Prof. George Haller
Professor, Institute for Mechanical Systems, ETH Zürich
Material barriers to diffusive and stochastic transport
Wednesday, April 24, 2019, 12:00pm to 1:00pm | Room 31-270

Observations of tracer transport in fluids generally reveal highly complex patterns shaped by an intricate network of transport barriers. The elements of this network appear to be universal for small diffusivities, independent of the tracer and its initial distribution. In this talk, I will first review prior, purely advective approaches to transport barrier detection. Next, I will discuss a mathematical theory to predict diffusive transport barriers and enhancers solely from the flow velocity, without reliance on expensive diffusive or stochastic simulations. This theory yields a simplified computational scheme for diffusive transport problems, such as the estimation of salinity redistribution for climate studies and the forecasting of oil spill spreads on the ocean surface. I will illustrate the results on turbulence simulations and observational ocean velocity data.

Professor Franz-Erich Wolter
Welfenlab, Institute of Man-Machine-Communication
Leibniz University Hannover, Germany, http://welfenlab.de/en/index.html

Differential Geometric Analysis of Multi-dimensional Quadratic Maps
Friday, April 8, 2019, 12:00pm to 13:00pm | Room 5-314

Multi-dimensional quadratic maps arise in engineering analysis (such as power flow analysis of electric grids). Differential geometric concepts provide new insights to the stability properties of such high-dimension quadratic maps and also yield accurate computational methods in the vicinity of singularities including local inversion of the map. The singular sets of such maps are of cardinal importance in analyzing their stability properties. The singular set of a map is defined by the zero set of the determinant of the Jacobean of the map and is alternatively sometimes referred to as the solution space boundary (SSB) in the context of power grids. This determinant is a high-degree multivariate polynomial in the components of the vector being mapped. The above zero set or SSB locally represents a hyper-surface for generic points. Local parametrization of the SSB hyper-surface is difficult to compute stably and accurately due to the high degree polynomials involved. Differential geometry methods using geodesic coordinates provide new ways to describe the local parametrization of the SSB, as well as sub-manifolds of the SSB resulting from non-linear constraints. A related paper can be found at: https://www.researchgate.net/publication/332093394

Prof. Nan Chen
Assistant Professor, Department of Mathematics, University of Wisconsin-Madison
A Nonlinear Conditional Gaussian Framework for Prediction, State Estimation and Uncertainty Quantification in Complex Dynamical System
Friday, April 5, 2019, 11:00am to 12:00pm | Room 5-314

A nonlinear conditional Gaussian framework for extreme events prediction, state estimation (data assimilation) and uncertainty quantification in complex dynamical systems will be introduced in this talk. Despite the conditional Gaussianity, the models within this framework remain highly nonlinear and are able to capture strongly non-Gaussian features such as intermittency and extreme events. The conditional Gaussian structure allows efficient and analytically solvable conditional statistics that facilitates the real-time data assimilation and prediction.
In the first part of this talk, the general framework of the nonlinear conditional Gaussian systems, including a gallery of examples in geophysics, fluids, engineering, neuroscience and material science, will be presented. This is followed by its wide applications in developing the physics-constrained data-driven nonlinear models and the stochastic mode reduction. In the second part, an efficient statistically accurate algorithm is developed for solving the Fokker-Planck equation in large dimensions, which is an extremely important and challenging topic in prediction, data assimilation and uncertainty quantification. This new efficient algorithm involves a novel hybrid strategy for different subspaces, a judicious block decomposition and statistical symmetry. Rigorous mathematical analysis shows that this method is able to overcome the curse of dimensionality. In the third part of this talk, a low-order model within the nonlinear conditional Gaussian framework is developed to predict the intermittent large-scale monsoon extreme events in nature. The nonlinear low-order model shows higher prediction skill than the operational models and it also succeeds in quantifying the uncertainty in prediction. Other applications of this nonlinear conditional Gaussian framework, such as assimilating multiscale turbulent ocean flows and parameter estimation, will be briefly mentioned at the end of the talk.

Prof. Charles Meneveau
Louis M. Sardella Professor, Department of Mechanical Engineering, Johns Hopkins University
Turbulence and Reduced Models for Large Wind Farms
Monday, March 18, 2019, 12:00pm to 1:00pm | Room 5-314

In this presentation we discuss several properties of the flow structure and turbulence in the wind turbine array boundary layer (WTABL). This particular type of shear flow develops when the atmospheric boundary layer interacts with an array of large wind turbines. Based on such understanding, we aim to develop reduced order, analytically tractable models. These are important engineering tools for wind energy, both for design and control purposes. After reviewing some basic tools to predict mean velocities for total power optimization, we will focus on two fluid mechanical themes relevant to wind farm control and inherent variability. We describe a simple (deterministic) dynamic wake model, its use for wind farm control, and its extensions to the case of yawed wind turbines based on a re-interpretation of lifting line theory adapted to the problem of yawed actuator disks. The second part deals with spectral characteristics of the fluctuations in power generated by an array of wind turbines in a wind farm. We show that modeling of the spatio-temporal structure of canonical turbulent boundary layers coupled with variants of the Kraichnan's random sweeping hypothesis can be used to develop analytical predictions of the frequency spectrum of power fluctuations of wind farms. The work to be presented arose from collaborations with Juliaan Bossuyt, Johan Meyers, Richard Stevens, Tony Martinez, Michael Wilczek, Carl Shapiro and Dennice Gayme. We are grateful for National Science Foundation financial support.

