Variational data assimilation using targetted random walks

The variational approach to data assimilation is a widely used methodology for both online prediction and for reanalysis (offline hindcasting). In either of these scenarios it can be important to assess uncertainties in the assimilated state. Ideally it would be desirable to have complete information concerning the Bayesian posterior distribution for unknown state, given data. The purpose of this paper is to show that complete computational probing of this posterior distribution is now within reach in the offline situation. In this paper we will introduce an MCMC method which enables us to directly sample from the Bayesian\ud posterior distribution on the unknown functions of interest, given observations. Since we are aware that these\ud methods are currently too computationally expensive to consider using in an online filtering scenario, we frame this in the context of offline reanalysis. Using a simple random walk-type MCMC method, we are able to characterize the posterior distribution using only evaluations of the forward model of the problem, and of the model and data mismatch. No adjoint model is required for the method we use; however more sophisticated MCMC methods are available\ud which do exploit derivative information. For simplicity of exposition we consider the problem of assimilating data, either Eulerian or Lagrangian, into a low Reynolds number (Stokes flow) scenario in a two dimensional periodic geometry. We will show that in many cases it is possible to recover the initial condition and model error (which we describe as unknown forcing to the model) from data, and that with increasing amounts of informative data, the uncertainty in our estimations reduces

The phase transition in random catalytic sets

The notion of (auto) catalytic networks has become a cornerstone in understanding the possibility of a sudden dramatic increase of diversity in biological evolution as well as in the evolution of social and economical systems. Here we study catalytic random networks with respect to the final outcome diversity of products. We show that an analytical treatment of this longstanding problem is possible by mapping the problem onto a set of non-linear recurrence equations. The solution of these equations show a crucial dependence of the final number of products on the initial number of products and the density of catalytic production rules. For a fixed density of rules we can demonstrate the existence of a phase transition from a practically unpopulated regime to a fully populated and diverse one. The order parameter is the number of final products. We are able to further understand the origin of this phase transition as a crossover from one set of solutions from a quadratic equation to the other.Comment: 7 pages, ugly eps files due to arxiv restriction

Emergence of the Shackleton Range from beneath the Antarctic Ice Sheet due to glacial erosion

This paper explores the long-term evolution of a subglacial fjord landscape in the Shackleton Range, Antarctica. We propose that prolonged ice-sheet erosion across a passive continental margin caused troughs to deepen and lower the surrounding ice-sheet surface, leaving adjacent mountains exposed. Geomorphological evidence suggests a change in the direction of regional ice flow accompanied emergence. Simple calculations suggest that isostatic compensation caused by the deepening of bounding ice-stream troughs lowered the ice-sheet surface relative to the mountains by ~800m. Use of multiple cosmogenic isotopes on bedrock and erratics (26Al, 10Be, 21Ne) provides evidence that overriding of the massif and the deepening of the adjacent troughs occurred earlier than the Quaternary. Perhaps this occurred in the mid-Miocene, as elsewhere in East Antarctica in the McMurdo Dry Valleys and the Lambert basin. The implication is that glacial erosion instigates feedback that can change ice-sheet thickness, extent, and direction of flow. Indeed, as the subglacial troughs evolve over millions of years, they increase topographic relief; and this changes the dynamics of the ice sheet. © 2013 Elsevier B.V

An experimental test of all theories with predictive power beyond quantum theory

According to quantum theory, the outcomes of future measurements cannot (in general) be predicted with certainty. In some cases, even with a complete physical description of the system to be measured and the measurement apparatus, the outcomes of certain measurements are completely random. This raises the question, originating in the paper by Einstein, Podolsky and Rosen, of whether quantum mechanics is the optimal way to predict measurement outcomes. Established arguments and experimental tests exclude a few specific alternative models. Here, we provide a complete answer to the above question, refuting any alternative theory with significantly more predictive power than quantum theory. More precisely, we perform various measurements on distant entangled photons, and, under the assumption that these measurements are chosen freely, we give an upper bound on how well any alternative theory could predict their outcomes. In particular, in the case where quantum mechanics predicts two equally likely outcomes, our results are incompatible with any theory in which the probability of a prediction is increased by more than ~0.19. Hence, we can immediately refute any already considered or yet-to-be-proposed alternative model with more predictive power than this.Comment: 13 pages, 4 figure

