Data Assimilation: A Mathematical Introduction

These notes provide a systematic mathematical treatment of the subject of data assimilation

Analysis of the 3DVAR Filter for the Partially Observed Lorenz '63 Model

The problem of effectively combining data with a mathematical model constitutes a major challenge in applied mathematics. It is particular challenging for high-dimensional dynamical systems where data is received sequentially in time and the objective is to estimate the system state in an on-line fashion; this situation arises, for example, in weather forecasting. The sequential particle filter is then impractical and ad hoc filters, which employ some form of Gaussian approximation, are widely used. Prototypical of these ad hoc filters is the 3DVAR method. The goal of this paper is to analyze the 3DVAR method, using the Lorenz '63 model to exemplify the key ideas. The situation where the data is partial and noisy is studied, and both discrete time and continuous time data streams are considered. The theory demonstrates how the widely used technique of variance inflation acts to stabilize the filter, and hence leads to asymptotic accuracy

Well-Posedness And Accuracy Of The Ensemble Kalman Filter In Discrete And Continuous Time

The ensemble Kalman filter (EnKF) is a method for combining a dynamical model with data in a sequential fashion. Despite its widespread use, there has been little analysis of its theoretical properties. Many of the algorithmic innovations associated with the filter, which are required to make a useable algorithm in practice, are derived in an ad hoc fashion. The aim of this paper is to initiate the development of a systematic analysis of the EnKF, in particular to do so in the small ensemble size limit. The perspective is to view the method as a state estimator, and not as an algorithm which approximates the true filtering distribution. The perturbed observation version of the algorithm is studied, without and with variance inflation. Without variance inflation well-posedness of the filter is established; with variance inflation accuracy of the filter, with resepct to the true signal underlying the data, is established. The algorithm is considered in discrete time, and also for a continuous time limit arising when observations are frequent and subject to large noise. The underlying dynamical model, and assumptions about it, is sufficiently general to include the Lorenz '63 and '96 models, together with the incompressible Navier-Stokes equation on a two-dimensional torus. The analysis is limited to the case of complete observation of the signal with additive white noise. Numerical results are presented for the Navier-Stokes equation on a two-dimensional torus for both complete and partial observations of the signal with additive white noise

Not all the bots are created equal:the Ordering Turing Test for the labelling of bots in MMORPGs

This article contributes to the research on bots in Social Media. It takes as its starting point an emerging perspective which proposes that we should abandon the investigation of the Turing Test and the functional aspects of bots in favor of studying the authentic and cooperative relationship between humans and bots. Contrary to this view, this article argues that Turing Tests are one of the ways in which authentic relationships between humans and bots take place. To understand this, this article introduces the concept of Ordering Turing Tests: these are sort of Turing Tests proposed by social actors for purposes of achieving social order when bots produce deviant behavior. An Ordering Turing Test is method for labeling deviance, whereby social actors can use this test to tell apart rule-abiding humans and rule-breaking bots. Using examples from Massively Multiplayer Online Role-Playing Games, this article illustrates how Ordering Turing Tests are proposed and justified by players and service providers. Data for the research comes from scientific literature on Machine Learning proposed for the identification of bots and from game forums and other player produced paratexts from the case study of the game Runescape

Non-Markovian master equation for a damped oscillator with time-varying parameters

We derive an exact non-Markovian master equation that generalizes the previous work [Hu, Paz and Zhang, Phys. Rev. D {\bf 45}, 2843 (1992)] to damped harmonic oscillators with time-varying parameters. This is achieved by exploiting the linearity of the system and operator solution in Heisenberg picture. Our equation governs the non-Markovian quantum dynamics when the system is modulated by external devices. As an application, we apply our equation to parity kick decoupling problems. The time-dependent dissipative coefficients in the master equation are shown to be modified drastically when the system is driven by $\pi$ pulses. For coherence protection to be effective, our numerical results indicate that kicking period should be shorter than memory time of the bath. The effects of using soft pulses in an ohmic bath are also discussed

Localized structures in Kagome lattices

We investigate the existence and stability of gap vortices and multi-pole gap solitons in a Kagome lattice with a defocusing nonlinearity both in a discrete case and in a continuum one with periodic external modulation. In particular, predictions are made based on expansion around a simple and analytically tractable anti-continuum (zero coupling) limit. These predictions are then confirmed for a continuum model of an optically-induced Kagome lattice in a photorefractive crystal obtained by a continuous transformation of a honeycomb lattice

Analysis and interpretation of high transverse entanglement in optical parametric down conversion

Quantum entanglement associated with transverse wave vectors of down conversion photons is investigated based on the Schmidt decomposition method. We show that transverse entanglement involves two variables: orbital angular momentum and transverse frequency. We show that in the monochromatic limit high values of entanglement are closely controlled by a single parameter resulting from the competition between (transverse) momentum conservation and longitudinal phase matching. We examine the features of the Schmidt eigenmodes, and indicate how entanglement can be enhanced by suitable mode selection methods.Comment: 4 pages, 4 figure