22,886 research outputs found
Bayesian linear inverse problems in regularity scales
We obtain rates of contraction of posterior distributions in inverse problems
defined by scales of smoothness classes. We derive abstract results for general
priors, with contraction rates determined by Galerkin approximation. The rate
depends on the amount of prior concentration near the true function and the
prior mass of functions with inferior Galerkin approximation. We apply the
general result to non-conjugate series priors, showing that these priors give
near optimal and adaptive recovery in some generality, Gaussian priors, and
mixtures of Gaussian priors, where the latter are also shown to be near optimal
and adaptive. The proofs are based on general testing and approximation
arguments, without explicit calculations on the posterior distribution. We are
thus not restricted to priors based on the singular value decomposition of the
operator. We illustrate the results with examples of inverse problems resulting
from differential equations.Comment: 34 page
Implicit iteration methods in Hilbert scales under general smoothness conditions
For solving linear ill-posed problems regularization methods are required
when the right hand side is with some noise. In the present paper regularized
solutions are obtained by implicit iteration methods in Hilbert scales. % By
exploiting operator monotonicity of certain functions and interpolation
techniques in variable Hilbert scales, we study these methods under general
smoothness conditions. Order optimal error bounds are given in case the
regularization parameter is chosen either {\it a priori} or {\it a posteriori}
by the discrepancy principle. For realizing the discrepancy principle, some
fast algorithm is proposed which is based on Newton's method applied to some
properly transformed equations
Loop quantum cosmology and the k = - 1 RW model
The loop quantization of the negatively curved k=-1 RW model poses several
technical challenges. We show that the issues can be overcome and a successful
quantization is possible that extends the results of the k=0,+1 models in a
natural fashion. We discuss the resulting dynamics and show that for a universe
consisting of a massless scalar field, a bounce is predicted in the backward
evolution in accordance with the results of the k=0,+1 models. We also show
that the model predicts a vacuum repulsion in the high curvature regime that
would lead to a bounce even for matter with vanishing energy density. We
finally comment on the inverse volume modifications of loop quantum cosmology
and show that, as in the k=0 model, the modifications depend sensitively on the
introduction of a length scale which a priori is independent of the curvature
scale or a matter energy scale.Comment: Clarified some of the discussion and updated reference
Inflationary scalar spectrum in loop quantum cosmology
In the context of loop quantum cosmology, we consider an inflationary era
driven by a canonical scalar field and occurring in the semiclassical regime,
where spacetime is a continuum but quantum gravitational effects are important.
The spectral amplitude and index of scalar perturbations on an unperturbed de
Sitter background are computed at lowest order in the slow-roll parameters. The
scalar spectrum can be blue-tilted and far from scale invariance, and tuning of
the quantization ambiguities is necessary for agreement with observations. The
results are extended to a generalized quantization scheme including those
proposed in the literature. Quantization of the matter field at sub-horizon
scales can provide a consistency check of such schemes.Comment: 29 pages, 2 figures. v2: typos corrected, discussion improved and
extended, new section added. Conclusions are unchange
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