Page 369 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 369
TIONAL APPROXIMATION FOR DATA-DRIVEN MODELING AND COMPLEXITY
REDUCTION OF LINEAR AND NONLINEAR DYNAMICAL SYSTEMS (MS-69)
terms of the spectral data of the associated Koopman operator. The main tasks of the analysis
are approximations of eigenfunctions and eigenvalues of the operator, and computation of a
spatio-temporal representation of the data snapshots, based on the computed eigenvalues and
eigenfunctions. The latter includes possibly ill-conditioned least squares problem.
Exceptional performances motivated developments of several modifications that make the
DMD an attractive method for identification, analysis and model reduction of nonlinear systems
in data driven settings.
In this talk, we will present our recent results on the numerical aspects of the DMD/Koopman
analysis. We show how the state of the art numerical linear algebra can be deployed to improve
the numerical performances in cases that are usually considered notoriously ill-conditioned.
The numerical framework is based on algorithms that also apply e.g. to matrix rational approx-
imations in modeling by Vector Fitting.
Barycentric Hermite interpolation and its application to data-driven
model reduction
Ion Victor Gosea, gosea@mpi-magdeburg.mpg.de
Max Planck Institute for Dynamics of Complex Technical Systems, Germany
The barycentric representation of rational interpolants offers some advantages over other clas-
sical rational formulations, one of which being the numerical stability of the barycentric for-
mula. Here, we concentrate on the Hermite interpolation problem, for which not only measure-
ments of the underlying function are available, but also of the function’s derivatives. We revisit
two model reduction algorithms based on rational approximation, i.e., the Loewner framework
(Mayo/Antoulas ’07) and IRKA/TF-IRKA (Gugercin/Antoulas /Beattie ’08, Beattie/Gugercin
’12); we show how Hermitian interpolation is connected to these methods. Moreover, we
present an extension of the recent AAA algorithm (Nakatsukasa/Sete/Trefethen ’18) that is
adapted to satisfy Hermitian conditions (interpolating not only functions values, as the original
AAA, but also values of derivatives). The new variant also uses least-squares fitting on the data
set to construct a rational interpolant in barycentric representation.
Algorithms for identification and reduction of nonlinear dynamical
systems from time-domain data
Dimitrios S. Karachalios, karachalios@mpi-magdeburg.mpg.de
Max-Planck Institute - Magdeburg, Germany
Coauthors: Ion Victor Gosea, Athanasios C. Antoulas
We propose a comparison of algorithms that use time-domain data to fit models of dynamical
systems that explain the available measurements. The eigensystem realization algorithm (ERA)
and the Loewner framework (LF) constitute data-driven methods for identifying and reduc-
ing linear and nonlinear dynamical systems. The ERA uses the system’s invariants known as
Markov parameters into a matrix of Hankel structure. Similarly, the LF encodes the invariants
of a system into the Loewner matrix structure through frequency-domain measurements. These
can be estimated from time-domain data by employing spectral transforms. For both methods
under consideration, the singular value decomposition (SVD) provides a trade-off between the
367
REDUCTION OF LINEAR AND NONLINEAR DYNAMICAL SYSTEMS (MS-69)
terms of the spectral data of the associated Koopman operator. The main tasks of the analysis
are approximations of eigenfunctions and eigenvalues of the operator, and computation of a
spatio-temporal representation of the data snapshots, based on the computed eigenvalues and
eigenfunctions. The latter includes possibly ill-conditioned least squares problem.
Exceptional performances motivated developments of several modifications that make the
DMD an attractive method for identification, analysis and model reduction of nonlinear systems
in data driven settings.
In this talk, we will present our recent results on the numerical aspects of the DMD/Koopman
analysis. We show how the state of the art numerical linear algebra can be deployed to improve
the numerical performances in cases that are usually considered notoriously ill-conditioned.
The numerical framework is based on algorithms that also apply e.g. to matrix rational approx-
imations in modeling by Vector Fitting.
Barycentric Hermite interpolation and its application to data-driven
model reduction
Ion Victor Gosea, gosea@mpi-magdeburg.mpg.de
Max Planck Institute for Dynamics of Complex Technical Systems, Germany
The barycentric representation of rational interpolants offers some advantages over other clas-
sical rational formulations, one of which being the numerical stability of the barycentric for-
mula. Here, we concentrate on the Hermite interpolation problem, for which not only measure-
ments of the underlying function are available, but also of the function’s derivatives. We revisit
two model reduction algorithms based on rational approximation, i.e., the Loewner framework
(Mayo/Antoulas ’07) and IRKA/TF-IRKA (Gugercin/Antoulas /Beattie ’08, Beattie/Gugercin
’12); we show how Hermitian interpolation is connected to these methods. Moreover, we
present an extension of the recent AAA algorithm (Nakatsukasa/Sete/Trefethen ’18) that is
adapted to satisfy Hermitian conditions (interpolating not only functions values, as the original
AAA, but also values of derivatives). The new variant also uses least-squares fitting on the data
set to construct a rational interpolant in barycentric representation.
Algorithms for identification and reduction of nonlinear dynamical
systems from time-domain data
Dimitrios S. Karachalios, karachalios@mpi-magdeburg.mpg.de
Max-Planck Institute - Magdeburg, Germany
Coauthors: Ion Victor Gosea, Athanasios C. Antoulas
We propose a comparison of algorithms that use time-domain data to fit models of dynamical
systems that explain the available measurements. The eigensystem realization algorithm (ERA)
and the Loewner framework (LF) constitute data-driven methods for identifying and reduc-
ing linear and nonlinear dynamical systems. The ERA uses the system’s invariants known as
Markov parameters into a matrix of Hankel structure. Similarly, the LF encodes the invariants
of a system into the Loewner matrix structure through frequency-domain measurements. These
can be estimated from time-domain data by employing spectral transforms. For both methods
under consideration, the singular value decomposition (SVD) provides a trade-off between the
367