Prof. Christian Lessig
Assistant Professor, Otto-von-Guerick-Universität Magdeburg
Divergence Free Polar Wavelets
Friday, May 18, 2018, 11:00am to 12:00pm | Room 1-150

Divergence free vector fields play an important role in many systems in science and engineering, making their analysis and numerical representation important in fields ranging from climate science to medical imaging. We introduce a tight frame of divergence free wavelets that resolves the different scales and orientations that occur in these vector fields. Our wavelets are thereby divergence free in the ideal, analytic sense, have an intuitive correspondence to natural phenomena, closed form expressions in frequency and space, a multi-resolution structure, and fast transforms. Our construction also provides well defined directional selectivity that, among other things, models the behavior of solenoidal vector fields in the vicinity of boundaries. With suitable window functions, this provides (up to a logarithmic factor) an optimal approximation rate for piecewise continuous divergence-free vector fields in two dimensions. We demonstrate the numerical practicality and efficiency of our construction for the representation of solenoidal vector fields and the simulation of the Navier-Stokes equation.

Prof. George Tsironis
Professor, Department of Physics, University of Crete & Visiting Fellow, SEAS, Harvard University
Wave propagation in complex media: extreme events, branching and chimeras
Thursday, November 2, 2017, 2:00pm to 3:00pm | Room 5-314

Wave propagation in complex media involves extreme phenomena such as branching and rogue-type waves. We focus on both discrete and continuous systems; in the discrete case the systems are nonlinear [1] while in the continuous one are made of gradient index lenses that provide strong scattering [2]. We investigate the statistics of extreme events and find the features that lead to rogue wave formation. In the continuous case of a disordered linear medium we find that while nonlinearity does not seem to be very important, strong scattering is a necessary condition for extreme events [3]. We also investigate the onset of branching in electronic flows and show that in the ultra-relativistic case, applied to doped graphene, a specific relation for the first passage to the branching regime as a function of surface disorder exists that may be discernible experimentally. Finally we discuss the onset of partial coherence in complex media and show that turbulent chimeras may result in the discrete limit [4].

Prof. Roman Garnett
Assistant Professor, Department of Computer Science and Engineering, Washington University in St. Louis
Bayesian optimization for automating model selection
Thursday, October 26, 2017, 3:00pm to 4:00pm | Room 5-314

We discuss a problem of enormous practical importance that is often ignored: the selection of appropriate prior mean/covariance functions for Gaussian process models. Despite the success of kernel-based nonparametric methods, kernel selection still requires considerable expertise, and is often described as a "black art." We present a sophisticated method for automatically searching for an appropriate kernel from an infinite space of potential choices. Previous efforts in this direction have focused on traversing a kernel grammar, only examining the data via computation of marginal likelihood. Our proposed search method is based on Bayesian optimization in model space, where we reason about model evidence as a function to be maximized. We explicitly reason about the data distribution and how it induces similarity between potential model choices in terms of the explanations they can offer for observed data. In this light, we construct a novel kernel between models to explain a given dataset (a "kernel kernel"). Our method is capable of finding a model that explains a given dataset well without any human assistance, often with fewer computations of model evidence than previous approaches, a claim we demonstrate empirically.

Professor, Department of Statistics and Operations Research, University of North Carolina - Chapel Hill
Some extreme value problems arising with ship motions
Thursday, September 21, 2017, 3:00pm to 4:00pm | Room 5-314

The focus of the talk is on estimating probabilities of two extreme events of interest in Naval Architecture: a ship motion (e.g. roll) exceeding some critical large value and capsizing of a ship, both in irregular waves. These events are rare and usually not observed in collected data. Extreme Value Theory provides statistical tools and some justification for constructing probability estimates of rare events by extrapolating into distribution tails where data are not available, for example, by using the generalized Pareto distribution (GPD) in the peaks-over-threshold approach. These statistical tools will be examined for the rare events of interest on ship motion data generated by high-fidelity computer codes, and in the context of qualitative physical models for ship motions such as 1-DOF nonlinear random oscillators. This will lead to several lessons on how stochastic dynamics and Naval Architecture can guide the use of statistical methods for extreme value analysis related to ship motions. The talk is based on joint work with D. Glotzer (UNC), T. Sapsis (MIT), and V. Belenky, K. Weems and other researchers (NSWC, Carderock Division).

Dr. Maziar Raissi
Postdoc, Division of Applied Mathematics, Brown University
Numerical Gaussian Processes (Physics Informed Learning Machines) [Slides]
Thursday, September 14, 2017, 11:00am to Noon | Room 5-314

We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes, by construction, are designed to deal with cases where: (1) all we observe are noisy data on black-box initial conditions, and (2) we are interested in quantifying the uncertainty associated with such noisy data in our solutions to time-dependent partial differential equations. Our method circumvents the need for spatial discretization of the differential operators by proper placement of Gaussian process priors. This is an attempt to construct structured and data-efficient learning machines, which are explicitly informed by the underlying physics that possibly generated the observed data. The effectiveness of the proposed approach is demonstrated through several benchmark problems involving linear and nonlinear time-dependent operators. In all examples, we are able to recover accurate approximations of the latent solutions, and consistently propagate uncertainty, even in cases involving very long time integration.

