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

Solution of the Holstein polaron anisotropy problem

We study Holstein polarons in three-dimensional anisotropic materials. Using a variational exact diagonalization technique we provide highly accurate results for the polaron mass and polaron radius. With these data we discuss the differences between polaron formation in dimension one and three, and at small and large phonon frequency. Varying the anisotropy we demonstrate how a polaron evolves from a one-dimensional to a three-dimensional quasiparticle. We thereby resolve the issue of polaron stability in quasi-one-dimensional substances and clarify to what extent such polarons can be described as one-dimensional objects. We finally show that even the local Holstein interaction leads to an enhancement of anisotropy in charge carrier motion.Comment: 6 pages, 7 figures; extended version accepted for publication in Phys. Rev.

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

Draft crystal structure of the vault shell at 9-A resolution.

Vaults are the largest known cytoplasmic ribonucleoprotein structures and may function in innate immunity. The vault shell self-assembles from 96 copies of major vault protein and encapsulates two other proteins and a small RNA. We crystallized rat liver vaults and several recombinant vaults, all among the largest non-icosahedral particles to have been crystallized. The best crystals thus far were formed from empty vaults built from a cysteine-tag construct of major vault protein (termed cpMVP vaults), diffracting to about 9-A resolution. The asymmetric unit contains a half vault of molecular mass 4.65 MDa. X-ray phasing was initiated by molecular replacement, using density from cryo-electron microscopy (cryo-EM). Phases were improved by density modification, including concentric 24- and 48-fold rotational symmetry averaging. From this, the continuous cryo-EM electron density separated into domain-like blocks. A draft atomic model of cpMVP was fit to this improved density from 15 domain models. Three domains were adapted from a nuclear magnetic resonance substructure. Nine domain models originated in ab initio tertiary structure prediction. Three C-terminal domains were built by fitting poly-alanine to the electron density. Locations of loops in this model provide sites to test vault functions and to exploit vaults as nanocapsules

Data Assimilation: A Mathematical Introduction

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

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

Detection of bottom ferromagnetic electrode oxidation in magnetic tunnel junctions by magnetometry measurements

Surface oxidation of the bottom ferromagnetic (FM) electrode, one of the major detrimental factors to the performance of a Magnetic Tunnel Junction (MTJ), is difficult to avoid during the fabrication process of the MTJ's tunnel barrier. Since Co rich alloys are commonly used for the FM electrodes in MTJs, over-oxidation of the tunnel barrier results in the formation of a CoO antiferromagnetic (AF) interface layer which couples with the bottom FM electrode to form a typical AF/FM exchange bias (EB) system. In this work, surface oxidation of the CoFe and CoFeB bottom electrodes was detected via magnetometry measurements of exchange-bias characterizations including the EB field, training effect, uncompensated spin density, and coercivity. Variations of these parameters were found to be related to the surface oxidation of the bottom electrode, among them the change of coercivity is most sensitive. Annealed samples show evidence for an oxygen migration back to the MgO tunnel barrier by annealing.Comment: 5 pages, 4 figues, submitted to J. Appl. Phy

A model for preferential concentration

The preferential concentration of inertial particles in a turbulent velocity field occurs when the particle and fluid time constants are commensurate. We propose a straightforward mathematical model for this phenomenon and use the model to study various scaling limits of interest and to study numerically the effect of interparticle collisions. The model comprises Stokes’ law for the particle motions, and a Gaussian random field for the velocity. The primary advantages of the model are its amenability to mathematical analysis in various interesting scaling limits and the speed at which numerical simulations can be performed. The scaling limits corroborate experimental evidence about the lack of preferential concentration for a large and small Stokes number and make new predictions about the possibility of preferential concentration at large times and lead to stochastic differential equations governing this phenomenon. The effect of collisions is found to be negligible for the most part, although in some cases they have an interesting antidiffusive effect

The Impact of Accretion Disk Winds on the Optical Spectra of Cataclysmic Variables

Many high-state non-magnetic cataclysmic variables (CVs) exhibit blue-shifted absorption or P-Cygni profiles associated with ultraviolet (UV) resonance lines. These features imply the existence of powerful accretion disk winds in CVs. Here, we use our Monte Carlo ionization and radiative transfer code to investigate whether disk wind models that produce realistic UV line profiles are also likely to generate observationally significant recombination line and continuum emission in the optical waveband. We also test whether outflows may be responsible for the single-peaked emission line profiles often seen in high-state CVs and for the weakness of the Balmer absorption edge (relative to simple models of optically thick accretion disks). We find that a standard disk wind model that is successful in reproducing the UV spectra of CVs also leaves a noticeable imprint on the optical spectrum, particularly for systems viewed at high inclination. The strongest optical wind-formed recombination lines are H$\alpha$ and He II $\lambda4686$. We demonstrate that a higher-density outflow model produces all the expected H and He lines and produces a recombination continuum that can fill in the Balmer jump at high inclinations. This model displays reasonable verisimilitude with the optical spectrum of RW Trianguli. No single-peaked emission is seen, although we observe a narrowing of the double-peaked emission lines from the base of the wind. Finally, we show that even denser models can produce a single-peaked H$\alpha$ line. On the basis of our results, we suggest that winds can modify, and perhaps even dominate, the line and continuum emission from CVs.Comment: 15 pages, 13 figures. Accepted to MNRA

Symmetrization Inequalities for Composition Operators of Carathéodory Type

Let F:(0, ∞) × [0, ∞) → R be a function of Carathéodory type. We establish the inequality $$\int_{\mathbb{R}^{N}} F( | x |, u(x) ) dx \leq \int_{\mathbb{R}^{N} } F( | x |, u^{\ast}(x)) dx.$$ where u* denotes the Schwarz symmetrization of u, under hypotheses on F that seem quite natural when this inequality is used to obtain existence results in the context of elliptic partial differential equations. We also treat the case where RN is replaced by a set of finite measure. The identity $$\int_{\mathbb{R}^{N}} G(u(x)) dx = \int_{\mathbb{R}^{N}} G(u^{\ast}(x)) dx$$ is also discussed under the assumption that G: [0,∞) → R is a Borel function. 2000 Mathematics Subject Classification 26D20, 42C20, 46E3