We study the transport properties of nonautonomous chaotic dynamical systems over a finite-time duration. We are particularly interested in those regions that remain coherent and relatively nondispersive over finite periods of time, despite the chaotic nature of the system. We develop a novel probabilistic methodology based upon transfer operators that automatically detect maximally coherent sets. The approach is very simple to implement, requiring only singular vector computations of a matrix of transitions induced by the dynamics. We illustrate our new methodology on an idealized stratospheric flow and in two and three-dimensional analyses of European Centre for Medium Range Weather Forecasting (ECMWF) reanalysis data.
This paper presents an approach for simultaneous estimation of the state and unknown parameters in a sequential data assimilation framework. The state augmentation technique, in which the state vector is augmented by the model parameters, has been investigated in many previous studies and some success with this technique has been reported in the case where model parameters are additive. However, many geophysical or climate models contains non-additive parameters such as those arising from physical parametrization of sub-grid scale processes, in which case the state augmentation technique may become ineffective since its inference about parameters from partially observed states based on the cross covariance between states and parameters is inadequate if states and parameters are not linearly correlated. In this paper, we propose a two-stages filtering technique that runs particle filtering (PF) to estimate parameters while updating the state estimate using Ensemble Kalman filter (ENKF; these two "sub-filters" interact. The applicability of the proposed method is demonstrated using the Lorenz-96 system, where the forcing is parameterized and the amplitude and phase of the forcing are to be estimated jointly with the states. The proposed method is shown to be capable of estimating these model parameters with a high accuracy as well as reducing uncertainty while the state augmentation technique fails.
It has been known that noise in a stochastically perturbed dynamical system can destroy what was the original zero-noise case barriers in the phase space (pseudobarrier). Noise can cause the basin hopping. We use the Frobenius-Perron operator and its finite rank approximation by the Ulam-Galerkin method to study transport mechanism of a noisy map. In order to identify the regions of high transport activity in the phase space and to determine flux across the pseudobarriers, we adapt a new graph theoretical method which was developed to detect active pseudobarriers in the original phase space of the stochastic dynamic. Previous methods to identify basins and basin barriers require a priori knowledge of a mathematical model of the system, and hence cannot be applied to observed time series data of which a mathematical model is not known. Here we describe a novel graph method based on optimization of the modularity measure of a network and introduce its application for determining pseudobarriers in the phase space of a multi-stable system only known through observed data. © 2007 Elsevier Ltd. All rights reserved.
Given a flow on a surface, we consider the problem of connecting two distinct trajectories by a curve of extremal (absolute) instantaneous flux. We develop a complete classification of flux optimal curves, accounting for the possibility of the flux having spatially and temporally varying weight. This weight enables modelling the flux of non-equilibrium distributions of tracer particles, pollution concentrations, or active scalar fields such as vorticity. Our results are applicable to all smooth autonomous flows, area preserving or not. © 2013 Elsevier Ltd.
We expose a phenomenon that can occur in the process of joint state and parameter estimation. Such estimation is crucial in tuning parameters for climate models and offline parameterizations used in the models. We show how a bimodal distribution can temporarily appear during this process and that a scheme relying on linear and Gaussian approximations may cause it to get trapped in the wrong mode and hence lead to faulty estimation. We propose a practical and effective resolution using a two-stage filtering process.
The flow field in a cylindrical container driven by a flat bladed impeller was investigated using particle image velocimetry (PIV). Three Reynolds numbers (0.02, 8, 108) were investigated for different impeller locations within the cylinder. The results showed that vortices were formed at the tips of the blades and rotated with the blades. As the blades were placed closer to the wall the vortices interacted with the induced boundary layer on the wall to enhance both regions of vorticity. Finite time lyapunov exponents (FTLE) were used to determine the lagrangian coherent structure (LCS) fields for the flow. These structures highlighted the regions where mixing occurred as well as barriers to fluid transport. Mixing was estimated using zero mass particles convected by numeric integration of the experimentally derived velocity fields. The mixing data confirmed the location of high mixing regions and barriers shown by the LCS analysis. The results indicated that mixing was enhanced within the region described by the blade motion as the blade was positioned closed to the cylinder wall. The mixing average within the entire tank was found to be largely independent of the blade location and flow Reynolds number. © 2011 American Society of Mechanical Engineers.
