Forthcoming events in this series
Abstract: We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control and random dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems -with a reasonable expectation of success - once the nonlinear system has been mapped into a high or infinite dimensional Reproducing Kernel Hilbert Space. In particular, we develop computable, non-parametric estimators approximating controllability and observability energy/Lyapunov functions for nonlinear systems, and study the ellipsoids they induce. It is then shown that the controllability energy estimator provides a key means for approximating the invariant measure of an ergodic, stochastically forced nonlinear system. We also apply this approach to the problem of model reduction of nonlinear control systems.
In all cases the relevant quantities are estimated from simulated or observed data. These results collectively argue that there is a reasonable passage from linear dynamical systems theory to a data-based nonlinear dynamical systems theory through reproducing kernel Hilbert spaces. This is a joint work with J. Bouvrie (MIT).
Inexact hardware trades reduced numerical precision against a reduction
in computational cost. A reduction of computational cost would allow
weather and climate simulations at higher resolution. In the first part
of this talk, I will introduce the concept of inexact hardware and
provide results that show the great potential for the use of inexact
hardware in weather and climate simulations. In the second part of this
talk, I will discuss how rounding errors can be assessed if the forecast
uncertainty and the chaotic behaviour of the atmosphere is acknowledged.
In the last part, I will argue that rounding errors do not necessarily
degrade numerical models, they can actually be beneficial. This
conclusion will be based on simulations with a model of the
one-dimensional Burgers' equation.
The Preisach model of hysteresis has a long history, a convenient algorithmic form and "nice" mathematical properties (for a given value of nice) that make it suitable for use in differential equations and other dynamical systems. The difficulty lies in the fact that the "parameter" for the Preisach model is infinite dimensional—in full generality it is a measure on the half-plane. Applications of the Preisach model (two interesting examples are magnetostrictive materials and vadose zone hydrology) require methods to specify a measure based on experimental or observed data. Current approaches largely rely on direct measurements of experimental samples, however in some cases these might not be sufficient or direct measurements may not be practical. I will present the Preisach model in all its glory, along with some history and applications, and introduce an open inverse problem of fiendish difficulty.
In this talk we aim to filter the Majda-McLaughlin-Tabak(MMT) model, which is a one-dimensional prototypical turbulence system. Due to its inherent high dimensionality, we first try to find a low dimensional dynamical system whose statistical property is similar to the original complexity system. This dimensional reduction, called stochastic parametrization, is clearly well-known method but the value of current work lies in the derivation of an analytic closure for the parameters. We then discuss the necessity of the accurate filtering algorithm for this effective dynamics, and introduce the particle filter using the cubature on Wiener space and the recombination skill.
This will be a discussion on Stochastic Parameterisation, led by Hannah.
The background for the multitarget tracking problem is presented
along with a new framework for solution using the theory of random
finite sets. A range of applications are presented including
submarine tracking with active SONAR, classifying underwater entities
from audio signals and extracting cell trajectories from biological
data.
This work aims to present a new approach called Flow Curvature Method
that applies Differential Geometry to Dynamical Systems. Hence, for a
trajectory curve, an integral of any n-dimensional dynamical system
as a curve in Euclidean n-space, the curvature of the trajectory or
the flow may be analytically computed. Then, the location of the
points where the curvature of the flow vanishes defines a manifold
called flow curvature manifold. Such a manifold being defined from
the time derivatives of the velocity vector field, contains
information about the dynamics of the system, hence identifying the
main features of the system such as fixed points and their stability,
local bifurcations of co-dimension one, centre manifold equation,
normal forms, linear invariant manifolds (straight lines, planes,
hyperplanes).
In the case of singularly perturbed systems or slow-fast dynamical
systems, the flow curvature manifold directly provides the slow
invariant manifold analytical equation associated with such systems.
Also, starting from the flow curvature manifold, it will be
demonstrated how to find again the corresponding dynamical system,
thus solving the inverse problem.
Moreover, the concept of curvature of trajectory curves applied to
classical dynamical systems such as Lorenz and Rossler models
enabled to highlight one-dimensional invariant sets, i.e. curves
connecting fixed points which are zero-dimensional invariant sets.
Such "connecting curves" provide information about the structure of
the attractors and may be interpreted as the skeleton of these
attractors. Many examples are given in dimension three and more.