NumerikHead of Research Unit Prof. Dr.Marcus J. GroteOverviewMembersPublicationsProjects & CollaborationsProjects & Collaborations OverviewMembersPublicationsProjects & Collaborations Projects & Collaborations 14 foundShow per page10 10 20 50 SPACE-TIME ADAPTIVE EXPLICIT TIME-STEPPING FOR WAVE PROPAGATION Research Project | 1 Project MembersNo Description available Local Time-Stepping Methods: Stability, Convergence and Advanced Applications Research Project | 4 Project MembersDie effiziente und detailtreue Computersimulation von Wellenphänomenen ist zentral bei vielen Anwendungen wie z.B. der medizinischen Bildgebung, der Seismik, dem Mobilfunk oder auch der zerstörungsfreien Materialprüfung. Numerische Verfahren zur Simulation von Wellenphänomenen basieren auf einer Raum- und Zeitdiskretisierung der Wellengleichung. Für die räumliche Diskretisierung sind finite Elemente Verfahren bestens geeignet, besonders weil sie auch in beliebig komplizierten Geometrien einsetzbar sind. Für die zeitliche Diskretisierung sind explizite Zeitschrittverfahren besonders effizient und lassen sich auch gut auf parallelen Hochleistungsrechnern einsetzen. Dabei benutzen Standardverfahren den gleichen Zeitschritt im gesamten Rechengebiet, der wegen der CFL Stabilitätsbedingung durch die kleinste Maschenweite bestimmt ist. Dies ist bei lokal verfeinerten Gittern höchst ineffizient. Um diese CFL Stabilitätsschranke zu umgehen, wurden in den letzten Jahren verschiedene lokale Zeitschrittverfahren (Local Time Stepping = LTS) entwickelt, die in den kleineren Elementen kleinere Zeitschritte und in den grösseren Elementen grössere Zeitschritte erlauben. Auch wenn lokale Zeitschrittverfahren sich in der Praxis schon bewährt haben, gibt es bisher noch keine allgemeine Konvergenztheorie. Bei der Zeitintegration der Wellengleichung unterscheidet man i.A. zwei Klassen von expliziten Zeitschrittverfahren: Leap-Frog (LF) Verfahren, die auf der klassischen Formulierung 2. Ordnung basieren, und Runge-Kutta (RK) Verfahren, die auf der Umformulierung als System 1. Ordnung basieren. Für LF Verfahren konnten wir 2019 zum ersten Mal optimale Konvergenzraten für ein auf LF basierendes LTS Verfahren beweisen, jedoch nicht unter einer optimalen (CFL) Stabilitätsschranke. Es geht nun darum, das LTS-LF Verfahren leicht zu modifizieren, damit optimale Konvergenzraten auch unter einer optimalen CFL Bedingung bewiesen werden können. Ausserdem möchten wir versuchen, diese Konvergenztheorie auch auf RK basierende LTS Verfahren zu erweitern. Ziel dieses Projekts ist es deshalb, einerseits eine rigorose Konvergenztheorie für explizite lokale Zeitschritte zu etablieren und andererseits deren Nutzen sowohl für Simulationen bei Unsicherheiten in der Wellengeschwindigkeit als auch für zeitharmonische Simulationen zu demonstrieren. Adaptive Spectral Decompositions for Inverse Medium Problems Research Project | 3 Project MembersInverse medium problems involve the reconstruction of a spatially varying unknown medium from available observations by exploring a restricted search space of possible solutions. Standard grid-based representations are very general but all too often computationally prohibitive due to the high dimension of the search space. Adaptive spectral decompositions instead expand the unknown medium in a basis of eigenfunctions of a judicious elliptic operator, which depends itself on the medium. We combine the AS decomposition with standard inexact Newton-type methods for the solution of time-harmonic and time-dependent wave scattering problems. By repeatedly adapting both the eigenfunction basis and its dimension, the resulting adaptive spec- tral inversion (ASI) method substantially reduces the dimension of the search space during the nonlinear optimization. Rigorous estimates of the AS decomposition are proved for a general piecewise constant medium. Numerical results illustrate the accuracy and efficiency of the ASI method for time-harmonic inverse scattering problems, including realistic subsurface models from geophysics. A Gauss-Jacobi Kernel Compression Scheme for Fractional Differential Equations Research Project | 1 Project MembersA scheme for approximating the kernel $w$ of the fractional $a$-integral by a