Dr. Antoine Blanchard,
PhD Student, Department of Aerospace Engineering, University of Illinois at Urbana-Champaign
Laminar vortex-induced vibration of a circular cylinder with nonlinear energy sinks

Thursday, Mar 23, 2017, 1:00pm to 2:00pm | Room 1-371

We consider two-dimensional laminar flow past a linearly-sprung cylinder undergoing rectilinear vortex-induced vibration (VIV) normal to the mean flow, with an attached "nonlinear energy sink" (NES). We show that a "rotational NES" (consisting of a mass allowed to rotate about the cylinder axis, and whose rotational motion is linearly damped by a viscous damper) gives rise to a variety of phenomena not seen in VIV with no NES, including partial stabilization of the vortex street formed downstream of the cylinder, drag reduction, and coexistence of multiple long-time solutions. We also show that a "translational NES" (consisting of a mass allowed to translate in the direction of travel of the cylinder, and whose rectilinear motion is restrained by an essentially cubic spring and a linear viscous damper) leads to complete and partial VIV-suppression mechanisms, relaxation cycles, as well as Hopf and Shilnikov bifurcations. For both configurations, the computational results are investigated analytically using reduced-order models of the fluid-structure interaction, thus providing insight into the nonlinear dynamics of the infinite-dimensional coupled system.

Prof. George Haller
Professor, Institute for Mechanical Systems, ETH Zürich
Exact Nonlinear Model Reduction in Mechanical Systems
Monday, January 23, 2017, 11:00am to Noon | Room 5-314

We discuss two recent methods that enable a mathematically exact reduction of multi-degree-of-freedom, nonlinear mechanical systems to lower-dimensional models. The first method is based on a reduction to spectral submanifolds, which are invariant manifolds arising from modal subspaces of the linearized oscillations near an equilibrium. The second method is based on a reduction onto a global slow manifold that enslaves stiffer vibration modes to softer ones. We show applications to both mechanical model problems and to experimental vibration data for beam oscillations.

Prof. Amin Chabchoub
Assistant Professor, Department of Mechanical Engineering, Aalto University (former Helsinki University of Technology), Finland
Rogue Waves within the Framework of Weakly Nonlinear Evolution Equations – Applicability and Limitations
Wednesday, November 16, 2016, 3:00pm to 4:00pm | Room 5-314

Extreme ocean waves, also referred to as freak or rogue waves (RWs), are known to appear without warning and have a disastrous impact on ships and offshore structures as a consequence of the substantially large wave heights they can reach. Studies on RWs have recently attracted scientific interest due to the interdisciplinary universal nature of the modulation instability (MI) of weakly nonlinear waves as well as for the sake of accurate modeling and prediction of these mysterious extremes. Indeed, exact solutions of the nonlinear Schrödinger equation provide advanced backbone models that can be used to describe the dynamics of RWs in time and space, providing therefore deterministic prototypes that can be investigated to reveal novel insights of MI. A wide range of recently conducted experiments and numerical simulations on breathers as well as prediction aspects will be discussed. Furthermore, the existence of such localized structures in realistic sea state conditions will be presented as well.

Prof. Gerassimos Athanassoulis
School of Naval Architecture and Marine Engineering, National Technical University of Athens, Greece and
Research Center for High Performance Computing, ITMO University, St. Petersburg, Russia
A Hamiltonian approach to the nonlinear water-wave problem over bathymetry based on a new implementation of the DtN operator
Tuesday, May 10, 2016, 2:00pm to 3:00pm | Room 5-314

The fully nonlinear water-wave problem, over arbitrary smooth bathymetry, is formulated as a nonlinear, nonlocal Hamiltonian system, in the spirit of Zakharov [1] and Craig & Sulem [2] approach. The nonlocal coefficients of the Hamiltonian equations, which are usually expressed in terms of the Dirichlet-to-Neumann (DtN) operator [2], [3], are represented and implemented differently in this work. They are expressed by means of the first mode of a novel, rapidly convergent series expansion of the substrate wave potential, solved in the instantaneous geometrical configuration [4], [5]. This expansion is based on a local vertical basis consisting of the usual vertical modes (providing an L2-local vertical basis), plus two additional modes, which make it an H2-local (Sobolev) vertical basis at each horizontal position [5], [6], [7]. Thus, a highly accurate representation/implementation of the boundary derivatives of the wave potential is possible, even for very steep waves and bathymetries. Besides, the expansion is rapidly convergent, with exponential decay rate for the few first modes, and O(n-4) decay rate asymptotically. As a consequence, only a few modes, up to 6 or 7, are enough to provide a highly accurate representation of the nonlocal coefficients of the Hamiltonian equations. The latter are solved in the time domain by the classical, fourth-order Runge-Kutta method, providing nice solutions to various configurations. The excitation to simulations of wave propagation over intermediate-depth water waves is applied to a generation-absorption entrance layer [8].

