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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5G Phased Arrays

International Summer School on 5G Phased Arrays

Understanding of phased array operation requires multi- disciplinary approach, which is based on the antenna array, microwave circuit and signal processing theories. By bringing these three areas together, the school provides integral approach to phased array front-ends for 5G communication systems.

At the school the phased array foundations will be considered from antenna, RF technology and signal processing points of view. Realization of 5G capabilities such as high data-rate communication link to moving objects will be discussed. The education will be concluded by a design project.

The summer school is open for all young specialists and researchers from both industry and academia. The attendees should have basic knowledge about EM, electrical circuits and signal processing (graduate courses on electromagnetic waves, electrical circuits including microwave (RF) circuits, and signal processing).


  • Foundations of antenna arrays
  • Antenna array topologies for 5G applications
  • Analog and digital beamforming in antenna arrays
  • Front-end architecture and performance
  • 5G applications and system requirements

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PRORISC 2018 Conference

Annual conference on Integrated Circuit (IC) design, organized within the three technical Dutch universities Twente, Delft and Eindhoven

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SAFE 2018 Conference

Annual conference on Micro-systems, Materials, Technology and RF-devices, organized within the three technical Dutch universities of Twente, Delft and Eindhoven.

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MSc SS Thesis Presentation

Direction-of-Arrival Estimation using an Unsynchronized Array of Acoustic Vector Sensors

Bart Coonen

Direction-Of-Arrival (DOA) estimation of acoustic signals is of great interest in various applications including battlefield acoustic and noise localization. Acoustic sensors are employed in an array configuration to estimate DOAs based on the time differences of arrival DOAs. However, the acoustic sensors in the network have all their own Data AcQuisition (DAQ) unit with independent clocks than, it might not be possible to perfectly synchronize the network which affects the performance of the time differences of arrival reliably.  

In this thesis  we consider the issue of clock synchronization errors in a network where Acoustic Vector Sensors (AVSs) are used. AVSs are shown to be advantageous in terms of direction finding compared to conventional Acoustic Pressure Sensors (APSs) due to their directional particle velocity measurement capability. Initiallity the measurement model for AVSs is presented. After that the behavior of the clocks is incorporated in the measurement model of the full array setup. Subsequently, the effects of the clocks on the MVDR DOA estimation method is discussed.

  The model with clock errors is used in the development of three new DOA-estimation methods. The first two techniques are eigenstructure methods that are capable of finding the DOAs regardless of the accuracy of the synchronization. However, to find the DOAs with high accuracy in a real-time application these methods are not due to their high computational cost. Alternatively, the third proposed algorithm takes the DOA estimate from previous methods with low accuracy as its input. The algorithm estimates the DOA in an iterative fashion with high accuracy based on these estimates with low accuracy.  

Finally, measurements are conducted in a controlled environment in order to show that these methods are usable in practical situations.

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Signal Processing Seminar

Compressive Covariance Sensing

Geert Leus

There are many engineering applications that rely on frequency or angular spectrum sensing, such as cognitive radio, radio astronomy, radar, seismic acquisition, and so on. Many of these applications do not require the reconstruction of the full signal, and can perfectly rely on an estimate of the power spectral density (PSD), or in other words, the second-order statistics of the signal. However, the large bandwidths of the involved signals lead to high sampling rates and thus high sampling costs, which can be prevented by a direct compression step carried out in the analog domain (e.g., by means of an analog-to-information converter, multi-coset sampling, analog beamforming, antenna selection, etc.). This leads to the problem of sensing the PSD or covariance using compressive observations, labeled as compressive covariance sensing (CCS). In this tutorial we will give an overview of the state-of-the-art in CCS and present its connections to compressive sensing (CS). We focus on the design constraints of the compression matrices, which are completely different as in classical CS, and elaborate on the estimation/detection techniques to sense the covariance using compressive measurements. In this context, both non-uniform and random sampling are discussed. We further elaborate on distributed CCS, where compressive measurements in one domain are fused in the dual domain, i.e., temporal compressive measurements are gathered at different spatial sensors or spatial compressive measurements from different time slots are combined. Finally, connections to super resolution techniques such as atomic norm minimization are discussed. We end this tutorial by sketching some open issues and presenting the concluding remarks.

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