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8 - Further Applications
from Part Three - Complementary Topics
Jean Bernard Lasserre, Edouard Pauwels, Institut de Recherche en Informatique, Toulouse, Mihai Putinar, University of California, Santa Barbara
Book:

The Christoffel–Darboux Kernel for Data Analysis

Published online:

31 March 2022

Print publication:

07 April 2022, pp 115-124
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Summary

Outside the immediate statistical applications, two notable implications of the Christoffel-Darboux kernel are sketched: on the effective semialgebraic approximation of nonsmooth functions and on the spectral analysis of Koopman's operator attached to some intricate dynamical systems.

13 - Data-Driven Methods and Reduced-Order Modeling
from Part III - Least Squares and Optimization
Kevin W. Cassel, Illinois Institute of Technology
Book:

Matrix, Numerical, and Optimization Methods in Science and Engineering

Published online:

18 February 2021

Print publication:

04 March 2021, pp 642-698
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Summary

Reduced-order modeling is an active area of research by which simplified models of experimental or numerical data can be generated that are faithful to the behavior of the unerlying system.These methods are based on Galerkin projection, which is motivated by variational methods, or some other method of weighted residuals and allow for the projection of any governing differential equation onto an appropriate set of basis vectors or functions.These basis vectors or functions can be obtained using proper-orthogonal decomposition (POD) or one of its extensions or alternatives.Galerkin projection and POD are applied to continuous and discrete data sets.

Refined scales of weak-mixing dynamical systems: typical behaviour
Part of
SILAS L. CARVALHO, CÉSAR R. DE OLIVEIRA
Journal:

Ergodic Theory and Dynamical Systems / Volume 40 / Issue 12 / December 2020

Published online by Cambridge University Press:

26 June 2019, pp. 3296-3309

Print publication:

December 2020
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We study sets of measure-preserving transformations on Lebesgue spaces with continuous measures taking into account extreme scales of variations of weak mixing. It is shown that the generic dynamical behaviour depends on subsequences of time going to infinity. We also present corresponding generic sets of (probability) invariant measures with respect to topological shifts over finite alphabets and Axiom A diffeomorphisms over topologically mixing basic sets.