This book connects many concepts in dynamical systems with mathematical tools from areas such as graph theory and ergodic theory.
In this paper, we present an approach to approximate the Frobenius-Perron transfer operator from a sequence of time-ordered images, that is, a movie dataset. Unlike time-series data, successive images do not provide a direct access to a trajectory of a point in a phase space; more precisely, a pixel in an image plane. Therefore, we reconstruct the velocity field from image sequences based on the infinitesimal generator of the Frobenius-Perron operator. Moreover, we relate this problem to the well-known optical flow problem from the computer vision community and we validate the continuity equation derived from the infinitesimal operator as a constraint equation for the optical flow problem. Once the vector field and then a discrete transfer operator are found, then, in addition, we present a graph modularity method as a tool to discover basin structure in the phase space. Together with a tool to reconstruct a velocity field, this graph-based partition method provides us with a way to study transport behavior and other ergodic properties of measurable dynamical systems captured only through image sequences.
We describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems. In the autonomous setting, such objects are variously known as almost-invariant sets, metastable sets, persistent patterns, or strange eigenmodes, and have proved to be important in a variety of applications. In this current work, we explain how to extend existing autonomous approaches to the nonautonomous setting. We call the new time-dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent. The new methods are illustrated via detailed examples in both discrete and continuous time.
We conduct Observing System Simulation Experiments (OSSEs) with Lagrangian data assimilation (LaDA) in two-layer point-vortex systems, where the trajectories of passive tracers (drifters or floats) are observed on one layer that is coupled to another layer with different dynamics. Depending on the initial position of the observed tracers, the model studied here can exhibit nonlinear features that cause the standard Kalman filter and its variants to fail. For this reason, we adopt a Monte Carlo approach known as particle filtering, which takes the nonlinear dynamics into account. The main objective of this paper is to understand the effects of drifter placement and layer coupling on the precision skill of assimilating Lagrangian data into multi-layered models. Therefore, we analyze the quality of the assimilated vortex estimates by assimilating path data from passive tracers launched at different locations, on different layers and in systems with various coupling strengths between layers. We consider two cases: vortices placed on different layers (heton) and on the same layer (non-heton). In both cases we find that launch location, launch layer and coupling strength all play a significant role in assimilation precision skill. However, the specifics of the interplay of these three factors are quite different for the heton case versus the non-heton case. © 2014.
A novel probabilistic methodology is applied to identify optimally coherent structures associated with Agulhas Rings, within a time varying velocity field in the South Atlantic Ocean, as simulated by an eddy-permitting ocean general model. It is shown that this technique provides a way of identifying the three-dimensional shape of a particular Ring in the upper ocean and tracking its evolution over space and time. Based on this three-dimensional representation we can accurately measure the amount of water mass remaining in an Agulhas Ring over time and consequently how much heat or salt is released from the structure as it decays. Identification techniques based on relative vorticity or the Okubo-Weiss parameter have previously been developed for a surface snapshot. Extending these methods in the vertical direction in the upper ocean and comparing the decay of all three-dimensional structures obtained by different methods, we demonstrate that our technique is able to define structures that are more coherent over time than classical methods. While our investigation concentrates on a single Agulhas Ring located in the Cape-Basin from May 2000 over 6. months, the technique may be extended to examine multiple Rings and other coherent structures that are involved in the Agulhas leakage. © 2012 Elsevier Ltd.
