Nonlinear stability and ergodicity of ensemble based Kalman filters

The ensemble Kalman filter (EnKF) and ensemble square root filter (ESRF) are data assimilation methods used to combine high dimensional, nonlinear dynamical models with observed data. Despite their widespread usage in climate science and oil reservoir simulation, very little is known about the long-time behavior of these methods and why they are effective when applied with modest ensemble sizes in large dimensional turbulent dynamical systems. By following the basic principles of energy dissipation and controllability of filters, this paper establishes a simple, systematic and rigorous framework for the nonlinear analysis of EnKF and ESRF with arbitrary ensemble size, focusing on the dynamical properties of boundedness and geometric ergodicity. The time uniform boundedness guarantees that the filter estimate will not diverge to machine infinity in finite time, which is a potential threat for EnKF and ESQF known as the catastrophic filter divergence. Geometric ergodicity ensures in addition that the filter has a unique invariant measure and that initialization errors will dissipate exponentially in time. We establish these results by introducing a natural notion of observable energy dissipation. The time uniform bound is achieved through a simple Lyapunov function argument, this result applies to systems with complete observations and strong kinetic energy dissipation, but also to concrete examples with incomplete observations. With the Lyapunov function argument established, the geometric ergodicity is obtained by verifying the controllability of the filter processes; in particular, such analysis for ESQF relies on a careful multivariate perturbation analysis of the covariance eigen-structure.Comment: 38 page

Improved linear response for stochastically driven systems

The recently developed short-time linear response algorithm, which predicts the average response of a nonlinear chaotic system with forcing and dissipation to small external perturbation, generally yields high precision of the response prediction, although suffers from numerical instability for long response times due to positive Lyapunov exponents. However, in the case of stochastically driven dynamics, one typically resorts to the classical fluctuation-dissipation formula, which has the drawback of explicitly requiring the probability density of the statistical state together with its derivative for computation, which might not be available with sufficient precision in the case of complex dynamics (usually a Gaussian approximation is used). Here we adapt the short-time linear response formula for stochastically driven dynamics, and observe that, for short and moderate response times before numerical instability develops, it is generally superior to the classical formula with Gaussian approximation for both the additive and multiplicative stochastic forcing. Additionally, a suitable blending with classical formula for longer response times eliminates numerical instability and provides an improved response prediction even for long response times

Moisture - Gravity Wave Interactions in a Multiscale Environment

Starting from the conservation laws for mass, momentum and energy together with a three species, bulk microphysic model, a model for the interaction of internal gravity waves and deep convective hot towers is derived by using multiscale asymptotic techniques. From the resulting leading order equations, a closed model is obtained by applying weighted averages to the smallscale hot towers without requiring further closure approximations. The resulting model is an extension of the linear, anelastic equations, into which moisture enters as the area fraction of saturated regions on the microscale with two way coupling between the large and small scale. Moisture reduces the effective stability in the model and defines a potential temperature sourceterm related to the net effect of latent heat release or consumption by microscale up- and downdrafts. The dispersion relation and group velocity of the system is analyzed and moisture is found to have several effects: It reduces energy transport by waves, increases the vertical wavenumber but decreases the slope at which wave packets travel and it introduces a lower horizontal cutoff wavenumber, below which modes turn into evanescent. Further, moisture can cause critical layers. Numerical examples for steady-state and time-dependent mountain waves are shown and the effects of moisture on these waves are investigated

Structural, magnetic, dielectric and mechanical properties of (Ba,Sr)MnO3_3 ceramics

Ceramic samples, produced by conventional sintering method in ambient air, 6H-SrMnO$_3$(6H-SMO), 15R-BaMnO$_3$(15R-BMO), 4H-Ba$_{0.5}$Sr$_{0.5}$MnO$_3$(4H-BSMO) were studied. In the XRD measurements for SMO the new anomalies of the lattice parameters at 600-800 K range and the increasing of thermal expansion coefficients with a clear maximum in a vicinity at 670 K were detected. The N$\acute{e}$el phase transition for BSMO was observed at $T_N$=250 K in magnetic measurements and its trace was detected in dielectric, FTIR, DSC, and DMA experiments. The enthalpy and entropy changes of the phase transition for BSMO at $T_N$ were determined as 17.5 J/mol and 70 mJ/K mol, respectively. The activation energy values and relaxation times characteristic for relaxation processes were determined from the Arrhenius law. Results of ab initio simulations showed that the contribution of the exchange correlation energy to the total energy is about 30%.Comment: 12 pages, 12 figure

