781 research outputs found
Spectral radii of asymptotic mappings and the convergence speed of the standard fixed point algorithm
Important problems in wireless networks can often be solved by computing
fixed points of standard or contractive interference mappings, and the
conventional fixed point algorithm is widely used for this purpose. Knowing
that the mapping used in the algorithm is not only standard but also
contractive (or only contractive) is valuable information because we obtain a
guarantee of geometric convergence rate, and the rate is related to a property
of the mapping called modulus of contraction. To date, contractive mappings and
their moduli of contraction have been identified with case-by-case approaches
that can be difficult to generalize. To address this limitation of existing
approaches, we show in this study that the spectral radii of asymptotic
mappings can be used to identify an important subclass of contractive mappings
and also to estimate their moduli of contraction. In addition, if the fixed
point algorithm is applied to compute fixed points of positive concave
mappings, we show that the spectral radii of asymptotic mappings provide us
with simple lower bounds for the estimation error of the iterates. An immediate
application of this result proves that a known algorithm for load estimation in
wireless networks becomes slower with increasing traffic.Comment: Paper accepted for presentation at ICASSP 201
Euclidean algorithms are Gaussian
This study provides new results about the probabilistic behaviour of a class
of Euclidean algorithms: the asymptotic distribution of a whole class of
cost-parameters associated to these algorithms is normal. For the cost
corresponding to the number of steps Hensley already has proved a Local Limit
Theorem; we give a new proof, and extend his result to other euclidean
algorithms and to a large class of digit costs, obtaining a faster, optimal,
rate of convergence. The paper is based on the dynamical systems methodology,
and the main tool is the transfer operator. In particular, we use recent
results of Dolgopyat.Comment: fourth revised version - 2 figures - the strict convexity condition
used has been clarifie
Spectral methods for CFD
One of the objectives of these notes is to provide a basic introduction to spectral methods with a particular emphasis on applications to computational fluid dynamics. Another objective is to summarize some of the most important developments in spectral methods in the last two years. The fundamentals of spectral methods for simple problems will be covered in depth, and the essential elements of several fluid dynamical applications will be sketched
Constellation Design for Channels Affected by Phase Noise
In this paper we optimize constellation sets to be used for channels affected
by phase noise. The main objective is to maximize the achievable mutual
information of the constellation under a given power constraint. The mutual
information and pragmatic mutual information of a given constellation is
calculated approximately assuming that both the channel and phase noise are
white. Then a simulated annealing algorithm is used to jointly optimize the
constellation and the binary labeling. The performance of optimized
constellations is compared with conventional constellations showing
considerable gains in all system scenarios.Comment: 5 pages, 6 figures, submitted to IEEE Int. Conf. on Communications
(ICC) 201
On the computation of geometric features of spectra of linear operators on Hilbert spaces
Computing spectra is a central problem in computational mathematics with an
abundance of applications throughout the sciences. However, in many
applications gaining an approximation of the spectrum is not enough. Often it
is vital to determine geometric features of spectra such as Lebesgue measure,
capacity or fractal dimensions, different types of spectral radii and numerical
ranges, or to detect essential spectral gaps and the corresponding failure of
the finite section method. Despite new results on computing spectra and the
substantial interest in these geometric problems, there remain no general
methods able to compute such geometric features of spectra of
infinite-dimensional operators. We provide the first algorithms for the
computation of many of these longstanding problems (including the above). As
demonstrated with computational examples, the new algorithms yield a library of
new methods. Recent progress in computational spectral problems in infinite
dimensions has led to the Solvability Complexity Index (SCI) hierarchy, which
classifies the difficulty of computational problems. These results reveal that
infinite-dimensional spectral problems yield an intricate infinite
classification theory determining which spectral problems can be solved and
with which type of algorithm. This is very much related to S. Smale's
comprehensive program on the foundations of computational mathematics initiated
in the 1980s. We classify the computation of geometric features of spectra in
the SCI hierarchy, allowing us to precisely determine the boundaries of what
computers can achieve and prove that our algorithms are optimal. We also
provide a new universal technique for establishing lower bounds in the SCI
hierarchy, which both greatly simplifies previous SCI arguments and allows new,
formerly unattainable, classifications
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