2,872 research outputs found
Topological Conformal Dimension
We investigate a quasisymmetrically invariant counterpart of the topological
Hausdorff dimension of a metric space. This invariant, called the topological
conformal dimension, gives a lower bound on the topological Hausdorff dimension
of quasisymmetric images of the space. We obtain results concerning the
behavior of this quantity under products and unions, and compute it for some
classical fractals. The range of possible values of the topological conformal
dimension is also considered, and we show that this quantity can be fractional.Comment: 16 pages, revised after referee's reports. To appear in Conformal
Geometry and Dynamic
Multi-scale control variate methods for uncertainty quantification in kinetic equations
Kinetic equations play a major rule in modeling large systems of interacting
particles. Uncertainties may be due to various reasons, like lack of knowledge
on the microscopic interaction details or incomplete informations at the
boundaries. These uncertainties, however, contribute to the curse of
dimensionality and the development of efficient numerical methods is a
challenge. In this paper we consider the construction of novel multi-scale
methods for such problems which, thanks to a control variate approach, are
capable to reduce the variance of standard Monte Carlo techniques
Asymptotic preserving Implicit-Explicit Runge-Kutta methods for non linear kinetic equations
We discuss Implicit-Explicit (IMEX) Runge Kutta methods which are
particularly adapted to stiff kinetic equations of Boltzmann type. We consider
both the case of easy invertible collision operators and the challenging case
of Boltzmann collision operators. We give sufficient conditions in order that
such methods are asymptotic preserving and asymptotically accurate. Their
monotonicity properties are also studied. In the case of the Boltzmann
operator, the methods are based on the introduction of a penalization technique
for the collision integral. This reformulation of the collision operator
permits to construct penalized IMEX schemes which work uniformly for a wide
range of relaxation times avoiding the expensive implicit resolution of the
collision operator. Finally we show some numerical results which confirm the
theoretical analysis
Fluid Solver Independent Hybrid Methods for Multiscale Kinetic equations
In some recent works [G. Dimarco, L. Pareschi, Hybrid multiscale methods I.
Hyperbolic Relaxation Problems, Comm. Math. Sci., 1, (2006), pp. 155-177], [G.
Dimarco, L. Pareschi, Hybrid multiscale methods II. Kinetic equations, SIAM
Multiscale Modeling and Simulation Vol 6., No 4,pp. 1169-1197, (2008)] we
developed a general framework for the construction of hybrid algorithms which
are able to face efficiently the multiscale nature of some hyperbolic and
kinetic problems. Here, at variance with respect to the previous methods, we
construct a method form-fitting to any type of finite volume or finite
difference scheme for the reduced equilibrium system. Thanks to the coupling of
Monte Carlo techniques for the solution of the kinetic equations with
macroscopic methods for the limiting fluid equations, we show how it is
possible to solve multiscale fluid dynamic phenomena faster with respect to
traditional deterministic/stochastic methods for the full kinetic equations. In
addition, due to the hybrid nature of the schemes, the numerical solution is
affected by less fluctuations when compared to standard Monte Carlo schemes.
Applications to the Boltzmann-BGK equation are presented to show the
performance of the new methods in comparison with classical approaches used in
the simulation of kinetic equations.Comment: 31 page
Fluid Simulations with Localized Boltzmann Upscaling by Direct Simulation Monte-Carlo
In the present work, we present a novel numerical algorithm to couple the
Direct Simulation Monte Carlo method (DSMC) for the solution of the Boltzmann
equation with a finite volume like method for the solution of the Euler
equations. Recently we presented in [14],[16],[17] different methodologies
which permit to solve fluid dynamics problems with localized regions of
departure from thermodynamical equilibrium. The methods rely on the
introduction of buffer zones which realize a smooth transition between the
kinetic and the fluid regions. In this paper we extend the idea of buffer zones
and dynamic coupling to the case of the Monte Carlo methods. To facilitate the
coupling and avoid the onset of spurious oscillations in the fluid regions
which are consequences of the coupling with a stochastic numerical scheme, we
use a new technique which permits to reduce the variance of the particle
methods [11]. In addition, the use of this method permits to obtain estimations
of the breakdowns of the fluid models less affected by fluctuations and
consequently to reduce the kinetic regions and optimize the coupling. In the
last part of the paper several numerical examples are presented to validate the
method and measure its computational performances
Effects of Transport Delays of Manual Control System Performance
Throughput or transport delays in manual control systems can cause degraded performance and lead to potentially unstable operation. With the expanding use of digital processors, throughput delays can occur in manual control systems in a variety of ways such as in digital flight control systems in real aircraft, and in equation of motion computers and computer generated images in simulators. Research has shown the degrading effect of throughput delays on subjective opinion and system performance and dynamic response. A generic manual control system model is used to provide a relatively simple analysis of and explanation for the effects of various types of delays. The consequence of throughput delays of some simple system architectures is also discussed
Implicit-Explicit multistep methods for hyperbolic systems with multiscale relaxation
We consider the development of high order space and time numerical methods
based on Implicit-Explicit (IMEX) multistep time integrators for hyperbolic
systems with relaxation. More specifically, we consider hyperbolic balance laws
in which the convection and the source term may have very different time and
space scales. As a consequence the nature of the asymptotic limit changes
completely, passing from a hyperbolic to a parabolic system. From the
computational point of view, standard numerical methods designed for the
fluid-dynamic scaling of hyperbolic systems with relaxation present several
drawbacks and typically lose efficiency in describing the parabolic limit
regime. In this work, in the context of Implicit-Explicit linear multistep
methods we construct high order space-time discretizations which are able to
handle all the different scales and to capture the correct asymptotic behavior,
independently from its nature, without time step restrictions imposed by the
fast scales. Several numerical examples confirm the theoretical analysis
Asymptotic-Preserving Monte Carlo methods for transport equations in the diffusive limit
We develop a new Monte Carlo method that solves hyperbolic transport
equations with stiff terms, characterized by a (small) scaling parameter. In
particular, we focus on systems which lead to a reduced problem of parabolic
type in the limit when the scaling parameter tends to zero. Classical Monte
Carlo methods suffer of severe time step limitations in these situations, due
to the fact that the characteristic speeds go to infinity in the diffusion
limit. This makes the problem a real challenge, since the scaling parameter may
differ by several orders of magnitude in the domain. To circumvent these time
step limitations, we construct a new, asymptotic-preserving Monte Carlo method
that is stable independently of the scaling parameter and degenerates to a
standard probabilistic approach for solving the limiting equation in the
diffusion limit. The method uses an implicit time discretization to formulate a
modified equation in which the characteristic speeds do not grow indefinitely
when the scaling factor tends to zero. The resulting modified equation can
readily be discretized by a Monte Carlo scheme, in which the particles combine
a finite propagation speed with a time-step dependent diffusion term. We show
the performance of the method by comparing it with standard (deterministic)
approaches in the literature
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