Data-Driven System Reduction and Identification
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 efficient data-driven reduction methods.
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. For the book press:
Hybrid systems are a class of nonlinear systems which result from the interaction of continuous time dynamical subsystems with discrete events. Switched systems constitute a subclass of hybrid systems, in the sense that the discrete dynamics is simplified. A switched system is a dynamical system that consists of a finite 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.
In collaboration with researchers from the Baylor College of Medicine in Houston, the issue of oscillations in (mouse) genes was studied. The problem can be regarded as data-driven modeling 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 significance of the disruption of such oscillations for human disease. Access to the paper press:
The Loewner framework has also been applied to approximating non-rational functions that arise from linear PDEs, e.g., describing a vibrating beam. Additionally, the approximation quality of the proposed method has been compared with other approximation methods such as the AAA (Adaptive-Antoulas-Anderson) algorithm, vector fitting or IRKA ( Iterative Rational Krylov Algorithm). The goal is to devise and improve methods that best exploit the information provided by data (measurements). This can be used to describe the above embedded minimal information in an efficient way, making use of the data-driven Loewner framework.
1. Connections between the Loewner framework and the behavioral approach of systems and control.
2. A posteriori error bounds in reduced-order modeling for which the main tool is the use of the dual system.
3. Addressing issues such as transfer functions of rectangular systems, stability, and DAE structure preservation.
4. Research toward discovering how worms implement control mechanism. For access to the presentation press: