Multivariate Rational Approximation in Action: From Data-driven Modeling to Nonlinear Eigenvalue Problems

In this dissertation, we develop data-driven reduced-order modeling and model order reduction techniques that rely on multivariate approximation methods as a fundamental tool. The investigated techniques aim to capture the complex behavior of physical systems based on given experimental measurements, simulation data, or the description of an underlying high-fidelity model. Many of the presented results are related to multivariate rational approximation methods, in particular the parametric adaptive Antoulas–Anderson (p-AAA) algorithm. The algorithm is inspired by the adaptive Antoulas-Anderson (AAA) algorithm for univariate rational approximation and combines rational interpolation and least-squares approximation into an effective framework. The key limitations of the p-AAA algorithm are its computational scalability and the requirement that the sampling data forms a grid. Two important contributions of this dissertation are tackling these limitations: First, we introduce a version of the p-AAA algorithm that improves scalability by incorporating low-rank tensor decompositions into the framework. Second, we generalize p-AAA to arbitrary data sets by deriving novel formulations of least-squares problems that incorporate interpolation constraints on scattered data sets. Several numerical experiments demonstrate the effectiveness of our proposed algorithms for reduced-order modeling problems and beyond. Further, novel theoretical results related to parametric nonlinear eigenvalue problems are presented. In particular, we propose an extension of the Keldysh decomposition for matrix-valued functions to the parametric setting. Key properties of the derived decomposition are established and illustrated via several examples. These theoretical developments motivate an effective new algorithm that tackles parametric nonlinear eigenvalue problems using multivariate rational interpolation methods and contour integration. As a final contribution of this dissertation, we introduce a model order reduction framework for linear systems with polynomial outputs. This framework is based on the controllability and observability energy functions of the underlying system. These are multivariate functions, and we discuss effective multivariate approximation algorithms that lead to a novel, scalable computational procedure. As for the other topics covered in this dissertation, numerical examples verify our theoretical developments.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2026

Author dc:creator

• Balicki, Linus

Chair dc:contributor.committeechair

• Gugercin, Serkan

Committee members dc:contributor.committeemember

• Beattie, Christopher A.
• Borggaard, Jeffrey T.
• Embree, Mark P.

• Rational approximation
• Model reduction
• Nonlinear eigenvalue problems
• System identification
• Tensors
• Low-rank tensor decompositions

Statement dc:rights

• In Copyright

Licence dc:rights.uri

• http://rightsstatements.org/vocab/InC/1.0/

Language dc:language.iso

en

Dc Identifier Other

vt_gsexam:45497

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/140927