Research

Research Topics

Nowadays, many processes in engineering and natural sciences can be modeled, studied and optimized based on large collections of measured data. The goal is to use the available resources to devise reliable and e cient data-driven reduction methods.

1. Nonlinear systems described by quadratic terms

The motivation behind considering the reduction of quadratic bilinear (QB) systems, is that the dynamics of many classical nonlinear PDEs can be expressed in terms of such nonlinearities without any approximation error. Examples include the Chafee-Infante, FitzHugh-Nagumo, Burgers', Stokes or Navier-Stokes equations.

2. Hybrid and linear switched systems

Hybrid systems are a class of nonlinear systems which result from the interaction of contin-uous time dynamical subsystems with discrete events. Switched systems constitute a subclass of hybrid systems, in the sense that the discrete dynamics is simpli ed. A switched system is a dynamical system that consists of a nite number of subsystems and a logical rule that orchestrates switching between these subsystems. The motivation behind the study of hybrid and switched systems stems from the fact that they represent useful models for the design of distributed embedded systems where discrete controls are applied to continuous processes.

3. Gene oscillations

In collaboration with researchers from the Baylor College of Medicine in Houston, the issue of oscillations in (mouse) genes was tackled. The problem can be regarded as data-driven mod-eling where the data is given in the time-domain. The mathematical insights provided by the so-called pencil method, led to new biological insights. In particular it was shown that several genes exhibit 12h oscillations (in addition to 24h oscillations) and these two sets of oscillations turn out to be independent from each other. This leads to questions regarding the signi cance of the disruption of such oscillations for human disease.

4. Approximation of non-rational functions

The Loewner framework has also been applied to this problem (the to-be-approximated functions arise e.g. from linear PDEs describing a vibrating beam). Additionally, the approxi-mation quality of the proposed method has been compared with other approximation methods (the AAA algorithm, vector tting or IRKA - Iterative Rational Krylov Algorithm). The goal is to devise/improve methods that best exploit the information provided by data (measurements) which are able to describe the above-embedded minimal information in an e cient way, making use of the data-driven Loewner framework.

5. Other Topics

5.1 Connections between the Loewner framework and the behavioral approach of systems and control. 5.2 A posteriori error bounds in reduced order modeling. The main tool is the use of the dual system. 5.3 Addressing issues such as transfer functions of rectangular systems, stability, DAE struc-ture preservation.