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