28,174 research outputs found
Strong convergence rates of probabilistic integrators for ordinary differential equations
Probabilistic integration of a continuous dynamical system is a way of
systematically introducing model error, at scales no larger than errors
introduced by standard numerical discretisation, in order to enable thorough
exploration of possible responses of the system to inputs. It is thus a
potentially useful approach in a number of applications such as forward
uncertainty quantification, inverse problems, and data assimilation. We extend
the convergence analysis of probabilistic integrators for deterministic
ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\
Comput.}, 2017), to establish mean-square convergence in the uniform norm on
discrete- or continuous-time solutions under relaxed regularity assumptions on
the driving vector fields and their induced flows. Specifically, we show that
randomised high-order integrators for globally Lipschitz flows and randomised
Euler integrators for dissipative vector fields with polynomially-bounded local
Lipschitz constants all have the same mean-square convergence rate as their
deterministic counterparts, provided that the variance of the integration noise
is not of higher order than the corresponding deterministic integrator. These
and similar results are proven for probabilistic integrators where the random
perturbations may be state-dependent, non-Gaussian, or non-centred random
variables.Comment: 25 page
Data Assimilation: A Mathematical Introduction
These notes provide a systematic mathematical treatment of the subject of
data assimilation
Cosmic Variance In the Transparency of the Intergalactic Medium After Reionization
Following the completion of cosmic reionization, the mean-free-path of
ionizing photons was set by a population of Ly-limit absorbers. As the
mean-free-path steadily grew, the intensity of the ionizing background also
grew, thus lowering the residual neutral fraction of hydrogen in ionization
equilibrium throughout the diffuse intergalactic medium (IGM). Ly-alpha photons
provide a sensitive probe for tracing the distribution of this residual
hydrogen at the end of reionization. Here we calculate the cosmic variance
among different lines-of-sight in the distribution of the mean Ly-alpha optical
depths. We find fractional variations in the effective post-reionization
optical depth that are of order unity on a scale of ~100 co-moving Mpc, in
agreement with observations towards high-redshift quasars. Significant
contributions to these variations are provided by the cosmic variance in the
density contrast on the scale of the mean-free-path for ionizing photons, and
by fluctuations in the ionizing background induced by delayed or enhanced
structure formation. Cosmic variance results in a highly asymmetric
distribution of transmission through the IGM, with fractional fluctuations in
Ly-alpha transmission that ar larger than in Ly-beta transmission.Comment: 7 pages 3 figures. Replaced with version accepted for publication in
Ap
Soft Handoff and Uplink Capacity in a Two-Tier CDMA System
This paper examines the effect of soft handoff on the uplink user capacity of
a CDMA system consisting of a single macrocell in which a single hotspot
microcell is embedded. The users of these two base stations operate over the
same frequency band. In the soft handoff scenario studied here, both macrocell
and microcell base stations serve each system user and the two received copies
of a desired user's signal are summed using maximal ratio combining. Exact and
approximate analytical methods are developed to compute uplink user capacity.
Simulation results demonstrate a 20% increase in user capacity compared to hard
handoff. In addition, simple, approximate methods are presented for estimating
soft handoff capacity and are shown to be quite accurate.Comment: To appear in IEEE Transactions on Wireless Communication
A Game-Theoretic Approach to Energy-Efficient Modulation in CDMA Networks with Delay Constraints
A game-theoretic framework is used to study the effect of constellation size
on the energy efficiency of wireless networks for M-QAM modulation. A
non-cooperative game is proposed in which each user seeks to choose its
transmit power (and possibly transmit symbol rate) as well as the constellation
size in order to maximize its own utility while satisfying its delay
quality-of-service (QoS) constraint. The utility function used here measures
the number of reliable bits transmitted per joule of energy consumed, and is
particularly suitable for energy-constrained networks. The best-response
strategies and Nash equilibrium solution for the proposed game are derived. It
is shown that in order to maximize its utility (in bits per joule), a user must
choose the lowest constellation size that can accommodate the user's delay
constraint. Using this framework, the tradeoffs among energy efficiency, delay,
throughput and constellation size are also studied and quantified. The effect
of trellis-coded modulation on energy efficiency is also discussed.Comment: Appeared in the Proceedings of the 2007 IEEE Radio and Wireless
Symposium, Long Beach, CA, January 9-11, 200
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