Probabilistic description of extreme events in intermittently unstable systems excited by correlated stochastic processes

In this work, we consider systems that are subjected to intermittent instabilities due to external stochastic excitation. These intermittent instabilities, though rare, have a large impact on the probabilistic response of the system and give rise to heavy-tailed probability distributions. By making appropriate assumptions on the form of these instabilities, which are valid for a broad range of systems, we formulate a method for the analytical approximation of the probability distribution function (pdf) of the system response (both the main probability mass and the heavy-tail structure). In particular, this method relies on conditioning the probability density of the response on the occurrence of an instability and the separate analysis of the two states of the system, the unstable and stable state. In the stable regime we employ steady state assumptions, which lead to the derivation of the conditional response pdf using standard methods for random dynamical systems. The unstable regime is inherently transient and in order to analyze this regime we characterize the statistics under the assumption of an exponential growth phase and a subsequent decay phase until the system is brought back to the stable attractor. The method we present allows us to capture the statistics associated with the dynamics that give rise to heavy-tails in the system response and the analytical approximations compare favorably with direct Monte Carlo simulations, which we illustrate for two prototype intermittent systems: an intermittently unstable mechanical oscillator excited by correlated multiplicative noise and a complex mode in a turbulent signal with fixed frequency, where multiplicative stochastic damping and additive noise model interactions between various modes.Comment: 29 pages, 15 figure

Linear theory for filtering nonlinear multiscale systems with model error

We study filtering of multiscale dynamical systems with model error arising from unresolved smaller scale processes. The analysis assumes continuous-time noisy observations of all components of the slow variables alone. For a linear model with Gaussian noise, we prove existence of a unique choice of parameters in a linear reduced model for the slow variables. The linear theory extends to to a non-Gaussian, nonlinear test problem, where we assume we know the optimal stochastic parameterization and the correct observation model. We show that when the parameterization is inappropriate, parameters chosen for good filter performance may give poor equilibrium statistical estimates and vice versa. Given the correct parameterization, it is imperative to estimate the parameters simultaneously and to account for the nonlinear feedback of the stochastic parameters into the reduced filter estimates. In numerical experiments on the two-layer Lorenz-96 model, we find that parameters estimated online, as part of a filtering procedure, produce accurate filtering and equilibrium statistical prediction. In contrast, a linear regression based offline method, which fits the parameters to a given training data set independently from the filter, yields filter estimates which are worse than the observations or even divergent when the slow variables are not fully observed

Derivation and analysis of simplified filters

Filtering is concerned with the sequential estimation of the state, and uncertainties, of a Markovian system, given noisy observations. It is particularly difficult to achieve accurate filtering in complex dynamical systems, such as those arising in turbulence, in which effective low-dimensional representation of the desired probability distribution is challenging. Nonetheless recent advances have shown considerable success in filtering based on certain carefully chosen simplifications of the underlying system, which allow closed form filters. This leads to filtering algorithms with significant, but judiciously chosen, model error. The purpose of this article is to analyze the effectiveness of these simplified filters, and to suggest modifications of them which lead to improved filtering in certain time-scale regimes. We employ a Markov switching process for the true signal underlying the data, rather than working with a fully resolved DNS PDE model. Such Markov switching models haven been demonstrated to provide an excellent surrogate test-bed for the turbulent bursting phenomena which make filtering of complex physical models, such as those arising in atmospheric sciences, so challenging

Assimilating Eulerian and Lagrangian data to quantify flow uncertainty in testbed oceanography models

Data assimilation is the act of merging observed data into a mathematical model. This act enables scientists from a wide range of disciplines to make predictions. For example, predictions of ocean circulations are needed to provide hurricane disaster maps. Alternatively, using ocean current predictions to adequately manage oil spills has significant practical applications. Predictions are uncertain and this uncertainty is encoded into a posterior probability distribution. This thesis aims to explore two overarching aspects of data assimilation, both of which address the influence of the mathematical model on the posterior distribution. The first aspect we study is model error. Error is always present in mathematical models. Therefore, characterising posterior flow information as function of model error is paramount in understanding the practical implications of predictions. In a model describing advective transport, we make observations of the underlying flow at fixed locations. We characterise the mean of the posterior distribution as a function of the error in the advection velocity parameter. When the error is zero, the model is perfect and we reconstruct the true underlying flow. Partial recovery of the true underlying flow occurs when the error is rational, the denominator of which dictates the number of Fourier modes present in the reconstruction. An irrational error leads to retrieval only of the spatial mean of the flow. The second aspect we study is the control of ocean drifters. Commonplace in oceanography is the collection of ocean drifter positions. Ocean drifters are devices that sit on the surface of the ocean and move with the flow, transmitting their position via GPS to stations on land. Using drifter data, it is possible to obtain a posterior on the underlying flow. This problem, however, is highly underdetermined. Through controlling an ocean drifter, we attempt to improve our knowledge of the underlying flow. We do this by instructing the drifter to explore parts of the flow currently uncharted, thereby obtaining fresh observations. The efficacy of a control is determined by its e↵ect on the variance of the posterior distribution. A smaller variance is interpreted as a better understanding of the flow. We show a systematic reduction in variance can be achieved by utilising controls that allow the drifter to navigate new or ‘interesting’ flow structures, a good example of which are eddies