There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

The description of coherent features of fluid flow is essential to our understanding of fluid-dynamical and transport processes. A method is introduced that is able to extract dynamic information from flow fields that are either generated by a (direct) numerical simulation or visualized/measured in a physical experiment. The extracted dynamic modes, which can be interpreted as a generalization of global stability modes, can be used to describe the underlying physical mechanisms captured in the data sequence or to project large-scale problems onto a dynamical system of significantly fewer degrees of freedom. The concentration on subdomains of the flow field where relevant dynamics is expected allows the dissection of a complex flow into regions of localized instability phenomena and further illustrates the flexibility of the method, as does the description of the dynamics within a spatial framework. Demonstrations of the method are presented consisting of a plane channel flow, flow over a two-dimensional cavity, wake flow behind a flexible membrane and a jet passing between two cylinders.

https://hal-polytechnique.archives-ouvertes.fr/hal-01020654/document

Dynamic mode decomposition of numerical and experimental data

The purpose of this paper is to provide a comprehensive presentation and interpretation of the Ensemble Kalman Filter (EnKF) and its numerical implementation. The EnKF has a large user group, and numerous publications have discussed applications and theoretical aspects of it. This paper reviews the important results from these studies and also presents new ideas and alternative interpretations which further explain the success of the EnKF. In addition to providing the theoretical framework needed for using the EnKF, there is also a focus on the algorithmic formulation and optimal numerical implementation. A program listing is given for some of the key subroutines. The paper also touches upon specific issues such as the use of nonlinear measurements, in situ profiles of temperature and salinity, and data which are available with high frequency in time. An ensemble based optimal interpolation (EnOI) scheme is presented as a cost-effective approach which may serve as an alternative to the EnKF in some applications. A fairly extensive discussion is devoted to the use of time correlated model errors and the estimation of model bias.

The Ensemble Kalman Filter: Theoretical formulation and practical implementation

Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.

https://www.pnas.org/content/pnas/113/15/3932.full.pdf

Discovering governing equations from data by sparse identification of nonlinear dynamical systems

http://cwrowley.princeton.edu/papers/Tu-JCP-2012.pdf

An improved algorithm for balanced POD through an analytic treatment of impulse response tails

We use Direct Lyapunov Exponents (DLE) to identify Lagrangian coherent structures in two different three-dimensional flows, including a single isolated hairpin vortex, and a fully developed turbulent flow These results are compared with commonly used Eulerian criteria for coherent vortices We find that, despite additional computational cost, the DLE method has several advantages over Eulerian methods, including greater detail and the ability to define structure boundaries without relying on a preselected threshold As a further advantage, the DLE method requires no velocity derivatives, which are often too noisy to be useful in the study of a turbulent flow We study the evolution of a single hairpin vortex into a packet of similar structures, and show that the birth of a secondary vortex corresponds to a loss of hyperbolicity of the Lagrangian coherent structures

/pdf/detection-of-lagrangian-coherent-structures-in-3d-turbulence-b1s8gz5csv.pdf

Detection of Lagrangian Coherent Structures in 3D Turbulence

Reduced-order models of the nonlinear complex Ginzburg-Landau (CGL) equation are computed using a nonlinear generalization of balanced truncation. The method involves Galerkin projection of the nonlinear dynamics onto modes determined by balanced truncation of a linearized system and is compared to a standard method using projection onto proper orthogonal decomposition (POD) modes computed from snapshots of nonlinear simulations. It is found that the nonlinear reduced- order models obtained using modes from linear balanced truncation capture very well the transient dynamics of the CGL equation and outperform POD models; i.e., a higher number of POD modes than linear balancing modes is typically necessary in order to capture the dynamics of the original system correctly. In addition, we find that the performance of POD models compares well to that of balanced truncation models when the degree of nonnormality in the system, in this case determined by the streamwise extent of a disturbance amplification region, is lower. Our findings therefore indicate that the superior performance of balanced truncation compared to POD/Galerkin models in capturing the input/output dynamics of linear systems extends to the case of a nonlinear system, both for the case of significant transient growth, which represents a basic model of boundary layer instabilities, and for a limit cycle case that represents a basic model of vortex shedding past a cylinder.

http://cwrowley.princeton.edu/papers/Ilak_CGL.pdf

Model Reduction of the Nonlinear Complex Ginzburg-Landau Equation ∗

Many of the tools of dynamical systems and control theory have gone largely unused for fluids, because the governing equations are so dynamically complex, both high-dimensional and nonlinear. Model reduction involves finding low-dimensional models that approximate the full high-dimensional dynamics. This paper compares three different methods of model reduction: proper orthogonal decomposition (POD), balanced truncation, and a method called balanced POD. Balanced truncation produces better reduced-order models than POD, but is not computationally tractable for very large systems. Balanced POD is a tractable method for computing approximate balanced truncations, that has computational cost similar to that of POD. The method presented here is a variation of existing methods using empirical Gramians, and the main contributions of the present paper are a version of the method of snapshots that allows one to compute balancing transformations directly, without separate reduction of the Gramians; and an output p...

Model Reduction for Fluids, Using Balanced Proper Orthogonal Decomposition

Techniques for feedback control of oscillations in the flow past a cavity are presented. Low-order models are obtained using two methods (empirical Galerkin models, and a simple nonlinear oscillator model), and validated against 2D direct numerical simulations of the flow, which is actuated by a body force at the leading edge of the cavity. The models are used to construct dynamic observers, which reconstruct the flow state from a single pressure sensor, and perform much better than static estimators commonly used for flow estimation. Several control approaches are compared, including simple proportional control with a phase lag, LQG control using Galerkin models, and a dynamic phasor approach based on the work of Noack et al (2003). All three controllers are implemented in the full simulation, and able to reduce the amplitude of oscillations. The LQG regulator requires careful tuning, and the closed-loop behavior often does not match that predicted by the model, but the dynamic phasor approach eliminates the oscillations completely in the full simulation, with a transient response that matches that predicted by the loworder model.

/pdf/model-based-control-and-estimation-of-cavity-flow-307qosx58v.pdf

Clarence W. Rowley

Papers

An improved algorithm for balanced POD through an analytic treatment of impulse response tails

Detection of Lagrangian Coherent Structures in 3D Turbulence

Model Reduction of the Nonlinear Complex Ginzburg-Landau Equation ∗

Model Reduction for Fluids, Using Balanced Proper Orthogonal Decomposition

Model-based control and estimation of cavity flow oscillations