linear combination of exponentials is proposed and studied. The scheme is based on the application of a composite Gauss-Jacobi quadrature rule to an integral representation of $w$. This results in an approximation of $w$ in an interval $[del,T]$, with $00$, and $delin(0,T)$. Local Time-Stepping Methods for Wave Propagation Research Project | 3 Project MembersWave phenomena are ubiquitous across science, technology and medicine. Typical applications in- clude ultrasound imaging, wireless communications and seismic tomography. In this proposal we shall analyze and further develop time integration methods for the numerical simulation of acoustic, electromagnetic or elastic wave phenomena. For the spatial discretization, we use either conforming finite elements or discontinuous Galerkin methods, which accomodate arbitrary meshes and geometry. For the time discretization, we consider recently derived local time-stepping (LTS) methods, which overcome the bottleneck due to local mesh refinement by taking smaller time-steps precisely where the smallest elements are located. Explicit LTS methods have already proved useful in many applications and shown nearly optimal speed-up on HPC architectures. Convergence (in the ODE sense) to the semi-discrete solution on a fixed mesh is fairly standard. However, a general convergence theory in the PDE sense, which establishes convergence to the (true) continuous solution as both the time-step and the mesh-size tend to zero, is still lacking. This proposal therefore aims at establishing a rigourous convergence theory for explicit LTS meth- ods, which fall into two separate categories. Hence this proposal consists of two separate projects. In the first project, we shall prove optimal space-time convergence of LTS methods based on energy conserving leap-frog (LF) methods. Moreover, we shall compare the accuracy of different fourth-order LTS methods in particular for long-time simulations. In the second project, we shall derive a complete space-time convergence theory for Runge-Kutta (RK) based LTS methods. Moreover, we shall demonstrate the usefulness of LTS methods also in time-harmonic regimes, when the controllability method is used for the solution of the Helmholtz equation. Advanced Methods in Computational Electromagnetics Research Project | 3 Project MembersMathematisch werden die verschiedensten Wellenphänomene im Grunde durch die klassische Wellengleichung und die Maxwell-Gleichungen beschrieben. Da sich deren Lösung nur in den einfachsten "akademischen" Fällen mit Bleistift und Papier bestimmen lässt, greift man in der Praxis auf Methoden der Numerischen Mathematik zurück, mit denen sich näherungsweise Wellenphänomene auch in kompliziertesten Situationen auf dem Computer simulieren lassen. Um eine möglichst detailtreue Darstellung des Phänomens zu erreichen, sind eine hohe Auflösung und ein riesiger Lösungsraum erforderlich, die nur mit modernsten numerischen Methoden und den neusten Hochleistungsrechnern zu bewältigen sind. In diesem Projekt werden wir neuartige numerische Verfahren zur Ort- und Zeitdiskretisierung obiger Gleichungen entwickeln. Für die Ortdiskretisierung werden wir auf diskontinuierliche Galerkin (DG) Finite Elemente (FE) Methoden zurückgreifen, die besonders flexibel und effizient sind und teils in früheren Projekten entwickelt wurden. Für die Zeitdiskretisierung werden wir neue explizite lokale Zeitschrittverfahren entwickeln, die auf Runge-Kutta Methoden basieren und es ermöglichen, die CFL-Stabilitätsgrenze zu umgehen. Dadurch werden bedeutend grössere Zeitschritte möglich, die wiederum zu schnelleren Computersimulationen führen. Meistens ist nicht nur die direkte Simulation (die Lösung des Vorwärtsproblems) gewünscht, sondern auch deren Anwendung bei der Lösung des inversen Problems, um z.B. anhand von Messdaten ein unbekanntes Medium zerstörungsfrei sichtbar zu machen (z.B. Ultraschall in der Medizin). Falls mehrere Hindernisse zur Überlagerung der Streufelder führen, wird es notwendig, die verschieden gestreuten Wellen von einander zu trennen. Dafür werden wir ein neues Verfahren entwickeln und erproben, das auf der Time Reversed Absorbing Conditions (TRAC) Methode basiert. Advanced Methods for Computational