions optimal
prediction yields a valid approximation to the original system.Comment: 22 pages, 10 figure
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