[:is] Math colloquium
Fyrirlesari: Peter Giesl, University of Sussex, UK
Titill: Existence and construction of a contraction metric as solution of a matrix-valued PDE.
Staðsetning: Via Zoom. Link to be sent.
Tími: Föstudag 2.október kl.10:00
Ágrip:
A contraction metric is a Riemannian metric, with respect to which the distance between adjacent solutions of an ordinary differential equation (ODE) decreases.
A contraction metric can be used to prove existence and uniqueness of an equilibrium of an autonomous ODE and determine a subset of its basin of attraction without requiring information about its location. Moreover, a contraction metric is robust to small perturbations of the system.
We will prove a converse theorem, showing the existence of a contraction metric for an equilibrium by characterising it as a matrix-valued solution of a certain linear partial differential equation (PDE). This leads to a construction method by numerically solving the matrix-valued PDE using mesh-free collocation. We use and present a recent extension of mesh-free collocation of scalar-valued functions, solving linear PDEs, to matrix-valued ones. Finally, we briefly discuss a method to verify that the computed metric satisfies the conditions of a contraction metric.
This is partly work with Holger Wendland, Bayreuth as well as Sigurdur Hafstein and Iman Mehrabinezhad, Iceland.[:en]
Math colloquium
Speakers: Peter Giesl, University of Sussex, UK
Title: Existence and construction of a contraction metric as solution of a matrix-valued PDE.
Room: Via Zoom. Link to be sent.
Time: Friday 2nd October, 10:00am
Abstract:
A contraction metric is a Riemannian metric, with respect to which the distance between adjacent solutions of an ordinary differential equation (ODE) decreases.
A contraction metric can be used to prove existence and uniqueness of an equilibrium of an autonomous ODE and determine a subset of its basin of attraction without requiring information about its location. Moreover, a contraction metric is robust to small perturbations of the system.
We will prove a converse theorem, showing the existence of a contraction metric for an equilibrium by characterising it as a matrix-valued solution of a certain linear partial differential equation (PDE). This leads to a construction method by numerically solving the matrix-valued PDE using mesh-free collocation. We use and present a recent extension of mesh-free collocation of scalar-valued functions, solving linear PDEs, to matrix-valued ones. Finally, we briefly discuss a method to verify that the computed metric satisfies the conditions of a contraction metric.
This is partly work with Holger Wendland, Bayreuth as well as Sigurdur Hafstein and Iman Mehrabinezhad, Iceland.[:]