224 research outputs found
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
Fast algorithms for computing the Boltzmann collision operator
The development of accurate and fast numerical schemes for the five fold
Boltzmann collision integral represents a challenging problem in scientific
computing. For a particular class of interactions, including the so-called hard
spheres model in dimension three, we are able to derive spectral methods that
can be evaluated through fast algorithms. These algorithms are based on a
suitable representation and approximation of the collision operator. Explicit
expressions for the errors in the schemes are given and spectral accuracy is
proved. Parallelization properties and adaptivity of the algorithms are also
discussed.Comment: 22 page
Residual equilibrium schemes for time dependent partial differential equations
Many applications involve partial differential equations which admits
nontrivial steady state solutions. The design of schemes which are able to
describe correctly these equilibrium states may be challenging for numerical
methods, in particular for high order ones. In this paper, inspired by
micro-macro decomposition methods for kinetic equations, we present a class of
schemes which are capable to preserve the steady state solution and achieve
high order accuracy for a class of time dependent partial differential
equations including nonlinear diffusion equations and kinetic equations.
Extension to systems of conservation laws with source terms are also discussed.Comment: 23 pages, 12 figure
Wealth distribution and collective knowledge. A Boltzmann approach
We introduce and discuss a nonlinear kinetic equation of Boltzmann type which
describes the influence of knowledge in the evolution of wealth in a system of
agents which interact through the binary trades introduced in Cordier,
Pareschi, Toscani, J. Stat. Phys. 2005. The trades, which include both saving
propensity and the risks of the market, are here modified in the risk and
saving parameters, which now are assumed to depend on the personal degree of
knowledge. The numerical simulations show that the presence of knowledge has
the potential to produce a class of wealthy agents and to account for a larger
proportion of wealth inequality.Comment: 21 pages, 10 figures. arXiv admin note: text overlap with
arXiv:q-bio/0312018 by other author
Binary interaction algorithms for the simulation of flocking and swarming dynamics
Microscopic models of flocking and swarming takes in account large numbers of
interacting individ- uals. Numerical resolution of large flocks implies huge
computational costs. Typically for interacting individuals we have a cost
of . We tackle the problem numerically by considering approximated
binary interaction dynamics described by kinetic equations and simulating such
equations by suitable stochastic methods. This approach permits to compute
approximate solutions as functions of a small scaling parameter
at a reduced complexity of O(N) operations. Several numerical results show the
efficiency of the algorithms proposed
Mean field mutation dynamics and the continuous Luria-Delbr\"uck distribution
The Luria-Delbr\"uck mutation model has a long history and has been
mathematically formulated in several different ways. Here we tackle the problem
in the case of a continuous distribution using some mathematical tools from
nonlinear statistical physics. Starting from the classical formulations we
derive the corresponding differential models and show that under a suitable
mean field scaling they correspond to generalized Fokker-Planck equations for
the mutants distribution whose solutions are given by the corresponding
Luria-Delbr\"uck distribution. Numerical results confirming the theoretical
analysis are also presented
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