PhD Thesis Defence

Model Reduction of Wave Equations

Jörn Zimmerling

How do you look inside a box without opening it? How can we know whether or not a heart valve is functioning correctly without cutting a person open?

Imaging – the art of seeing the unseeable. A CT-scan at the doctor’s office, crack detection in the wing of an airplane, or medical ultrasound are all examples of imaging techniques that allow us to inspect the interior of an object or person and enable us to observe features that are not directly visible to the naked eye. Science continuously improves upon existing imaging methods and occasionally invents new ones leading to improved image quality and faster image acquisition.

Many imaging applications rely on acoustic, electromagnetic, or elastodynamic waves for imaging. These waves illuminate a penetrable object and an image is formed of its interior from measurements of the transmitted or scattered waves. Being able to efficiently compute wavefields in complex geometries is key in such wavefield imaging problems. To keep up with the developments within the imaging industry to move to larger domains, higher resolution, and larger data sets, new mathematical methods and algorithms need to be developed, since advancements in the computer industry cannot keep up with these demands.

This thesis is about reduced-order modeling of the equations that describe the dynamics of wave propagation. In reduced-order modeling, the aim is to systematically develop a small model that describes a complex system without losing information that is valuable for a specific application. Evaluating such a model is computationally much more efficient than direct evaluation of the unreduced system and in the context of imaging it can lighten the computational burden associated with imaging algorithms. The central question is, of course: How does one construct a model that describes the wave dynamics relevant for a particular application?

Wave equations are partial differential equations that interrelate the spatial and temporal variations of some physical wavefield quantity. When we discretize such equations in space, sparse systems of equations with hundreds of thousands or even millions of unknowns are obtained. Via projection onto a small subspace such a large-scale system can be reduced to a much smaller reduced system. The solution of this small system is called a reduced-order model. A properly constructed reduced-order model can be easily evaluated and gives an accurate wavefield description over a certain time or frequency interval or parameter range of interest.

In this thesis, we discuss different choices for the subspaces that are used for projection in model-order reduction. In particular, we show what types of subspaces are effective for wavefields that are localized and highly resonant and how to efficiently generate such subspaces by exploiting certain symmetry properties of the wave equations. We illustrate the effectiveness of the resulting reduced-order models by computing optical wavefield responses in three-dimensional metallic nano-resonators.

Not all wavefields are determined by a few resonances, of course. Waves can also travel over long distances without losing information; a property that is used by mobile phones every day. The reduction methods developed for resonating fields are not efficient for these types of propagation problems and require a different approach. In this thesis, we present a so-called phase-preconditioning reduction method, in which a specific subspace is generated that explicitly takes the large travel times of the waves into account. We demonstrate the effectiveness of this reduction approach using examples from geophysics, where waves with long travel times are frequently encountered or used to probe the subsurface of the Earth.

Finally, we show how reduced-order modeling techniques can be incorporated in advanced nonlinear imaging algorithms. Here, we focus on an imaging application in geophysics, where the goal is to retrieve the conductivity tensor of a bounded anomaly located in the subsurface of the Earth, based on measured electromagnetic field data that is collected on a borehole axis. We demonstrate that the use of reduced-order models in a nonlinear optimization framework that attempts to solve this imaging problem indeed leads to significant computational savings without sacrificing the quality of the imaging results. To illustrate the wide applicability of model-order reduction techniques in imaging, an additional example from nuclear geophysical imaging is presented as well.

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