55,683 research outputs found
Solving the stationary Liouville equation via a boundary element method
Intensity distributions of linear wave fields are, in the high frequency
limit, often approximated in terms of flow or transport equations in phase
space. Common techniques for solving the flow equations for both time dependent
and stationary problems are ray tracing or level set methods. In the context of
predicting the vibro-acoustic response of complex engineering structures,
reduced ray tracing methods such as Statistical Energy Analysis or variants
thereof have found widespread applications. Starting directly from the
stationary Liouville equation, we develop a boundary element method for solving
the transport equations for complex multi-component structures. The method,
which is an improved version of the Dynamical Energy Analysis technique
introduced recently by the authors, interpolates between standard statistical
energy analysis and full ray tracing, containing both of these methods as
limiting cases. We demonstrate that the method can be used to efficiently deal
with complex large scale problems giving good approximations of the energy
distribution when compared to exact solutions of the underlying wave equation
High-Dimensional Stochastic Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition
This paper presents a novel adaptive-sparse polynomial dimensional
decomposition (PDD) method for stochastic design optimization of complex
systems. The method entails an adaptive-sparse PDD approximation of a
high-dimensional stochastic response for statistical moment and reliability
analyses; a novel integration of the adaptive-sparse PDD approximation and
score functions for estimating the first-order design sensitivities of the
statistical moments and failure probability; and standard gradient-based
optimization algorithms. New analytical formulae are presented for the design
sensitivities that are simultaneously determined along with the moments or the
failure probability. Numerical results stemming from mathematical functions
indicate that the new method provides more computationally efficient design
solutions than the existing methods. Finally, stochastic shape optimization of
a jet engine bracket with 79 variables was performed, demonstrating the power
of the new method to tackle practical engineering problems.Comment: 18 pages, 2 figures, to appear in Sparse Grids and
Applications--Stuttgart 2014, Lecture Notes in Computational Science and
Engineering 109, edited by J. Garcke and D. Pfl\"{u}ger, Springer
International Publishing, 201
Double power series method for approximating cosmological perturbations
We introduce a double power series method for finding approximate analytical
solutions for systems of differential equations commonly found in cosmological
perturbation theory. The method was set out, in a non-cosmological context, by
Feshchenko, Shkil' and Nikolenko (FSN) in 1966, and is applicable to cases
where perturbations are on sub-horizon scales. The FSN method is essentially an
extension of the well known Wentzel-Kramers-Brillouin (WKB) method for finding
approximate analytical solutions for ordinary differential equations. The FSN
method we use is applicable well beyond perturbation theory to solve systems of
ordinary differential equations, linear in the derivatives, that also depend on
a small parameter, which here we take to be related to the inverse wave-number.
We use the FSN method to find new approximate oscillating solutions in linear
order cosmological perturbation theory for a flat radiation-matter universe.
Together with this model's well known growing and decaying M\'esz\'aros
solutions, these oscillating modes provide a complete set of sub-horizon
approximations for the metric potential, radiation and matter perturbations.
