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

Data Assimilation: A Mathematical Introduction

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

Determining White Noise Forcing From Eulerian Observations in the Navier Stokes Equation

The Bayesian approach to inverse problems is of paramount importance in quantifying uncertainty about the input to and the state of a system of interest given noisy observations. Herein we consider the forward problem of the forced 2D Navier Stokes equation. The inverse problem is inference of the forcing, and possibly the initial condition, given noisy observations of the velocity field. We place a prior on the forcing which is in the form of a spatially correlated temporally white Gaussian process, and formulate the inverse problem for the posterior distribution. Given appropriate spatial regularity conditions, we show that the solution is a continuous function of the forcing. Hence, for appropriately chosen spatial regularity in the prior, the posterior distribution on the forcing is absolutely continuous with respect to the prior and is hence well-defined. Furthermore, the posterior distribution is a continuous function of the data. We complement this theoretical result with numerical simulation of the posterior distribution

Longhorn Cavern, A Reconnaissance Survey

The recent opening of the Longhorn Cavern, twelve miles south of the town of Burnet, in the Central Mineral Region of Texas, as a state park, has given this fine cave known by local residents for almost a hundred years, state and national interest

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

Atomic layer deposition of ZnS nanotubes

We report on growth of high-aspect-ratio ($\gtrsim300$) zinc sulfide nanotubes with variable, precisely tunable, wall thicknesses and tube diameters into highly ordered pores of anodic alumina templates by atomic layer deposition (ALD) at temperatures as low as 75 $^{\circ}$C. Various characterization techniques are employed to gain information on the composition, morphology, and crystal structure of the synthesized samples. Besides practical applications, the ALD-grown tubes could be envisaged as model systems for the study of a certain class of size-dependent quantum and classical phenomena.Comment: 1 LaTeX source file, 8 eps figures, and the manuscript in PDF forma

Stability of Filters for the Navier-Stokes Equation

Data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the properties of the state of the system. Filters refer to a class of data assimilation algorithms designed to update the estimation of the state in a on-line fashion, as data is acquired sequentially. For linear problems subject to Gaussian noise filtering can be performed exactly using the Kalman filter. For nonlinear systems it can be approximated in a systematic way by particle filters. However in high dimensions these particle filtering methods can break down. Hence, for the large nonlinear systems arising in applications such as weather forecasting, various ad hoc filters are used, mostly based on making Gaussian approximations. The purpose of this work is to study the properties of these ad hoc filters, working in the context of the 2D incompressible Navier-Stokes equation. By working in this infinite dimensional setting we provide an analysis which is useful for understanding high dimensional filtering, and is robust to mesh-refinement. We describe theoretical results showing that, in the small observational noise limit, the filters can be tuned to accurately track the signal itself (filter stability), provided the system is observed in a sufficiently large low dimensional space; roughly speaking this space should be large enough to contain the unstable modes of the linearized dynamics. Numerical results are given which illustrate the theory. In a simplified scenario we also derive, and study numerically, a stochastic PDE which determines filter stability in the limit of frequent observations, subject to large observational noise. The positive results herein concerning filter stability complement recent numerical studies which demonstrate that the ad hoc filters perform poorly in reproducing statistical variation about the true signal

Packet narrowing and quantum entanglement in photoionization and photodissociation

The narrowing of electron and ion wave packets in the process of photoionization is investigated, with the electron-ion recoil fully taken into account. Packet localization of this type is directly related to entanglement in the joint quantum state of electron and ion, and to Einstein-Podolsky-Rosen localization. Experimental observation of such packet-narrowing effects is suggested via coincidence registration by two detectors, with a fixed position of one and varying position of the other. A similar effect, typically with an enhanced degree of entanglement, is shown to occur in the case of photodissociation of molecules