Prof. Stuart Anderson
Space and Atmospheric Physics Group, School of Physical Sciences, University of Adelaide
Observing the geophysical environment and its dynamics with decametric radar
Monday, May 9, 2016, 3:00pm to 4:00pm | Room 3-434

Radars operating at decametric wavelengths - popularly known as 'over-the-horizon' radars because of their ability to exploit non-line-of-sight propagation mechanisms – possess unique capabilities for observing the environment on a synoptic scale and measuring a variety of geophysical parameters. These remote sensing applications have been the subject of continuing research since the 1950's, with over 500 radars presently operating in the HF band, 3 – 30 MHz. The majority of these exploit the surface wave mode of radiowave propagation, some use the skywave mode, and a few employ line-of-sight or more exotic mechanisms.
By far the best known decametric radar observables are integral and bulk properties of the upper ocean, such as significant waveheight, dominant wave period and direction, and surface current velocity, but the ocean is not the only domain whose state variables and dynamics imprint themselves on the radar signals in ways which offer the prospect of information retrieval. Decametric radar signatures of conditions and phenomena in the lithosphere, the cryosphere, the atmosphere, the ionosphere, the magnetosphere and the heliosphere have all been subjects of inquiry by the radar community and, in many cases, these studies have yielded significant insights and sometimes unique windows onto the associated physics. Recently it has been demonstrated that observability can be extended to include quite subtle aspects of dynamical processes characterised by nonlinearity, non-adiabaticity, non-Gaussianity and other complex behaviour.
In this talk I shall describe the remote sensing abilities of decametric radar with reference to the observation process and its limitations, the electrodynamics of the interaction mechanisms, the intrinsically nonlinear and multiscale nature of the geophysical environment, the ill-posedness of the inversion problem, and the symbiosis of this sensing modality with other technologies.

Prof. Vassilis M Rothos
Lab of Nonlinear Mathematics, School of Mechanical Engineering,
Aristotle University of Thessaloniki and
Complex System Group at Institute of Applied & Computational Mathematics (IACM), Crete, GREECE
Localized Structures in Nonlinear Magnetic metamaterial Lattices
Tuesday, May 3, 2016, 1:00pm to 2:00pm | Room 5-233

This talk reviews results about the existence of spatially localized waves in nonlinear chains of coupled oscillators, and provides new results for the Klein-Gordon (KG) lattice and model of a one-dimensional magnetic metamaterial formed by a discrete array of nonlinear resonators. Localized solutions include solitary waves of permanent form and traveling breathers which appear time periodic in a system of reference moving at constant velocity. For KG lattices of magnetic metamaterials, we obtain a general criterion for spectral stability of multi-site breathers for a small coupling constant. For the metamaterial lattices we focus on periodic traveling wave due to the presence of periodic force. We employ topological and variational methods to study the existence and the stability of periodic waves. These localized structures are also computed and discussed numerically.

Prof. Linda Petzold,
Professor, Department of Computer Science and the Department of Mechanical Engineering, University of California Santa Barbara
The Emerging Roles and Computational Challenges of Stochasticity in Biological Systems

Thursday, April 21, 2016, 1:30pm to 2:30pm | Room 5-314

In recent years it has become increasingly clear that stochasticity plays an important role in many biological processes. Examples include bistable genetic switches, noise enhanced robustness of oscillations, and fluctuation enhanced sensitivity or "stochastic focusing".. Numerous cellular systems rely on spatial stochastic noise for robust performance. We examine the need for stochastic models, report on the state of the art of algorithms and software for modeling and simulation of stochastic biochemical systems, and identify some computational challenges.

Dr. Ed Habtour,
US Army Research Laboratory, Vehicle Technology Directorate
Exploiting Nonlinear Dynamic Parameters to Outsmart Fatigue in Rotorcrafts
Thursday, March 31, 2016, 1:30pm | Room 3-333

Military rotorcrafts are highly nonlinear systems that operate in complex environments. The common methods for evaluating the health of these systems are based on simplified fatigue tests and linear models. Consequently, engineers compensate by including safety and correction factors, which lead to a "safety multiplicative effect". These compromises come with penalties, such as increases in the vehicle's size, and weight. As a result, the US Army Research Laboratory (ARL) has developed a holistic approach to improve the sustainment of rotorcrafts through detecting precursors to fatigue damage using the aircraft global response. Analytical and experimental techniques are under development at ARL to understand the interplays between the components nonlinear dynamic parameters and the microstructural evolution to track the aging process prior to crack initiation. The presentation provides an overview of damage precursor research efforts at ARL, and an approach to quantify the interplays between the nonlinear macro- and micro-parameters. Our method exploit the nonlinear parameters sensitivities to precursors to track the structural health. ARL approach is a paradigm shift from costly manual maintenance to preemptively reporting precursors to damage. We believe including precursors' detection in the aircraft health monitoring systems is profoundly a new innovation in achieving bio-inspired health awareness that needs significant development.