Comparison with numerical solutions of the perturbation equations shows that
our approximations can be made accurate to within a typical error of 1%, or
better. We also set out a heuristic method for error estimation. A Mathematica
notebook which implements the double power series method is made available
online.Comment: 22 pages, 10 figures, 2 tables. Mathematica notebook available from
Github at https://github.com/AndrewWren/Double-power-series.gi
Methods for Estimating Capacities and Rates of Gaussian Quantum Channels
Optimization methods aimed at estimating the capacities of a general Gaussian
channel are developed. Specifically evaluation of classical capacity as maximum
of the Holevo information is pursued over all possible Gaussian encodings for
the lossy bosonic channel, but extension to other capacities and other Gaussian
channels seems feasible. Solutions for both memoryless and memory channels are
presented. It is first dealt with single use (single-mode) channel where the
capacity dependence from channel's parameters is analyzed providing a full
classification of the possible cases. Then it is dealt with multiple uses
(multi-mode) channel where the capacity dependence from the (multi-mode)
environment state is analyzed when both total environment energy and
environment purity are fixed. This allows a fair comparison among different
environments, thus understanding the role of memory (inter-mode correlations)
and phenomenon like superadditivity of the capacity. The developed methods are
also used for deriving transmission rates with heterodyne and homodyne
measurements at the channel output. Classical capacity and transmission rates
are presented within a unique framework where the rates can be treated as
logarithmic approximations of the capacity.Comment: 39 pages, 30 figures. New results and graphs were added. Errors and
misprints were corrected. To appear in IEEE Trans. Inf. T
Kinematic model-independent reconstruction of Palatini cosmology
A kinematic treatment to trace out the form of cosmology, within the
Palatini formalism, is discussed by only postulating the universe homogeneity
and isotropy. To figure this out we build model-independent approximations of
the luminosity distance through rational expansions. These approximants extend
the Taylor convergence radii computed for usual cosmographic series. We thus
consider both Pad\'e and the rational Chebyshev polynomials. They can be used
to accurately describe the universe late-time expansion history, providing
further information on the thermal properties of all effective cosmic fluids
entering the energy momentum tensor of Palatini's gravity. To perform our
numerical analysis, we relate the Palatini's Ricci scalar with the Hubble
parameter and thus we write down a single differential equation in terms of
the redshift . Therefore, to bound , we make use of the most recent
outcomes over the cosmographic parameters obtained from combined data surveys.
In particular our clue is to select two scenarios, i.e. Pad\'e and
Chebyshev approximations, since they well approximate the luminosity
distance at the lowest possible order. We find that best analytical matches to
the numerical solutions lead to with free parameters given by the
set for Pad\'e approximation,
whereas with for rational Chebyshev approximation. Finally, our results are
compared with the CDM predictions and with previous studies in the
literature. Slight departures from General Relativity are also discussed.Comment: 10 pages, 6 figures. Accepted for publication in Gen. Rel. Gra
Testing Two-Field Inflation
We derive semi-analytic formulae for the power spectra of two-field inflation
assuming an arbitrary potential and non-canonical kinetic terms, and we use
them both to build phenomenological intuition and to constrain classes of
two-field models using WMAP data. Using covariant formalism, we first develop a
framework for understanding the background field kinematics and introduce a
"slow-turn" approximation. Next, we find covariant expressions for the
evolution of the adiabatic/curvature and entropy/isocurvature modes, and we
discuss how the mode evolution can be inferred directly from the background
kinematics and the geometry of the field manifold. From these expressions, we
derive semi-analytic formulae for the curvature, isocurvature, and cross
spectra, and the spectral observables, all to second-order in the slow-roll and
slow-turn approximations. In tandem, we show how our covariant formalism
provides useful intuition into how the characteristics of the inflationary
Lagrangian translate into distinct features in the power spectra. In
particular, we find that key features of the power spectra can be directly read
off of the nature of the roll path, the curve the field vector rolls along with
respect to the field manifold. For example, models whose roll path makes a
sharp turn 60 e-folds before inflation ends tend to be ruled out because they
produce strong departures from scale invariance. Finally, we apply our
formalism to confront four classes of two-field models with WMAP data,
including doubly quadratic and quartic potentials and non-standard kinetic
terms, showing how whether a model is ruled out depends not only on certain
features of the inflationary Lagrangian, but also on the initial conditions.
Ultimately, models must possess the right balance of kinematical and dynamical
behaviors, which we capture in a set of functions that can be reconstructed
from spectral observables.Comment: Revised to match accepted PRD version: Improved discussion of
background kinematics and multi-field effects, added tables summarizing key
quantities and their links to observables, more detailed figures, fixed typos
in former equations (103) and (117). 49 PRD pages, 11 figure
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