Wavelet-based Dynamic Mode Decomposition in the Context of Extended Dynamic Mode Decomposition and Koopman Theory

Koopman theory is widely used for data-driven modeling of nonlinear dynamical systems. One of the well-known algorithms that stem from this approach is the Extended Dynamic Mode Decomposition (EDMD), a data-driven algorithm for uncontrolled systems. In this thesis, we will start by discussing the EDMD algorithm. We will discuss how this algorithm encompasses Dynamic Mode Decomposition (DMD), a widely used data-driven algorithm. Then we will extend our discussion to input-output systems and identify ways to extend the Koopman Operator Theory to input-output systems. We will also discuss how various algorithms can be identified as instances of this framework. Special care is given to Wavelet-based Dynamic Mode Decomposition (WDMD). WDMD is a variant of DMD that uses only the input and output data. WDMD does that by generating auxiliary states acquired from the Wavelet transform. We will show how the action of the Koopman operator can be simplified by using the Wavelet transform and how the WDMD algorithm can be motivated by this representation. We will also introduce a slight modification to WDMD that makes it more robust to noise.

Name thesis:degree_name

Master of Science

Level thesis:degree_level

masters

Discipline thesis:degree_discipline

Mathematics

Department dc:contributor.department

Mathematics

Grantor dc:publisher

Virginia Tech

Year dc:date.issued

2024

Author dc:creator

• Tilki, Cankat

Chair dc:contributor.committeechair

• Gugercin, Serkan

Committee members dc:contributor.committeemember

• Beattie, Christopher A.
• Embree, Mark P.
• Miedlar, Agnieszka

• Wavelets
• Data Driven Modeling
• Dynamic Mode Decomposition
• Koopman Theory

Statement dc:rights

• In Copyright

Licence dc:rights.uri

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

Handle dc:identifier.uri

https://hdl.handle.net/10919/120888

OAI identifier oai:identifier

oai:vtechworks.lib.vt.edu:10919/120888