Reducing the stress of drug administration:implications for the 3Rs

Restraint in animals is known to cause stress but is used during almost all scientific procedures in rodents, representing a major welfare and scientific issue. Administration of substances, a key part of most scientific procedures, almost always involves physical restraint of the animal. In this study, we developed a method to inject substances to rats using a non-restrained technique. We then compared the physiological, behavioral and emotional impacts of restrained versus non-restrained injection procedures. Our results highlight the negative welfare implications associated with physical restraint and demonstrate a method which can be used to avoid this. Our work shows how adopting strategies that avoid restraint can minimize a widespread source of stress in laboratory animals and improve welfare through refinement

Adiabatic limit and the slow motion of vortices in a Chern-Simons-Schr\"odinger system

We study a nonlinear system of partial differential equations in which a complex field (the Higgs field) evolves according to a nonlinear Schroedinger equation, coupled to an electromagnetic field whose time evolution is determined by a Chern-Simons term in the action. In two space dimensions, the Chern-Simons dynamics is a Galileo invariant evolution for A, which is an interesting alternative to the Lorentz invariant Maxwell evolution, and is finding increasing numbers of applications in two dimensional condensed matter field theory. The system we study, introduced by Manton, is a special case (for constant external magnetic field, and a point interaction) of the effective field theory of Zhang, Hansson and Kivelson arising in studies of the fractional quantum Hall effect. From the mathematical perspective the system is a natural gauge invariant generalization of the nonlinear Schroedinger equation, which is also Galileo invariant and admits a self-dual structure with a resulting large space of topological solitons (the moduli space of self-dual Ginzburg-Landau vortices). We prove a theorem describing the adiabatic approximation of this system by a Hamiltonian system on the moduli space. The approximation holds for values of the Higgs self-coupling constant close to the self-dual (Bogomolny) value of 1. The viability of the approximation scheme depends upon the fact that self-dual vortices form a symplectic submanifold of the phase space (modulo gauge invariance). The theorem provides a rigorous description of slow vortex dynamics in the near self-dual limit.Comment: Minor typos corrected, one reference added and DOI give

Surface pinning of fluctuating charge order: an "extraordinary" surface phase transition

We study the mean-field theory of charge-density wave (CDW) order in a layered system, including the effect of the long-range Coulomb interaction and of screening by uncondensed electrons. We particularly focus on the conditions necessary for an ``extraordinary'' transition, in which the surface orders at a higher temperature, and is more likely to be commensurate, than the bulk. We interpret recent experiments on NaCCOC as indicating the presence of commensurate CDW at the surface that is not present in the bulk. More generally, we show that poor screening of the Coulomb interaction tends to stabilize incommensurate order, possibly explaining why the CDW order in LSCO and NbSe2 remains incommensurate to T -> 0, despite the small magnitude of the incommensurability.Comment: 9 pages, no figures, 31 references; 1 new figure and minor editing of the tex

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

Absorbing systematic effects to obtain a better background model in a search for new physics

This paper presents a novel approach to estimate the Standard Model backgrounds based on modifying Monte Carlo predictions within their systematic uncertainties. The improved background model is obtained by altering the original predictions with successively more complex correction functions in signal-free control selections. Statistical tests indicate when sufficient compatibility with data is reached. In this way, systematic effects are absorbed into the new background model. The same correction is then applied on the Monte Carlo prediction in the signal region. Comparing this method to other background estimation techniques shows improvements with respect to statistical and systematical uncertainties. The proposed method can also be applied in other fields beyond high energy physics