4,291 research outputs found
Multilevel Monte Carlo for Random Degenerate Scalar Convection Diffusion Equation
We consider the numerical solution of scalar, nonlinear degenerate
convection-diffusion problems with random diffusion coefficient and with random
flux functions. Building on recent results on the existence, uniqueness and
continuous dependence of weak solutions on data in the deterministic case, we
develop a definition of random entropy solution. We establish existence,
uniqueness, measurability and integrability results for these random entropy
solutions, generalizing \cite{Mishr478,MishSch10a} to possibly degenerate
hyperbolic-parabolic problems with random data. We next address the numerical
approximation of random entropy solutions, specifically the approximation of
the deterministic first and second order statistics. To this end, we consider
explicit and implicit time discretization and Finite Difference methods in
space, and single as well as Multi-Level Monte-Carlo methods to sample the
statistics. We establish convergence rate estimates with respect to the
discretization parameters, as well as with respect to the overall work,
indicating substantial gains in efficiency are afforded under realistic
regularity assumptions by the use of the Multi-Level Monte-Carlo method.
Numerical experiments are presented which confirm the theoretical convergence
estimates.Comment: 24 Page
The evolution of the Mira variable R Hydrae
The Mira variable R Hydrae is well known for its declining period, which Wood
& Zarro (1981) attributed to a possible recent thermal pulse. Here we
investigate the long-term period evolution, covering 340 years, going back to
its discovery in AD 1662. Wavelets are used to determine both the period and
semi-amplitude. We show that the period decreased linearly between 1770 and
1950; since 1950 the period has stabilized at 385 days. The semi-amplitude
closely follows the period evolution. Detailed analysis of the oldest data
shows that before 1770 the period was about 495 days. We find no evidence for
an increasing period during this time as found by Wood & Zarro. IRAS data shows
that the mass loss dropped dramatically around AD 1750. The decline agrees with
the mass-loss formalism from Vassiliadis & Wood, but is much larger than
predicted by the Bloecker mass-loss law. An outer detached IRAS shell suggests
that R Hya has experienced such mass-loss interruptions before. The period
evolution can be explained by a thermal pulse occuring around AD 1600, or by an
non-linear instability leading to an internal relaxation of the stellar
structure. The elapsed time between the mass-loss decline giving rise to the
outer detached shell, and the recent event, of approximately 5000 yr suggests
that only one of these events could be due to a thermal pulse. Further
monitoring of R Hya is recommended, as both models make strong predictions for
the future period evolution. R Hya-type events, on time scales of 10^2-10^3 yr,
could provide part of the explanation for the rings seen around some AGB and
post-AGB stars.Comment: 13 pages. MNRAS, accepted for publicatio
The quest for the solar g modes
Solar gravity modes (or g modes) -- oscillations of the solar interior for
which buoyancy acts as the restoring force -- have the potential to provide
unprecedented inference on the structure and dynamics of the solar core,
inference that is not possible with the well observed acoustic modes (or p
modes). The high amplitude of the g-mode eigenfunctions in the core and the
evanesence of the modes in the convection zone make the modes particularly
sensitive to the physical and dynamical conditions in the core. Owing to the
existence of the convection zone, the g modes have very low amplitudes at
photospheric levels, which makes the modes extremely hard to detect. In this
paper, we review the current state of play regarding attempts to detect g
modes. We review the theory of g modes, including theoretical estimation of the
g-mode frequencies, amplitudes and damping rates. Then we go on to discuss the
techniques that have been used to try to detect g modes. We review results in
the literature, and finish by looking to the future, and the potential advances
that can be made -- from both data and data-analysis perspectives -- to give
unambiguous detections of individual g modes. The review ends by concluding
that, at the time of writing, there is indeed a consensus amongst the authors
that there is currently no undisputed detection of solar g modes.Comment: 71 pages, 18 figures, accepted by Astronomy and Astrophysics Revie
Generalized Entropies
We study an entropy measure for quantum systems that generalizes the von
Neumann entropy as well as its classical counterpart, the Gibbs or Shannon
entropy. The entropy measure is based on hypothesis testing and has an elegant
formulation as a semidefinite program, a type of convex optimization. After
establishing a few basic properties, we prove upper and lower bounds in terms
of the smooth entropies, a family of entropy measures that is used to
characterize a wide range of operational quantities. From the formulation as a
semidefinite program, we also prove a result on decomposition of hypothesis
tests, which leads to a chain rule for the entropy.Comment: 21 page
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