303 research outputs found
Martingale approximations and anisotropic Banach spaces with an application to the time-one map of a Lorentz gas
In this paper, we show how the Gordin martingale approximation method fits
into the anisotropic Banach space framework. In particular, for the time-one
map of a finite horizon planar periodic Lorentz gas, we prove that Holder
observables satisfy statistical limit laws such as the central limit theorem
and associated invariance principles.Comment: Final version, to appear in Nonlinearity. Corrected some minor typos
from previous versio
Multidimensional potential Burgers turbulence
We consider the multidimensional generalised stochastic Burgers equation in
the space-periodic setting:
under the assumption that
is a gradient. Here is strongly convex and satisfies a growth
condition, is small and positive, while is a random forcing term,
smooth in space and white in time. For solutions of this equation,
we study Sobolev norms of averaged in time and in ensemble: each
of these norms behaves as a given negative power of . These results yield
sharp upper and lower bounds for natural analogues of quantities characterising
the hydrodynamical turbulence, namely the averages of the increments and of the
energy spectrum. These quantities behave as a power of the norm of the relevant
parameter, which is respectively the separation in the physical
space and the wavenumber in the Fourier space. Our bounds do not
depend on the initial condition and hold uniformly in . We generalise the
results obtained for the one-dimensional case in \cite{BorW}, confirming the
physical predictions in \cite{BK07,GMN10}. Note that the form of the estimates
does not depend on the dimension: the powers of
are the same in the one- and the multi-dimensional setting.Comment: arXiv admin note: substantial text overlap with arXiv:1201.556
Certified dimension reduction in nonlinear Bayesian inverse problems
We propose a dimension reduction technique for Bayesian inverse problems with
nonlinear forward operators, non-Gaussian priors, and non-Gaussian observation
noise. The likelihood function is approximated by a ridge function, i.e., a map
which depends non-trivially only on a few linear combinations of the
parameters. We build this ridge approximation by minimizing an upper bound on
the Kullback--Leibler divergence between the posterior distribution and its
approximation. This bound, obtained via logarithmic Sobolev inequalities,
allows one to certify the error of the posterior approximation. Computing the
bound requires computing the second moment matrix of the gradient of the
log-likelihood function. In practice, a sample-based approximation of the upper
bound is then required. We provide an analysis that enables control of the
posterior approximation error due to this sampling. Numerical and theoretical
comparisons with existing methods illustrate the benefits of the proposed
methodology
On rate of convergence of finite difference scheme for degenerate parabolic-hyperbolic PDE with Levy noise
In this article, we consider a semi discrete finite difference scheme for a
degenerate parabolic-hyperbolic PDE driven by L\'evy noise in one space
dimension. Using bounded variation estimations and a variant of classical
Kru\v{z}kov's doubling of variable approach, we prove that expected value of
the -difference between the unique entropy solution and approximate
solution converges at a rate of , where is
the spatial mesh size
- …