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, and 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 methods use waves to illuminate a penetrable object, and then forman image of its interior from measurements of transmitted or scattered waves. In such imaging problems efficient computation of wavefields in complex geometries is key. New mathematical methods and algorithms are needed to keep up with the demands of the imaging industry – advancements in the computer industry alone cannot respond to the shift towards larger domains, higher resolution, and larger data sets. 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 to a particular application? Wave equations are partial differential equations that interrelate the spatial and temporal variations of a particular 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 largescale 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 which 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 into 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 collected on a borehole axis. We demonstrate that the use of reduced-order models in a nonlinear optimization framework does indeed lead to significant computational savings without sacrificing the quality of imaging results. To illustrate the wide applicability of model-order reduction techniques in imaging, an additional example from nuclear geophysical imaging is also presented. @en

We present a reduced-order model (ROM) methodology for inverse scattering problems in which the ROMs are data-driven, i.e. they are constructed directly from data gathered by sensors. Moreover, the entries of the ROM contain localised information about the coefficients of the wave equation. We solve the inverse problem by embedding the ROM in physical space. Such an approach is also followed in the theory of ‘optimal grids,’ where the ROMs are interpreted as two-point finite-difference discretisations of an underlying set of equations of a first-order continuous system on this special grid. Here, we extend this line of work to wave equations and introduce a new embedding technique, which we call Krein embedding, since it is inspired by Krein’s seminal work on vibrations of a string. In this embedding approach, an adaptive grid and a set of medium parameters can be directly extracted from a ROM and we show that several limitations of optimal grid embeddings can be avoided. Furthermore, we show how Krein embedding is connected to classical optimal grid embedding and that convergence results for optimal grids can be extended to this novel embedding approach. Finally, we also briefly discuss Krein embedding for open domains, that is, semi-infinite domains that extend to infinity in one direction.@en

Determining the electromagnetic field response of photonic and plasmonic resonators is a formidable task in general. Field expansions in terms of quasi-normal modes (QNMs) are often used, since only a few of these modes are typically required for an accurate field description. We show that by exploiting the structure of Maxwell’s equations, conjugate-symmetric frequency-domain field expansions can be efficiently computed via a Lanczos-type algorithm. Dominant QNMs can be identified a posteriori with error control and without a priori mode selection. Discrete QNM approximations of resonating nanostructures are presented and the spontaneous decay rate of a quantum emitter is also considered.@en

In this paper we present a Krylov subspace model-order reduction technique for time- and frequency-domain electromagnetic wave fields in linear dispersive media. Starting point is a self-consistent first-order form of Maxwell's equations and the constitutive relation. This form is discretized on a standard staggered Yee grid, while the extension to infinity is modeled via a recently developed global complex scaling method. By applying this scaling method, the time- or frequency-domain electromagnetic wave field can be computed via a so-called stability-corrected wave function. Since this function cannot be computed directly due to the large order of the discretized Maxwell system matrix, Krylov subspace reduced-order models are constructed that approximate this wave function. We show that the system matrix exhibits a particular physics-based symmetry relation that allows us to efficiently construct the time- and frequency-domain reduced-order models via a Lanczos-type reduction algorithm. The frequency-domain models allow for frequency sweeps meaning that a single model provides field approximations for all frequencies of interest and dominant field modes can easily be determined as well. Numerical experiments for two- and three-dimensional configurations illustrate the performance of the proposed reduction method.@en

In this paper we present a Krylov subspace model-order reduction technique for time- and frequency-domain electromagnetic wave fields in linear dispersive media. Starting point is a self-consistent first-order form of Maxwell's equations and the constitutive relation. This form is discretized on a standard staggered Yee grid, while the extension to infinity is modeled via a recently developed global complex scaling method. By applying this scaling method, the time- or frequency-domain electromagnetic wave field can be computed via a so-called stability-corrected wave function. Since this function cannot be computed directly due to the large order of the discretized Maxwell system matrix, Krylov subspace reduced-order models are constructed that approximate this wave function. We show that the system matrix exhibits a particular physics-based symmetry relation that allows us to efficiently construct the time- and frequency-domain reduced-order models via a Lanczos-type reduction algorithm. The frequency-domain models allow for frequency sweeps meaning that a single model provides field approximations for all frequencies of interest and dominant field modes can easily be determined as well. Numerical experiments for two- and three-dimensional configurations illustrate the performance of the proposed reduction method.@en