Dr. Andreas Damianou,
Research Associate, Institute for Translational Neuroscience, Robotics group, University of Sheffield
System identification and control with (deep) Gaussian processes

Thursday, Feb 11, 2016, 1:00pm to 2:00pm | Room 5-314

Work in Gaussian processes (GPs) is setting a new paradigm for data-driven modeling in engineering fields, such as control, dynamical systems and robotics. In control and systems identification, GP-based approaches often outperform traditional NAR(MA)X and Kalman filtering schemes. The attractive properties of GPs in these settings include their Bayesian, non-parametric nature and principled uncertainty quantification/propagation. In this talk I will give a brief introduction to non-parametric modelling with GPs and review work which applies them in the control and dynamical systems domain. I will then introduce recent, powerful approaches obtained by combining GPs with latent variable and deep learning techniques.

Prof. Ioannis Kougioumtzoglou,
Assistant Professor, Dept. of Civil Engineering & Engineering Mechanics, Columbia University, USA
Compressive Sensing and Path Integral Techniques for Uncertainty Modeling and Propagation in Complex Dynamic Systems

Thursday, Dec 3, 2015, 3:00pm to 4:00pm | Room 3-350

Abstract

"...The ubiquity of uncertainty in computational estimates of reality and the necessity for its quantification..." has been recently recognized by the National Academies / Research Council. In this regard, a large portion of the engineering mechanics/dynamics community has focused on multi-scale/physics problems with stochastic media properties, random excitations and uncertain initial/boundary conditions. Two main challenges associated with uncertainty treatment relate to the (A) modeling, and the (B) propagation of the uncertainties.

Prof. Predrag Cvitanović
Professor, School of Physics, Georgia Tech
Noise is your friend, or: How well can we resolve state space?

Monday, Nov 16, 2015, 3:00pm to 4:00pm | Room 5-314

All physical systems are affected by some noise that limits the resolution that can be attained in partitioning their state space. What is the best resolution possible for a given physical system?

It turns out that for nonlinear dynamical systems the noise itself is highly nonlinear, with the effective noise different for different regions of system's state space. The best obtainable resolution thus depends on the observed state, the interplay of local stretching/contraction with the smearing due to noise, as well as the memory of its previous states. We show how that is computed, orbit by orbit. But noise also associates to each a finite state space volume, thus helping us by both smoothing out what is deterministically a fractal strange attractor, and restricting the computation to a set of unstable periodic orbits of finite period. By computing the local eigenfunctions of the Fokker-Planck evolution operator, forward operator along stable linearized directions and the adjoint operator along the unstable directions, we determine the 'finest attainable' partition for a given hyperbolic dynamical system and a given weak additive noise. The space of all chaotic spatiotemporal states is infinite, but noise kindly coarse-grains it into a finite set of resolvable states.

Dr Katerina Konakli
ETH Zurich, Chair of Risk, Safety & Uncertainty Quantification
Low-rank tensor approximations versus polynomial chaos expansions for uncertainty propagation and reliability analysis
Thursday, October 8, 2015, 2:30pm to 3:30pm | Room 5-314

Modern engineering faces the challenge of uncertainty propagation through increasingly complex computational models. A remedy is to substitute expensive-to-evaluate models with so-called meta-models, which possess similar statistical properties, while maintaining simple functional forms. Polynomial chaos expansions have proven an effi- cient meta-modeling technique in a wide range of applications, but suf- fer from the curse of dimensionality. A promising alternative for build- ing meta-models with polynomial bases in high-dimensional spaces is the newly emerged technique of low-rank tensor approximations. In this talk, open questions in the construction of such approximations will first be discussed. In the sequel, the newly emerged approach will be confronted with polynomial chaos expansions in applications involving models of different dimensionality. Special emphasis will be given on the estimation of the response distribution at the tails, which is critical for evaluating rare-event probabilities in reliability analysis.

Dr Heyrim Cho

Division of Applied Mathematics - Brown University
High-dimensional Numerical schemes and Dimension Reduction techniques for Uncertainty Quantification based on Probability Density Functions
Thursday, May 14, 2015, 4:00pm to 5:00pm | Room 3-133

Probability density functions (PDFs) provide the entire statistical structure of the solution to stochastic systems. In this talk, we introduce the joint response-excitation PDF approach that enables us to do stochastic simulations based on PDFs with various type of randomness involving non-Gaussian non-Markovian colored noise. We develop efficient numerical algorithms to solve this system in high-dimensions. In particular, we develop high-dimensional numerical schemes by using ANOVA approximation and separated series expansion. Alternatively, we employ dimension reduction techniques such as conditional moment closures and Mori-Zwanzig approach to obtain reduced order equations. These methodologies can be applied in general to stochastic systems to overcome high-dimensionality. The effectiveness of our approach is demonstrated in various stochastic dynamical systems and stochastic PDEs, including Lorenz 96 system and Burgers equation yielding multiple interacting shock waves at random space-time locations.

Prof. Dimitris Giannakis
Assistant Professor, Courant Insitute of Mathematical Sciences, NYU
Extracting and predicting spatiotemporal patterns from data with dynamics-adapted kernels
Friday, March 6, 2015, 2:00pm to 3:00pm | Room 1-390

Kernel methods provide an attractive way of extracting features from data by biasing their geometry in a controlled manner. In this talk, we discuss a family of kernels for dynamical systems featuring an explicit dependence on the generator of the dynamics operating in the phase-space manifold, estimated empirically through finite differences of time-ordered data samples. The associated diffusion operator for data analysis is adapted to the dynamics in that it generates diffusions along the integral curves of the dynamical vector field. We present applications to dimension reduction and timescale separation in toy dynamical systems and comprehensive climate models. We also discuss a technique for analog forecasting based on these kernels. In this nonparametric forecasting technique (originally introduced by Lorenz in 1969), kernels are used to create weighted ensembles of states (analogs) with high similarity to the initial data from a record of historical observations, and the future values of observables are predicted from the historical evolution of the ensemble.

