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

/pdf/extending-dynamic-mode-decomposition-a-data-driven-2ugfne3xq8.pdf

Extending Dynamic Mode Decomposition: A Data--Driven Approximation of the Koopman Operator

The formation and accretion of ice on the leading edge of an airfoil can be detrimental to aerodynamic performance. Furthermore, the geometric shape of leading edge ice profiles can vary significantly depending on a wide range of physical parameters, which can translate into a wide variability in aerodynamic performance. The purpose of this work is to explore the variability in airfoil aerodynamic performance that results from variability in leading edge ice shape profile. First, we demonstrate how to identify a low-dimensional set of parameters that governs ice shape from a database of ice shapes using Proper Orthogonal Decomposition (POD). Then, we investigate the effects of uncertainty in the POD coefficients. This is done by building a global response surface surrogate using Polynomial Chaos Expansions (PCE). To construct this surrogate efficiently, we use adaptive sparse grid sampling of the POD parameter space. We then analyze the data from a statistical standpoint.

Data-Driven Low-Dimensional Modeling and Uncertainty Quantification for Airfoil Icing

A data-driven procedure for identifying the dominant transport barriers in a time-varying flow from limited quantities of Lagrangian data is presented. Our approach partitions state space into pairs of coherent sets, which are sets of initial conditions chosen to minimize the number of trajectories that "leak" from one set to the other under the influence of a stochastic flow field during a pre-specified interval in time. In practice, this partition is computed by posing an optimization problem, which once solved, yields a pair of functions whose signs determine set membership. From prior experience with synthetic, "data rich" test problems and conceptually related methods based on approximations of the Perron-Frobenius operator, we observe that the functions of interest typically appear to be smooth. As a result, given a fixed amount of data our approach, which can use sets of globally supported basis functions, has the potential to more accurately approximate the desired functions than other functions tailored to use compactly supported indicator functions. This difference enables our approach to produce effective approximations of pairs of coherent sets in problems with relatively limited quantities of Lagrangian data, which is usually the case with real geophysical data. We apply this method to three examples of increasing complexity: the first is the double gyre, the second is the Bickley Jet, and the third is data from numerically simulated drifters in the Sulu Sea.

Identifying Finite-Time Coherent Sets from Limited Quantities of Lagrangian Data

We demonstrate the use of the harmonic resolvent to study the physics of subharmonic flow structures in the proximity of time-periodic solutions of the Navier-Stokes equation. As an illustrative example, we study the vortex pairing phenomenon in a periodically forced axisymmetric jet. While it is well-known that vortex pairing in jets and mixing layers arises from the interaction between subharmonic perturbations and the fundamental vortex shedding mode, here we provide additional insight. In particular, we uncover the nature of the strong cross-frequency interactions in the flow, and we compute the spatiotemporal mode that drives the pairing mechanism. 

Clarence W. Rowley

Papers

Extending Dynamic Mode Decomposition: A Data--Driven Approximation of the Koopman Operator

Data-Driven Low-Dimensional Modeling and Uncertainty Quantification for Airfoil Icing

Identifying Finite-Time Coherent Sets from Limited Quantities of Lagrangian Data

Data-Driven Low-Dimensional Modeling and Uncertainty Quantification for Airfoil Icing

Analysis of the dynamics of subharmonic flow structures via the harmonic resolvent: Application to vortex pairing in an axisymmetric jet