Page 371 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 371
TIONAL APPROXIMATION FOR DATA-DRIVEN MODELING AND COMPLEXITY
REDUCTION OF LINEAR AND NONLINEAR DYNAMICAL SYSTEMS (MS-69)
[3] D. C. Sorensen, M. Embree. A DEIM induced CUR factorization, SIAM Journal on Scien-
tific Computing, 38(3): A1454–A1482, 2016.
[4] S. Lefteriu, A. C. Antoulas. A New Approach to Modeling Multiport Systems From
Frequency-Domain Data, IEEE Transactions on Computer-Aided Design of Integrated Cir-
cuits and Systems, 29(1):14-27, Jan. 2010.
Input-tailored moment matching – a system-theoretic model reduction
method for nonlinear systems
Björn Liljegren Sailer, bjoern.sailler@uni-trier.de
Trier University, Germany
Coauthor: Nicole Marheineke
We suggest a new moment matching method for quadratic-bilinear dynamical systems. Most
system-theoretic model order reduction methods for nonlinear systems rely on multivariate fre-
quency representations derived from the Volterra series expansion of the solution. Similarly,
our approach relies on variational expansions, but we consider instead univariate frequency
representations tailored towards user-pre-defined families of inputs. Then moment matching
corresponds to a one-dimensional interpolation problem, not to multi-dimensional interpola-
tion as for the multivariate approaches, i.e., it also involves fewer interpolation frequencies to
be chosen. The resulting moment matching problems are approached exploiting the inherent
low-rank tensor structure.
In addition, our approach allows for the incorporation of more general input relations in
the state equations – not only affine-linear ones as in existing system-theoretic methods – in an
elegant way.
The use of rational approximation for linearization of models that are
nonlinear in the frequency
Karl Meerbergen, Karl.Meerbergen@kuleuven.be
KU Leuven, Belgium
Coauthors: Elke Deckers, Stijn Jonckheere
Finite element models for the analysis of vibrations typically have a quadratic dependency on
the frequency. This makes the finite element method suitable for eigenvalue computations and
time integration, by a formulation as a first order system, which we call a linearization.
The study of new damping materials often leads to nonlinear frequency dependencies, some-
times represented by a rational functions but, often, by truly nonlinear functions. In classical
analyses, vibrations are studied in the frequency domain. The nonlinear frequency dependency
is an issue for algorithms for fast frequency sweeping. In the context of numerical algorithms
for digital twins, time integration of mathematical models is required, which is not straightfor-
ward for models that are not linear or polynomial in the frequency.
We will discuss rational approximation and linearization of nonlinear frequency dependen-
cies and their use for fast frequency sweeping and time integration. In particular, we use the
AAA rational approximation and the associated linearization based on the barycentric Lagrange
formulation of rational functions, which is successfully used for solving nonlinear eigenvalue
369
REDUCTION OF LINEAR AND NONLINEAR DYNAMICAL SYSTEMS (MS-69)
[3] D. C. Sorensen, M. Embree. A DEIM induced CUR factorization, SIAM Journal on Scien-
tific Computing, 38(3): A1454–A1482, 2016.
[4] S. Lefteriu, A. C. Antoulas. A New Approach to Modeling Multiport Systems From
Frequency-Domain Data, IEEE Transactions on Computer-Aided Design of Integrated Cir-
cuits and Systems, 29(1):14-27, Jan. 2010.
Input-tailored moment matching – a system-theoretic model reduction
method for nonlinear systems
Björn Liljegren Sailer, bjoern.sailler@uni-trier.de
Trier University, Germany
Coauthor: Nicole Marheineke
We suggest a new moment matching method for quadratic-bilinear dynamical systems. Most
system-theoretic model order reduction methods for nonlinear systems rely on multivariate fre-
quency representations derived from the Volterra series expansion of the solution. Similarly,
our approach relies on variational expansions, but we consider instead univariate frequency
representations tailored towards user-pre-defined families of inputs. Then moment matching
corresponds to a one-dimensional interpolation problem, not to multi-dimensional interpola-
tion as for the multivariate approaches, i.e., it also involves fewer interpolation frequencies to
be chosen. The resulting moment matching problems are approached exploiting the inherent
low-rank tensor structure.
In addition, our approach allows for the incorporation of more general input relations in
the state equations – not only affine-linear ones as in existing system-theoretic methods – in an
elegant way.
The use of rational approximation for linearization of models that are
nonlinear in the frequency
Karl Meerbergen, Karl.Meerbergen@kuleuven.be
KU Leuven, Belgium
Coauthors: Elke Deckers, Stijn Jonckheere
Finite element models for the analysis of vibrations typically have a quadratic dependency on
the frequency. This makes the finite element method suitable for eigenvalue computations and
time integration, by a formulation as a first order system, which we call a linearization.
The study of new damping materials often leads to nonlinear frequency dependencies, some-
times represented by a rational functions but, often, by truly nonlinear functions. In classical
analyses, vibrations are studied in the frequency domain. The nonlinear frequency dependency
is an issue for algorithms for fast frequency sweeping. In the context of numerical algorithms
for digital twins, time integration of mathematical models is required, which is not straightfor-
ward for models that are not linear or polynomial in the frequency.
We will discuss rational approximation and linearization of nonlinear frequency dependen-
cies and their use for fast frequency sweeping and time integration. In particular, we use the
AAA rational approximation and the associated linearization based on the barycentric Lagrange
formulation of rational functions, which is successfully used for solving nonlinear eigenvalue
369