Prof. Alexey Miroshnikov
Visiting Assistant Professor of Department of Mathematics - University of Massachusetts Amherst
On the properties of weak solutions describing dynamic cavitation in nonlinear elasticity

Thursday, Nov 13, 2014, 12:00pm to 1:00pm | Room 1-390

In this work we study the problem of dynamic cavity formation in isotropic compressible nonlinear elastic media. Cavitating solutions were introduced by J.M. Ball [1982, Phil. Trans. R. Soc. Lond. A] in elastostatics and by K.A. Pericak-Spector and S. Spector [1988, Arch. Rational Mech. Anal.] in elastodynamics. They turn out to decrease the total mechanical energy and provide a striking example of non-uniqueness of entropy weak solutions (in the sense of hyperbolic conservation laws) for polyconvex energies. In our work we established various further properties of cavitating solutions. For the equations of radial elasticity we construct self-similar weak solutions that describe a cavity emanating from a state of uniform deformation. For dimensions d = 2,3 we show that cavity formation is necessarily associated with a unique precursor shock. We also study the bifurcation diagram and do a detailed analysis of the singular asymptotics associated to cavity initiation as a function of the cavity speed of the self-similar profiles. We show that for stress-free cavities the critical stretching associated with dynamically cavitating solutions coincides with the critical stretching in the bifurcation diagram of equilibrium elasticity.

Prof. Leonid Manevitch
Semenov Institute of Chemical Physics, Russian Academy of Sciences
Energy exchange, localization and transfer in nonlinear oscillatory chains and nanostructures (resonance non-stationary dynamics)
Wednesday, October 29, 2014, 4:00pm to 5:00pm | Room 3-333

A new approach to non-stationary nonlinear dynamics, based on the concept of Limiting Phase Trajectories (LPTs) is presented. The systems under consideration are finite nonlinear oscillatory chains which can be identified, e.g., as the dynamical models of mechanical structures, polymer macromolecules or carbon nanotubes (CNTs). The LPT describes the most intensive energy exchange between weakly coupled parts of the system which can be considered as effective particles (EPs). They represent the excitations alternative to Nonlinear Normal Modes (NNMs) which are not involved into the processes with the energy exchange and demonstrate the wave-like behavior. It is possible to speak about distinctive wave-particle duality (WPD) in the framework of classical mechanics, and manifestation of particle-like or wave-like behavior depends on the initial conditions or on the type of attractor (it may be NNM as well as LPT).

Postdoctoral Fellow, Georgia Institute of Technology
A variational theory of shearless transport barriers in unsteady dynamical systems
Tuesday, October 28, 2014, 12:00pm to 1:00pm | Room 5-314

The theory of Lagrangian Coherent Structures (LCSs) has advanced significantly over recent years, and now covers both hyperbolic and elliptic material surfaces in unsteady flow. Parabolic (i.e., jet-type) LCSs have, however, remained outside the reach of the theory, despite their significance in oceanic and atmospheric transport. Here I discuss a new variational approach to general shearless transport barriers in two-dimensional unsteady flows, which covers both hyperbolic and parabolic LCSs. I also describe a computational implementation of this new theory, and show applications to model flows and geophysical data sets.

"Wigner measures for singular and nonlinear problems: the scalar case"
Professor Agis Athanasoulis, Department of Mathematics
University of Leicester, UK
Tuesday, September 16, 2014, 12:00pm | Room 5-314

Wigner measures (WMs) have been successfully used as a parameter-free tool to provide homogenized descriptions of wave problems. Notable applications are the efficient simulation of large linear wave fields, and the painless resolution of linear caustics. However, their applicability to non-linear problems has been very limited. In this talk we discuss the role of smoothness of the underlying flow as a limiting factor in the applicability of WMs. Non-smooth flows are ill-posed for measures, and new phenomena are possible in that regime. For example, single wavepackets may be "split" cleanly into several new wavepackets. We introduce a modification of the WM approach, and show that it can capture successfully some of these new phenomena. The motivation behind this work is to develop methods applicable to non-linear problems as well. Some first such applications are also explored.