Chemotactic Collapse and Mesenchymal Morphogenesis

We study the effect of chemotactic signaling among mesenchymal cells. We show that the particular physiology of the mesenchymal cells allows one-dimensional collapse in contrast to the case of bacteria, and that the mesenchymal morphogenesis represents thus a more complex type of pattern formation than those found in bacterial colonies. We finally compare our theoretical predictions with recent in vitro experiments

A Variational Principle Based Study of KPP Minimal Front Speeds in Random Shears

Variational principle for Kolmogorov-Petrovsky-Piskunov (KPP) minimal front speeds provides an efficient tool for statistical speed analysis, as well as a fast and accurate method for speed computation. A variational principle based analysis is carried out on the ensemble of KPP speeds through spatially stationary random shear flows inside infinite channel domains. In the regime of small root mean square (rms) shear amplitude, the enhancement of the ensemble averaged KPP front speeds is proved to obey the quadratic law under certain shear moment conditions. Similarly, in the large rms amplitude regime, the enhancement follows the linear law. In particular, both laws hold for the Ornstein-Uhlenbeck process in case of two dimensional channels. An asymptotic ensemble averaged speed formula is derived in the small rms regime and is explicit in case of the Ornstein-Uhlenbeck process of the shear. Variational principle based computation agrees with these analytical findings, and allows further study on the speed enhancement distributions as well as the dependence of enhancement on the shear covariance. Direct simulations in the small rms regime suggest quadratic speed enhancement law for non-KPP nonlinearities.Comment: 28 pages, 14 figures update: fixed typos, refined estimates in section

Effect of non-zero constant vorticity on the nonlinear resonances of capillary water waves

The influence of an underlying current on 3-wave interactions of capillary water waves is studied. The fact that in irrotational flow resonant 3-wave interactions are not possible can be invalidated by the presence of an underlying current of constant non-zero vorticity. We show that: 1) wave trains in flows with constant non-zero vorticity are possible only for two-dimensional flows; 2) only positive constant vorticities can trigger the appearance of three-wave resonances; 3) the number of positive constant vorticities which do trigger a resonance is countable; 4) the magnitude of a positive constant vorticity triggering a resonance can not be too small.Comment: 6 pages, submitte

A relative entropy rate method for path space sensitivity analysis of stationary complex stochastic dynamics

We propose a new sensitivity analysis methodology for complex stochastic dynamics based on the Relative Entropy Rate. The method becomes computationally feasible at the stationary regime of the process and involves the calculation of suitable observables in path space for the Relative Entropy Rate and the corresponding Fisher Information Matrix. The stationary regime is crucial for stochastic dynamics and here allows us to address the sensitivity analysis of complex systems, including examples of processes with complex landscapes that exhibit metastability, non-reversible systems from a statistical mechanics perspective, and high-dimensional, spatially distributed models. All these systems exhibit, typically non-gaussian stationary probability distributions, while in the case of high-dimensionality, histograms are impossible to construct directly. Our proposed methods bypass these challenges relying on the direct Monte Carlo simulation of rigorously derived observables for the Relative Entropy Rate and Fisher Information in path space rather than on the stationary probability distribution itself. We demonstrate the capabilities of the proposed methodology by focusing here on two classes of problems: (a) Langevin particle systems with either reversible (gradient) or non-reversible (non-gradient) forcing, highlighting the ability of the method to carry out sensitivity analysis in non-equilibrium systems; and, (b) spatially extended Kinetic Monte Carlo models, showing that the method can handle high-dimensional problems