3,068 research outputs found
Numerical approximation of high-dimensional Fokker-Planck equations with polynomial coefficients
This paper is concerned with the numerical solution of high-dimensional Fokker-
Planck equations related to multi-dimensional diffusion with polynomial coefficients
or Pearson diffusions. Classification of multi-dimensional Pearson diffusion follows
from the classification of one-dimensional Pearson diffusion. There are six important
classes of Pearson diffusion - three of them possess an infinite system of moments
(Gaussian, Gamma, Beta) while the other three possess a finite number of moments
(inverted Gamma, Student and Fisher-Snedecor). Numerical approximations to the
solution of the Fokker-Planck equation are generated using the spectral method.
The use of an adaptive reduced basis technique facilitates a significant reduction in
the number of degrees of freedom required in the approximation through the determination
of an optimal basis using the singular value decomposition (SVD). The
basis functions are constructed dynamically so that the numerical approximation
is optimal in the current finite-dimensional subspace of the solution space. This is
achieved through basis enrichment and projection stages. Numerical results with different
boundary conditions are presented to demonstrate the accuracy and efficiency
of the numerical scheme
Statistics of non-linear stochastic dynamical systems under L\'evy noises by a convolution quadrature approach
This paper describes a novel numerical approach to find the statistics of the
non-stationary response of scalar non-linear systems excited by L\'evy white
noises. The proposed numerical procedure relies on the introduction of an
integral transform of Wiener-Hopf type into the equation governing the
characteristic function. Once this equation is rewritten as partial
integro-differential equation, it is then solved by applying the method of
convolution quadrature originally proposed by Lubich, here extended to deal
with this particular integral transform. The proposed approach is relevant for
two reasons: 1) Statistics of systems with several different drift terms can be
handled in an efficient way, independently from the kind of white noise; 2) The
particular form of Wiener-Hopf integral transform and its numerical evaluation,
both introduced in this study, are generalizations of fractional
integro-differential operators of potential type and Gr\"unwald-Letnikov
fractional derivatives, respectively.Comment: 20 pages, 5 figure
Improved estimation of Fokker-Planck equations through optimisation
An improved method for the description of hierarchical complex systems by
means of a Fokker-Planck equation is presented. In particular the
limited-memory Broyden-Fletcher-Goldfarb-Shanno algorithm for constraint
problems (L-BFGS-B) is used to minimize the distance between the numerical
solutions of the Fokker-Planck equation and the empirical probability density
functions and thus to estimate properly the drift and diffusion term of the
Fokker-Planck equation. The optimisation routine is applied to a time series of
velocity measurements obtained from a turbulent helium gas jet in order to
demonstrate the benefits and to quantify the improvements of this new
optimisation routine
Logarithmic Gradient Transformation and Chaos Expansion of Ito Processes
Since the seminal work of Wiener, the chaos expansion has evolved to a
powerful methodology for studying a broad range of stochastic differential
equations. Yet its complexity for systems subject to the white noise remains
significant. The issue appears due to the fact that the random increments
generated by the Brownian motion, result in a growing set of random variables
with respect to which the process could be measured. In order to cope with this
high dimensionality, we present a novel transformation of stochastic processes
driven by the white noise. In particular, we show that under suitable
assumptions, the diffusion arising from white noise can be cast into a
logarithmic gradient induced by the measure of the process. Through this
transformation, the resulting equation describes a stochastic process whose
randomness depends only upon the initial condition. Therefore the stochasticity
of the transformed system lives in the initial condition and thereby it can be
treated conveniently with the chaos expansion tools
Finite element approximation of high-dimensional transport-dominated diffusion problems
High-dimensional partial differential equations with nonnegative characteristic form arise in numerous mathematical models in science. In problems of this kind, the computational challenge of beating the exponential growth of complexity as a function of dimension is exacerbated by the fact that the problem may be transport-dominated. We develop the analysis of stabilised sparse finite element methods for such high-dimensional, non-self-adjoint and possibly degenerate partial differential equations.\ud
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(Presented as an invited lecture under the title "Computational multiscale modelling: Fokker-Planck equations and their numerical analysis" at the Foundations of Computational Mathematics conference in Santander, Spain, 30 June - 9 July, 2005.
Corrections to Einstein's relation for Brownian motion in a tilted periodic potential
In this paper we revisit the problem of Brownian motion in a tilted periodic
potential. We use homogenization theory to derive general formulas for the
effective velocity and the effective diffusion tensor that are valid for
arbitrary tilts. Furthermore, we obtain power series expansions for the
velocity and the diffusion coefficient as functions of the external forcing.
Thus, we provide systematic corrections to Einstein's formula and to linear
response theory. Our theoretical results are supported by extensive numerical
simulations. For our numerical experiments we use a novel spectral numerical
method that leads to a very efficient and accurate calculation of the effective
velocity and the effective diffusion tensor.Comment: 29 pages, 7 figures, submitted to the Journal of Statistical Physic
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