"Differential Geometric Analysis of Multi-dimensional Quadratic Maps"
Professor Franz-Erich Wolter, Welfenlab, Institute of Man-Machine-Communication
Leibniz University Hannover, Germany, http://welfenlab.de/en/index.html
Monday, April 8, 2019, 12:00-13:00, Room 5-314

Multi-dimensional quadratic maps arise in engineering analysis (such as power flow analysis of electric grids). Differential geometric concepts provide new insights to the stability properties of such high-dimension quadratic maps and also yield accurate computational methods in the vicinity of singularities including local inversion of the map. The singular sets of such maps are of cardinal importance in analyzing their stability properties. The singular set of a map is defined by the zero set of the determinant of the Jacobean of the map and is alternatively sometimes referred to as the solution space boundary (SSB) in the context of power grids. This determinant is a high-degree multivariate polynomial in the components of the vector being mapped. The above zero set or SSB locally represents a hyper-surface for generic points. Local parametrization of the SSB hyper-surface is difficult to compute stably and accurately due to the high degree polynomials involved. Differential geometry methods using geodesic coordinates provide new ways to describe the local parametrization of the SSB, as well as sub-manifolds of the SSB resulting from non-linear constraints. A related paper can be found at: https://www.researchgate.net/publication/332093394

Francesco Romeo, Associate Professor, Dept. Of Structural and Geotechnical Engineering - Sapienza University of Rome
"Transient Dynamics of a Bistable Nonlinear Energy Sink Coupled System"
Monday, August 25, 2014, 4:00pm to 4:00pm | Room 3-434

The dynamics of a two-degree-of-freedom system composed of a grounded linear oscillator coupled to a lightweight mass by means of a spring with both a strongly nonlinear and a negative linear component is described. Numerical and analytical studies are presented aiming to assess the influence of this combined coupling on both the conservative and the dissipative transient dynamics. In particular, these studies are focused on passive nonlinear targeted energy transfer from the impulsively excited linear oscillator to the nonlinear bistable lightweight attachment. It is shown that the main feature of the proposed configuration is the ability of assuring broadband efficient energy transfer over a broad range of input energy. Due to the bistability of the attachment, such favorable behavior is triggered by different nonlinear dynamic mechanisms depending on the energy level.

Dr. Ravi Kumar
MathWorks
Influence of local internal nonlinear attachments on the global dynamics of circular cylinder undergoing vortex-induced vibration
Thursday, May 8, 2014, 4:00pm | Room 5-217
Circular cylinder undergoing "vortex-induced vibration" (VIV) is a well-known nonlinear fluid-structure interaction phenomenon. An additional element known as "nonlinear energy sink" (NES), consisting of a small mass, a linear damper, and an essentially nonlinear spring, is attached to the rigid circular cylinder undergoing VIV in a laminar incompressible flow. The nonlinear interaction of the NES and fluid via rigid body motion of the cylinder leads to several interesting response regimes of the coupled system of flow-cylinder-NES in laminar VIV regime. The localized nonlinearity of NES leads "targeted energy transfer" (TET) resulting in partial suppression of VIV. Unlike a linear tuned-mass damper to suppress VIV, the NES alters the solution over compete VIV lock-in regime. A descriptive reduced-order model is developed using the computational data to study the global dynamics of the coupled system.

Dr Handy Zhang
Division of Applied Mathematics - Brown University
Numerical methods for stochastic partial differential equations with white noise: a spectral approach
Thursday, Apr 24, 2014, 4:00pm to 5:00pm | Room 5-134

Deterministic integration methods in random space, such as polynomial chaos methods and stochastic collocation methods, have been extensively use for numerical methods of stochastic partial differential equation with color noise for their high accuracy. However, these deterministic integration methods are only efficient for problems with low random dimensionality. Thus, these deterministic integration methods are only efficient for short time integration of stochastic partial differential equation with temporal white noise as the number of increments of Brownian motion increases with time steps. For linear stochastic partial differential equations, we apply deterministic integration methods using the linear property of these equations and recursive strategy in time for a longer-time numerical integration.

Department of Electrical & Computer Engineering and Mathematics - Northeastern University
Can Low Order Galerkin Models Work in Active Fluid Flow Control?
Thursday, Apr 17, 2014, 2:00pm to 3:00pm | Room 5-314

Active fluid flow control dates to Prandtl's groundbreaking work on shear layer flows during the first decade of the 20th century and has been the subject of intensive experimental and theoretical research over the past several decades. The potential benefits of success are of truly epic proportions. Yet to date there is no noticeable penetration of active flow control into the realm of engineered products. The talk concerns one largely outstanding hurdle, i.e., the development of low order models suitable for the design of feedback loops for active fluid flow control.

Prof. Yannis Kevrekidis
Professor of Chemical and Biological Engineering and PACM - Princeton University
Coarse-graining the dynamics of (and on) complex networks

Thursday, Apr 3, 2014, 4:00pm to 5:00pm | Room 1-390

Complex, large scale networks often dynamically evolve in time. One can discriminate several different forms for such an evolution: (a) dynamics ON networks, when the connectivity of the network is fixed, but properties of the nodes evolve (e.g. concentrations in a complex biochemical reaction network); (b) dynamics OF networks, where the connectivity of the network itself is evolving – edges either form or disappear in time; finally (c) both properties of the nodes and existence/weights of edges evolve, giving us dynamics "of and on" networks, sometimes termed, adaptive network dynamics.

Professor of Physics - Brown University
Direct Statistical Simulation of Flows by Expansions in Cumulants

Thursday, March 13, 2014, 4:00pm to 5:00pm | Room 5-134

Low-order statistics of model geophysical and astrophysical fluids may be directly accessed by solving the equations of motion for the statistics themselves as proposed by Lorenz in 1967.  I implement such Direct Statistical Simulation by systematic expansion in equal-time cumulants. Live simulations are performed using a barotropic model on the sphere to illustrate the approach. The first cumulant is the zonally averaged vorticity as a function of latitude, and the second and higher cumulants encode information about nonlocal teleconnections. No assumptions of homogeneity or isotropy are imposed.  Closure of the equations of motion at second order (CE2) is realizable and retains the eddy -- mean-flow interactions, but neglects eddy-eddy interactions. Eddy-eddy interactions appear at third (CE3) order, but care must be taken to maintain realizability with a non-negative probability distribution function.

