Model-driven reduced-order models

In model-driven reduced-order modeling one tries to find a small surrogate model that accurately describes the behavior of a large-scale dynamical system. Construction of such a surrogate model from a known large-scale dynamical system is model-driven. Preserving properties (e.g. passivity, symmetry etc.) is a key aspect of model-driven reduced-order modeling.

In my research the large-scale dynamical system is typically some wave or diffusion equation and the reduced-order model can be used as a solver for the PDE, a surrogate model during design optimization or to accelerate algorithms for imaging.

Applications include for instance:

• Design optimization of optical resonators
• Surrogate models for fast well logging
• Forward solvers for ground penetrating radars

Data-driven reduced-order models

In data-driven reduced-order modelling one tries to identify a mechanical system that explains measured data. One of the challenges is to incorporate prior information about the system (like passivity or the underlying structure) into the reduced-order model. In the context of inverse problems, we try to construct reduced order-models that have the same sparsity structure as a discretization of the underlying partial-differential equation.

Inverse Problems

In inverse problems one tries to reconstruct the coefficients of a PDE from observations of the PDE variable. We encounter inverse problems in a wide variety of everyday applications like: medical imaging, nondestructive testing, radar, or geophysical exploration. In short it is the science of figuring out what is in a box without opening the box.

Inverse Problems via ROM embedding

A reduced-order model that explains the measured data and has the right structure can be interpreted as a coarse discretization of the underlying PDE. Thus, in the context of inverse problems, a carefully constructed ROM can be embedded back into the physical space and the coefficients of this ROM-PDE discretization can be interpreted as averages of the coefficients of the underlying PDE.

Resonance computation and expansions

The sound of a music instrument is mainly determined by the acoustical resonances it supports, and the frequencies it is excited at. In engineering and science wave propagation in resonating structures is often of interest. Computation of resonances can be a formidable task, if the resonator is "open", i.e. waves can leave the resonator and propagate outwards. However, efficient algorithms can be designed for such configurations using symmetries in the underlying equations. Furthermore, just like in the case for a music instrument, the response of such a system can (sometimes) be efficiently described by those resonances, leading to low rank resonance expansions.

Resonances in nano-photonics

One field of research were this idea resonances is nano-photonics. At optical frequencies many noble metals are dispersive and the electrical permittivity can become negative in a frequency-band, which is the foundation of plasmonics. Many devices in for instance biosensing have been proposed using plasmonic resonators.

We develop efficient solvers for field responses and resonance modes of such dispersive resonators.