The "edge" of the Antarctic polar vortex is known to behave as a barrier to the meridional (poleward) transport of ozone during the austral winter. This chemical isolation of the polar vortex from the middle and low latitudes produces an ozone minimum in the vortex region, intensifying the ozone hole relative to that which would be produced by photochemical processes alone. Observational determination of the vortex edge remains an active field of research. In this paper, we obtain objective estimates of the structure of the polar vortex by introducing a technique based on transfer operators that aims to find regions with minimal external transport. Applying this technique to European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-40 three-dimensional velocity data, we produce an improved three-dimensional estimate of the vortex location in the upper stratosphere where the vortex is most pronounced. This computational approach has wide potential application in detecting and analyzing mixing structures in a variety of atmospheric, oceanographic, and general fluid dynamical settings.
We introduce a data assimilation method to estimate model parameters with observations of
passive tracers by directly assimilating Lagrangian Coherent Structures. Our approach differs
from the usual Lagrangian Data Assimilation approach, where parameters are estimated based
on tracer trajectories. We employ the Approximate Bayesian Computation (ABC) framework to
avoid computing the likelihood function of the coherent structure, which is usually unavailable.
We solve the ABC by a Sequential Monte Carlo (SMC) method, and use Principal Component
Analysis (PCA) to identify the coherent patterns from tracer trajectory data. Our new method
shows remarkably improved results compared to the bootstrap particle filter when the physical
model exhibits chaotic advection.
Froyland G, Santitissadeekorn N (2017) Optimal mixing enhancement, SIAM Journal on Applied Mathematics 77 (4) pp. 1444-1470
Society for Industrial and Applied Mathematics
We introduce a general-purpose method for optimising the mixing rate of advective
fluid flows. An existing velocity field is perturbed in a C
1 neighborhood to maximize the
mixing rate for flows generated by velocity fields in this neighborhood. Our numerical
approach is based on the infinitesimal generator of the flow and is solved by standard
linear programming methods. The perturbed flow may be easily constrained to preserve
the same steady state distribution as the original flow, and various natural geometric
constraints can also be simply applied. The same technique can also be used to optimize
the mixing rate of advection-diffusion flow models by manipulating the drift term in a
A number of models ? such as the Hawkes process and log Gaussian Cox process
? have been used to understand how crime rates evolve in time and/or
space. Within the context of these models and actual crime data, parameters
are often estimated using maximum likelihood estimation (MLE) on batch
data, but this approach has several limitations such as limited tracking in
real-time and uncertainty quantification. For practical purposes, it would be
desirable to move beyond batch data estimation to sequential data assimilation.
A novel and general Bayesian sequential data assimilation algorithm is
developed for joint state-parameter estimation for an inhomogeneous Poisson
process by deriving an approximating Poisson-Gamma ?Kalman? filter
that allows for uncertainty quantification. The ensemble-based implementation
of the filter is developed in a similar approach to the ensemble Kalman
filter, making the filter applicable to large-scale real world applications unlike
nonlinear filters such as the particle filter. The filter has the advantage
that it is independent of the underlying model for the process intensity,
and can therefore be used for many different crime models, as well as other
application domains. The performance of the filter is demonstrated on synthetic
data and real Los Angeles gang crime data and compared against a
very large sample-size particle filter, showing its effectiveness in practice. In addition the forecast skill of the Hawkes model is investigated for a forecast
system using the Receiver Operating Characteristic (ROC) to provide a useful
indicator for when predictive policing software for a crime type is likely
to be useful. The ROC and Brier scores are used to compare and analyse
the forecast skill of sequential data assimilation and MLE. It is found that
sequential data assimilation produces improved probabilistic forecasts over
The primary focus is a sequential data assimilation method for count data modelled by an inhomogeneous Poisson process. In particular, a quadratic approximation technique similar to the extended Kalman filter is applied to develop a sub-optimal, discrete-time, filtering algorithm, called the extended Poisson-Kalman filter (ExPKF), where only the mean and covariance are sequentially updated using count data via the Poisson likelihood function. The performance of ExPKF is investigated in several synthetic experiments where the true solution is known. In numerical examples, ExPKF provides a good estimate of the ?true? posterior mean, which can be well-approximated by the particle filter (PF) algorithm in the very large sample size limit. In addition, the experiments demonstrates that the ExPKF algorithm can be conveniently used to track parameter changes; on the other hand, a non-filtering framework such as a maximum likelihood estimation (MLE) would require a statistical test for change points or implement time-varying parameters. Finally, to demonstrate the model on real-world data, the ExPKF is used to approximate the uncertainty of urban crime intensity and parameters for self-exciting crime models. The Chicago Police Department?s CLEAR (Citizen Law Enforcement Analysis and Reporting) system data is used as a case study for both univariate and multivariate Hawkes models. An improved goodness of fit measured by the Kolomogrov-Smirnov (KS) statistics is achieved by the filtered intensity. The potential of using filtered intensity to improve police patrolling prioritisation is also tested. By comparing with the prioritisation based on MLE-derived intensity and historical frequency, the result suggests an insignificant difference between them. While the filter is developed and tested in the context of urban crime, it has the potential to make a contribution to data assimilation in other application areas.
