24,989 research outputs found
The rational SPDE approach for Gaussian random fields with general smoothness
A popular approach for modeling and inference in spatial statistics is to
represent Gaussian random fields as solutions to stochastic partial
differential equations (SPDEs) of the form , where
is Gaussian white noise, is a second-order differential
operator, and is a parameter that determines the smoothness of .
However, this approach has been limited to the case ,
which excludes several important models and makes it necessary to keep
fixed during inference.
We propose a new method, the rational SPDE approach, which in spatial
dimension is applicable for any , and thus remedies
the mentioned limitation. The presented scheme combines a finite element
discretization with a rational approximation of the function to
approximate . For the resulting approximation, an explicit rate of
convergence to in mean-square sense is derived. Furthermore, we show that
our method has the same computational benefits as in the restricted case
. Several numerical experiments and a statistical
application are used to illustrate the accuracy of the method, and to show that
it facilitates likelihood-based inference for all model parameters including
.Comment: 28 pages, 4 figure
Computing covariant vectors, Lyapunov vectors, Oseledets vectors, and dichotomy projectors: a comparative numerical study
Covariant vectors, Lyapunov vectors, or Oseledets vectors are increasingly
being used for a variety of model analyses in areas such as partial
differential equations, nonautonomous differentiable dynamical systems, and
random dynamical systems. These vectors identify spatially varying directions
of specific asymptotic growth rates and obey equivariance principles. In recent
years new computational methods for approximating Oseledets vectors have been
developed, motivated by increasing model complexity and greater demands for
accuracy. In this numerical study we introduce two new approaches based on
singular value decomposition and exponential dichotomies and comparatively
review and improve two recent popular approaches of Ginelli et al. (2007) and
Wolfe and Samelson (2007). We compare the performance of the four approaches
via three case studies with very different dynamics in terms of symmetry,
spectral separation, and dimension. We also investigate which methods perform
well with limited data
The role of singularities in chaotic spectroscopy
We review the status of the semiclassical trace formula with emphasis on the
particular types of singularities that occur in the Gutzwiller-Voros zeta
function for bound chaotic systems. To understand the problem better we extend
the discussion to include various classical zeta functions and we contrast
properties of axiom-A scattering systems with those of typical bound systems.
Singularities in classical zeta functions contain topological and dynamical
information, concerning e.g. anomalous diffusion, phase transitions among
generalized Lyapunov exponents, power law decay of correlations. Singularities
in semiclassical zeta functions are artifacts and enters because one neglects
some quantum effects when deriving them, typically by making saddle point
approximation when the saddle points are not enough separated. The discussion
is exemplified by the Sinai billiard where intermittent orbits associated with
neutral orbits induce a branch point in the zeta functions. This singularity is
responsible for a diverging diffusion constant in Lorentz gases with unbounded
horizon. In the semiclassical case there is interference between neutral orbits
and intermittent orbits. The Gutzwiller-Voros zeta function exhibit a branch
point because it does not take this effect into account. Another consequence is
that individual states, high up in the spectrum, cannot be resolved by
Berry-Keating technique.Comment: 22 pages LaTeX, figures available from autho
Locally stationary long memory estimation
There exists a wide literature on modelling strongly dependent time series
using a longmemory parameter d, including more recent work on semiparametric
wavelet estimation. As a generalization of these latter approaches, in this
work we allow the long-memory parameter d to be varying over time. We embed our
approach into the framework of locally stationary processes. We show weak
consistency and a central limit theorem for our log-regression wavelet
estimator of the time-dependent d in a Gaussian context. Both simulations and a
real data example complete our work on providing a fairly general approach
On the rate of convergence to equilibrium for countable ergodic Markov chains
Using elementary methods, we prove that for a countable Markov chain of
ergodic degree the rate of convergence towards the stationary
distribution is subgeometric of order , provided the initial
distribution satisfies certain conditions of asymptotic decay. An example,
modelling a renewal process and providing a markovian approximation scheme in
dynamical system theory, is worked out in detail, illustrating the
relationships between convergence behaviour, analytic properties of the
generating functions associated to transition probabilities and spectral
properties of the Markov operator on the Banach space . Explicit
conditions allowing to obtain the actual asymptotics for the rate of
convergence are also discussed.Comment: 31 pages. to appear in Markov Processes and Related Field
Incompressible viscous fluid flows in a thin spherical shell
Linearized stability of incompressible viscous fluid flows in a thin
spherical shell is studied by using the two-dimensional Navier--Stokes
equations on a sphere. The stationary flow on the sphere has two singularities
(a sink and a source) at the North and South poles of the sphere. We prove
analytically for the linearized Navier--Stokes equations that the stationary
flow is asymptotically stable. When the spherical layer is truncated between
two symmetrical rings, we study eigenvalues of the linearized equations
numerically by using power series solutions and show that the stationary flow
remains asymptotically stable for all Reynolds numbers.Comment: 28 pages, 10 figure
- …