20,773 research outputs found
Discretization-related issues in the KPZ equation: Consistency, Galilean-invariance violation, and fluctuation--dissipation relation
In order to perform numerical simulations of the KPZ equation, in any
dimensionality, a spatial discretization scheme must be prescribed. The known
fact that the KPZ equation can be obtained as a result of a Hopf--Cole
transformation applied to a diffusion equation (with \emph{multiplicative}
noise) is shown here to strongly restrict the arbitrariness in the choice of
spatial discretization schemes. On one hand, the discretization prescriptions
for the Laplacian and the nonlinear (KPZ) term cannot be independently chosen.
On the other hand, since the discretization is an operation performed on
\emph{space} and the Hopf--Cole transformation is \emph{local} both in space
and time, the former should be the same regardless of the field to which it is
applied. It is shown that whereas some discretization schemes pass both
consistency tests, known examples in the literature do not. The requirement of
consistency for the discretization of Lyapunov functionals is argued to be a
natural and safe starting point in choosing spatial discretization schemes. We
also analyze the relation between real-space and pseudo-spectral discrete
representations. In addition we discuss the relevance of the Galilean
invariance violation in these consistent discretization schemes, and the
alleged conflict of standard discretization with the fluctuation--dissipation
theorem, peculiar of 1D.Comment: RevTex, 23pgs, 2 figures, submitted to Phys. Rev.
Sparse Quadrature for High-Dimensional Integration with Gaussian Measure
In this work we analyze the dimension-independent convergence property of an
abstract sparse quadrature scheme for numerical integration of functions of
high-dimensional parameters with Gaussian measure. Under certain assumptions of
the exactness and the boundedness of univariate quadrature rules as well as the
regularity of the parametric functions with respect to the parameters, we
obtain the convergence rate , where is the number of indices,
and is independent of the number of the parameter dimensions. Moreover, we
propose both an a-priori and an a-posteriori schemes for the construction of a
practical sparse quadrature rule and perform numerical experiments to
demonstrate their dimension-independent convergence rates
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
- …