Consider a body of marine sediment, which an organism burrows into. Advective transport of water induced by the organism?s activity is referred to as bioirrigation here. One consequence is that the spatial distribution of oxygen in the sediment can be disturbed. To better understand these effects, further research into estimating flows induced by bioirrigation is conducted given images depicting spatio- temporal distributions of tracers that are carried with flows induced by the organism Arenicola marina in aquaria which are ?narrow? in depth [1?3].
The multi- resolution Horn- Schunck method is employed here because it can cope with the sometimes fictitious ?large? displacements that fluid deformation seems to produce . But some of the recovered divergences here have unrealistically ?large? magnitudes (relative to those near the injection location) where they ought to be comparatively ?small? .
Quantifying error in flow fields is difficult when the true solution is unknown. One can subjectively define uncertainty in their components, and observations, to follow Gaussian distributions which are updated using Kalman filtering . Given a pair of synthetic images, posterior variances seem to get reduced most where angular errors are comparatively ?small? . So posterior variances are used to infer errors when the true solution is unknown here because they are independent of it .
Unrealistic ?large? divergence magnitudes (relative to those near injection locations) still appear where they ought to be comparatively ?small? . In line with previous research, one tries modelling flows induced by bioirrigating Arenicola marina as two- dimensional incompressible point sources. Only three parameters, namely the source strength, x- and y- coordinates, need estimating rather than flow components at each grid cell. A Markov chain Monte- Carlo method is employed for this task, instead of the Kalman filter, because the state being estimated is no longer proportional to observations. Although comparatively ?large? divergence magnitudes now only appear near locations of fluid injection, this approach seems computationally expensive on one?s Dell Optiplex 7010 computer.
Outflow appears to be induced at the sediment- water interface by a two- dimensional incompressible point source beneath it. One questions whether there should be a little inflow as well because when an organism burrows forwards, the volume that it previously occupied ought to refill with surrounding fluid. This could be accounted for here by considering an additional two- dimensional incompressible flow at the sediment- water interface, as well as a point source at the injection location. But more parameters would need estimating. In an attempt to reduce computation times, the simulations involving the Markov chain Monte- Carlo method are rerun using two iterative ensemble Kalman filters (respectively).
Given a sequence of empirical distribution data (e.g. a movie of a spatiotemporal process
such as a fluid
flow), this work develops an ensemble data assimilation method to estimate the
transition probability that represents a finite approximation of the Frobenius-Perron operator.
This allows a dynamical systems knowledge to be incorporated into a prior ensemble, which provides
sensible estimates in instances of limited observation. We demonstrate improved estimates over a constrained optimization approach (based on a quadratic programming problem) which does not impose a prior on the solution except for Markov properties. The estimated transition
probability then enables several probabilistic analysis of dynamical systems. We focus only on the identification of coherent patterns from the estimated Markov transition to demonstrate its application as a proof-of-concept. To the best of our knowledge, there have not been many
works on data-driven methods to identify coherent patterns from this type of data. While here the results are presented only in the context of dynamical systems applications, this work we present here has the potential to make a contribution in wider application areas that require the
estimation of transition probabilities from a time-ordered spatio-temporal distribution data.