Rational Krylov subspace (RKS) techniques are well-established and powerful tools for projection-based model reduction of time-invariant dynamic systems. For hyperbolic wavefield problems, such techniques perform well in configurations where only a few modes contribute to the field. RKS methods, however, are fundamentally limited by the Nyquist-Shannon sampling rate, making them unsuitable for the approximation of wavefields in configuration characterized by large travel times and propagation distances, since wavefield responses in such configurations are highly oscillatory in the frequency-domain. To overcome this limitation, we propose to precondition the RKSs by factoring out the rapidly varying frequency-domain field oscillations. The remaining amplitude-functions are generally slowly varying functions of source position and spatial coordinate and allow for a significant compression of the approximation subspace. Our one-dimensional analysis together with numerical experiments for large-scale two-dimensional acoustic models shows superior approximation properties of preconditioned RKS compared with the standard RKS model-order reduction. The preconditioned RKS results in a reduction of the frequency sampling well below the Nyquist-Shannon rate, a weak dependence of the RKS size on the number of inputs and outputs for multiple-input/multiple-output problems, and, most importantly, in a significant coarsening of the finite-difference grid used to generate the RKS. A prototype implementation indicates that the preconditioned RKS algorithm is competitive in the modern high performance computing environment.@en

Rational Krylov subspace (RKS) techniques are well-established and powerful tools for projection-based model reduction of time-invariant dynamic systems. For hyperbolic wavefield problems, such techniques perform well in configurations where only a few modes contribute to the field. RKS methods, however, are fundamentally limited by the Nyquist-Shannon sampling rate, making them unsuitable for the approximation of wavefields in configuration characterized by large travel times and propagation distances, since wavefield responses in such configurations are highly oscillatory in the frequency-domain. To overcome this limitation, we propose to precondition the RKSs by factoring out the rapidly varying frequency-domain field oscillations. The remaining amplitude-functions are generally slowly varying functions of source position and spatial coordinate and allow for a significant compression of the approximation subspace. Our one-dimensional analysis together with numerical experiments for large-scale two-dimensional acoustic models shows superior approximation properties of preconditioned RKS compared with the standard RKS model-order reduction. The preconditioned RKS results in a reduction of the frequency sampling well below the Nyquist-Shannon rate, a weak dependence of the RKS size on the number of inputs and outputs for multiple-input/multiple-output problems, and, most importantly, in a significant coarsening of the finite-difference grid used to generate the RKS. A prototype implementation indicates that the preconditioned RKS algorithm is competitive in the modern high performance computing environment.@en

In this paper we present a structure-preserving model-order reduction technique to efficiently compute electromagnetic wave fields on unbounded domains. As an approximation space, we take the span of the real and imaginary parts of frequency-domain solutions of Maxwell's equations. Reduced-order models for the electromagnetic field belong to this space and the expansion coefficients of these models are determined from a Galerkin condition. We show that the models constructed in this manner are structure-preserving and interpolate the electromagnetic field responses at the expansion frequencies. Moreover, for monostatic field responses (coinciding sources and receivers), the first-order derivative of a reduced-order model with respect to frequency interpolates this first-order derivative of the unreduced monostatic field response as well. A two-dimensional numerical example illustrates the performance of the proposed reduction method.@en

In this paper we present a structure-preserving model-order reduction technique to efficiently compute electromagnetic wave fields on unbounded domains. As an approximation space, we take the span of the real and imaginary parts of frequency-domain solutions of Maxwell's equations. Reduced-order models for the electromagnetic field belong to this space and the expansion coefficients of these models are determined from a Galerkin condition. We show that the models constructed in this manner are structure-preserving and interpolate the electromagnetic field responses at the expansion frequencies. Moreover, for monostatic field responses (coinciding sources and receivers), the first-order derivative of a reduced-order model with respect to frequency interpolates this first-order derivative of the unreduced monostatic field response as well. A two-dimensional numerical example illustrates the performance of the proposed reduction method.@en

In this paper we present a structure-preserving model-order reduction technique to efficiently compute electromagnetic wave fields on unbounded domains. As an approximation space, we take the span of the real and imaginary parts of frequency-domain solutions of Maxwell's equations. Reduced-order models for the electromagnetic field belong to this space and the expansion coefficients of these models are determined from a Galerkin condition. We show that the models constructed in this manner are structure-preserving and interpolate the electromagnetic field responses at the expansion frequencies. Moreover, for monostatic field responses (coinciding sources and receivers), the first-order derivative of a reduced-order model with respect to frequency interpolates this first-order derivative of the unreduced monostatic field response as well. A two-dimensional numerical example illustrates the performance of the proposed reduction method.@en