Prof. Oleg V. Gendelman
Faculty of Mechanical Engineering - Technion-Isreal Institute of Technology University
Exact solutions for Hamiltonian and forced/damped discrete breathers in vibro-impact chain
Tuesday, Feb 25, 2014, 12:00pm to 1:00pm | Room 5-314

Discrete breathers (DBs), or intrinsic localized modes (ILMs) are well-known in many mechanical and physical systems, including chains of mechanical oscillators, superconducting Josephson junctions, nonlinear magnetic metamaterials, electrical lattices, michromechanical cantilever arrays, antiferromagnets and  Bose – Einstein condensates. Generically, these response regimes appear due to interplay between discreetness of the system and its nonlinearity; therefore, analytic description of this sort of phenomena poses essential challenge.

Anastasios Matzavinos (Brown University) "A stochastic analysis of the motion of DNA nanomechanical bipeds"
Thursday, December 12, 2013, 4:00pm to 5:00pm | Room 5-234

Research in biological motors and recent advances in DNA nanofabrication technology have spurred a lot of interest in biomimetic nanomotor designs and DNA-based devices, such as nanomechanical switches and DNA templates for the growth of semiconductor nanocrystals, to name a few. Research activity in this area has been focused on designing and controlling dynamic DNA nanomachines that can be activated by and respond to specific chemical signals in their environment. In this talk, we formulate and analyze a Markov process modeling the motion of DNA nanomechanical walking devices. We consider a molecular biped restricted to a well-defined one-dimensional track and study its asymptotic behavior. Our main result is a functional central limit theorem for the biped with an explicit formula for the effective diffusion coefficient in terms of the parameters of the model. A law of large numbers and large deviation estimates are also obtained. Our approach is applicable to a variety of other biological motors such as myosin and motor proteins on polymer filaments.

This is joint work with Iddo Ben-Ari and Alexander Roitershtein.

Daniele Venturi (Brown University) "Convolutionless Nakajima-Zwanzig probability-density-function equations for stochastic analysis in large scale simulations"
Tuesday, December 3, 2013, 4:00pm to 5:00pm | Room 1-375

Determining the statistical properties of stochastic nonlinear systems is a problem of major interest in many areas of science and engineering. Even with recent theoretical and computational advancements, no broadly applicable technique has yet been developed for dealing with the challenging problems of high dimensionality, low regularity and random frequencies. In this talk we present a new framework for stochastic analysis in large scale simulations based on goal-oriented probability density function (PDF) methods. The key idea stems from techniques of irreversible statistical mechanics, and it relies on deriving evolution equations for the PDF of quantities of interest, e.g., functionals of the solution to systems of stochastic ordinary and partial differential equations. Such quantities could be low-dimensional objects in infinite dimensional phase spaces. We develop the goal-oriented PDF method in the context of the time-convolutionless Nakajima-Zwanzig-Mori formalism. We address the question of approximation of reduced-order density equations by multi-level coarse graining, perturbation series and operator cumulant resummation. Numerical examples are presented for stochastic resonance, stochastic advection-reaction and Burgers equations.

http://www.dam.brown.edu/people/venturi/Home.htm

Hessam Babaee (MIT) "Effect of Bifurcation in Dynamical Systems on Uncertainty Quantification of Jet in Crossflow"
Thursday, November 14, 2013, 4:00pm to 5:00pm | Room 5-234

In this study the effect of uncertainty of velocity ratio on jet in crossflow and particularly film cooling performance in gas turbines is studied. Direct numerical simulations using spectral/hp element have been combined with a stochastic collocation approach where the parametric space is discretized using Multi-Element general Polynomial Chaos (ME-gPC) method. Velocity ratio serves as a bifurcation parameter in a jet in a crossflow and the dynamical system is shown to have several bifurcations. As a result of the bifurcations, the target functional is observed to have low-regularity with respect to the parametric space. Due to the low-regularity of the response surface, ME-gPC is observed to be a computationally effective strategy to study the effect of uncertainty in a jet in a crossflow when velocity ratio is the random parameter.

Andrew J. Majda (NYU) "Data Driven Methods for Complex Turbulent Systems"
Thursday, October 31, 2013, 4:00pm to 5:00pm | Room 3-370

An important contemporary research topic is the development of physics constrained data driven methods for complex, large-dimensional turbulent systems such as the equations for climate change science. Three new approaches to various aspects of this topic are emphasized here: 1) the systematic development of physics constrained quadratic regression models with memory for low frequency components of complex systems; 2) Novel dynamic stochastic superresolution algorithms for real time filtering of turbulent systems; 3) New nonlinear Laplacian Spectral Analysis (NLSA) algorithms for large dimensional time series which capture both intermittency and low frequency variability unlike conventional EOF or principal component analysis. This is joint work with John Harlim (1, 2), Michal Branicki (2), and Dimitri Giannakis (3).