george-hallerProf. 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.

AminProf. 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.

Past Seminars

MakisProf. 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].

Stuart Anderson 2015Prof. 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.

Rothos VProf. 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.


Petzold
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.

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

Computations in Riemannian space involving geodesics with their singularities and computing and visualizing singularities of dynamical systems using numerical methods based on differential geometry
Friday, April 8, 2016, 4:00pm to 5:00pm | Room 5-314
Download the Seminar Slides

This talk will start with a brief overview on current research projects pursued at the speaker´s lab, the Welfenlab. Thereafter, the seminar will focus on two ongoing research projects. The first project deals with computations on geodesics in Riemannian manifolds with an emphasis on singular situations caused by singularities of the geodesic exponential map. In this part of the seminar, I will present computations of focal sets, geodesic joins, medial curves and geodesic Voronoi diagrams. The second project employs some insights from the first one; in this part, I will present new differential geometry-based methods for numerical computations, analysis and visualization with respect to special singularities of dynamical systems governed by differential algebraic equations. In this context, we present an apparently new technique showing that an implicitly-defined manifold can be numerically parameterized.

EdDr. 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.


Andreas
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,Ioannis Kougioumtzoglou photo
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.

PredragProf. 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.

 

KaterinaDr 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.


HeyrimCho SNU
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.


giannakis
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.

MiroshnikovProf. 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.

manevitchProf. 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). 

farazmandDr. Mohammad Farazmand
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.

Agis"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.

Franz"Shape and Image Cognition, Construction and Compression via Tools from Differential Geometry"
Professor Franz-Erich Wolter, Welfenlab, Institute of Man-Machine-Communication
Leibniz University Hannover, Germany, http://welfenlab.de/en/index.html
Thursday, September 11, 2014, 12:00pm to 1:00pm | Room 33-116
Download the Seminar Slides

We shall describe how concepts from differential geometry have been providing powerful tools creating major advances in geometric modeling, geometry processing and image analysis dealing with the topics presented in the title of this address. This talk includes a retrospective compiling contributions of the author's works showing how concepts from local and global differential geometry have introduced new methods into geometric modeling and shape interrogation and classification finally ending with modern state of the art research on geometry processing and image processing. A major part of this seminar starting with works in the late nineties at the author's laboratory is dedicated to discussing how "efficient finger prints" useful for indexing and clustering digital data collections can be derived from spectra of Laplace operators being naturally associated with geometric objects such as surfaces and solids as well as (colored) images including medical 2d- and 3d-images. Recently the latter works obtained particular attention in the area of medical imaging. Geometric aspects of the Laplacian operator lead to generalizations of the Laplacian operating on line bundles allowing to compute eigen-functions whose iso-surfaces yield smooth non-orientable Seifert surfaces spanning knot complements in three-dimensional manifolds. Next we focus on cut loci, the medial axis and its inverse in Euclidean and Riemannian worlds.

romeoFrancesco 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.

raviDr. 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.

HandyDr 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.

tudmorProf. Gilead Tadmor
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.

yannisProf. 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.

MarstonProf. Brad Marston
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.

GendelmanProf. 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.

matzavinos2Anastasios 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.

VenturiDaniele 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

HessamHessam 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.

AndyAndrew 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).