In this paper we present a structure-preserving model-order reduction technique to efficiently compute electromagnetic wave fields on unbounded domains. As an approximation space, we take the span of the real and imaginary parts of frequency-domain solutions of Maxwell's equations. Reduced-order models for the electromagnetic field belong to this space and the expansion coefficients of these models are determined from a Galerkin condition. We show that the models constructed in this manner are structure-preserving and interpolate the electromagnetic field responses at the expansion frequencies. Moreover, for monostatic field responses (coinciding sources and receivers), the first-order derivative of a reduced-order model with respect to frequency interpolates this first-order derivative of the unreduced monostatic field response as well. A two-dimensional numerical example illustrates the performance of the proposed reduction method.@en

We have developed several Krylov projection-based model-order reduction techniques to simulate electromagnetic wave propagation and diffusion in unbounded domains. Such techniques can be used to efficiently approximate transfer function field responses between a given set of sources and receivers and allow for fast and memory-efficient computation of Jacobians, thereby lowering the computational burden associated with inverse scattering problems. We found how general wavefield principles such as reciprocity, passivity, and the Schwarz reflection principle translate from the analytical to the numerical domain and developed polynomial, extended, and rational Krylov model-order reduction techniques that preserve these structures. Furthermore, we found that the symmetry of the Maxwell equations allows for projection onto polynomial and extended Krylov subspaces without saving a complete basis. In particular, short-term recurrence relations can be used to construct reduced-order models that are as memory efficient as time-stepping algorithms. In addition, we evaluated the differences between Krylov reduced-order methods for the full wave and diffusive Maxwell equations and we developed numerical examples to highlight the advantages and disadvantages of the discussed methods.@en

Optical resonators are widely used in modern photonics. Their spectral response and temporal dynamics are fundamentally driven by their natural resonances, the so-called quasinormal modes (QNMs), with complex frequencies. For optical resonators made of dispersive materials, the QNM computation requires solving a nonlinear eigenvalue problem. This raises a difficulty that is only scarcely documented in the literature. We review our recent efforts for implementing efficient and accurate QNM solvers for computing and normalizing the QNMs of micro- and nanoresonators made of highly dispersive materials. We benchmark several methods for three geometries, a two-dimensional plasmonic crystal, a two-dimensional metal grating, and a three-dimensional nanopatch antenna on a metal substrate, with the perspective to elaborate standards for the computation of resonance modes.@en

Optical resonators are widely used in modern photonics. Their spectral response and temporal dynamics are fundamentally driven by their natural resonances, the so-called quasinormal modes (QNMs), with complex frequencies. For optical resonators made of dispersive materials, the QNM computation requires solving a nonlinear eigenvalue problem. This raises a difficulty that is only scarcely documented in the literature. We review our recent efforts for implementing efficient and accurate QNM solvers for computing and normalizing the QNMs of micro- and nanoresonators made of highly dispersive materials. We benchmark several methods for three geometries, a two-dimensional plasmonic crystal, a two-dimensional metal grating, and a three-dimensional nanopatch antenna on a metal substrate, with the perspective to elaborate standards for the computation of resonance modes.@en

In this talk we present travel time or asymptotically corrected Krylov subspace methods to efficiently compute time- and frequency-domain wavefields in inhomogeneous structures. Fields characterized by large travel times can be effectively captured, by adding travel time information to parameter dependent Rational Krylov subspaces. This method provides reduced-order models of small order, which can be incorporated in wavefield inversion schemes. Numerical experiments will be presented that illustrate the performance of the method.@en

In this talk we present travel time or asymptotically corrected Krylov subspace methods to efficiently compute time- and frequency-domain wavefields in inhomogeneous structures. Fields characterized by large travel times can be effectively captured, by adding travel time information to parameter dependent Rational Krylov subspaces. This method provides reduced-order models of small order, which can be incorporated in wavefield inversion schemes. Numerical experiments will be presented that illustrate the performance of the method.@en

In this talk we present travel time or asymptotically corrected Krylov subspace methods to efficiently compute time- and frequency-domain wavefields in inhomogeneous structures. Fields characterized by large travel times can be effectively captured, by adding travel time information to parameter dependent Rational Krylov subspaces. This method provides reduced-order models of small order, which can be incorporated in wavefield inversion schemes. Numerical experiments will be presented that illustrate the performance of the method.@en

In this talk we present travel time or asymptotically corrected Krylov subspace methods to efficiently compute time- and frequency-domain wavefields in inhomogeneous structures. Fields characterized by large travel times can be effectively captured, by adding travel time information to parameter dependent Rational Krylov subspaces. This method provides reduced-order models of small order, which can be incorporated in wavefield inversion schemes. Numerical experiments will be presented that illustrate the performance of the method.@en