Electromagnetics Research Project | 1 Project MembersComputational electromagnetics (CEM) presents a number of challenges. First, electromagnetic fields tend to be highly oscillatory and wave-dominated ; moreover, they can develop singularities at material interfaces and boundaries. Second, the phenomena of interest typically involve complicated geometries, inhomogeneous media, and even nonlinear materials. Third, the underlying partial differential equations often need to be solved in an unbounded domain, which needs to be truncated by an artificial boundary to confine the region of interest to a finite computational domain. Thus, we seek efficient, reliable and inherently parallel computational methods which handle - complex geometry, corner singularities and material interfaces, - highly varying material properties, - wave-dominated behavior, - unbounded domains. To fully address the wide range of difficulties inherent to Maxwell's equations, great flexibility is needed from any computational approach. In continuation of our previous work, we shall pursue the development of novel explicit local time-stepping (LTS) methods and combine them with discon- tinuous Galerkin (DG) discretizations to achieve the required flexibility. These DG discretization techniques greatly facilitate the handling of material interfaces and non-matching grids; they also permit the coupling of different elements of arbitrary shapes and polynomial order. Starting from our new symmetric , interior penalty DG finite element discretization we have recently devised first explicit, energy conserving, high-order local time-stepping methods for electromagnetic wave propagation in non-conducting media. In our current work, we are studying new explicit LTS methods of arbitrarily high accuracy, which also handle lossy media or applied currents. While these LTS methods are based on classical multistep methods, we now wish to derive high-order explicit LTS methods starting from classical Runge-Kutta and low-storage Runge-Kutta methods for even greater flexibility and efficiency. Since wave phenomena typically occur in very large or truly unbounded regions, perfectly matched layers (PML) or nonreflecting boundary conditions (NBC) must be used to truncate the computational domain, yet without introducing spurious reflections. In our recent work we have derived a new PML for the wave equation in second-order form, which requires fewer auxiliary variables than previous fomulations and hence is more efficient. Currently we are extending that approach to Maxwell's equations and shall combine it with a DG discretization and our explicit local time-stepping methods. To handle situations of multiple scattering, where PML cannot be used, we have also derived the first local NBC for time-dependent multiple scattering problems. By using these new NBC, we now wish to include multiple-scattering effects into time-reversed absorbing conditions (TRAC), a completely new approach to recreate the past of a scattered field, but also to rapidly locate an unknown buried scatterer: both steps are crucial prior to any subsequent accurate numerical inversion procedure. Fast Methods for Frequency-Domain Full-Waveform Inversion in Strongly Heterogeneous Media Research Project | 4 Project MembersMany scientific and engineering problems - in such diverse areas as wave propagation in ultrasound to- mography, wireless communication, geophysical seismic imaging, and other areas such as atmospheric sciences, image registration, medicine, structural-fluid interactions, and chemical process industry - can be expressed in the form of a PDE-constrained optimization problem. For instance, the difficult task of non-destructively investigating a solid body, such as the Earth's interior, a piece of steel, or the human body only from surface measurements of propagating wave fields, can be formulated as a PDE-constrained optimization problem. Indeed, a distinct feature of waves propagating through a homogeneous medium is their ability to travel over long distances while retaining much of their shape and initial energy. Thus waves are ubiquitous for remote-sensing of well-defined bodies (e.g., micro-cracks, land mines) or more general inhomogeneities (e.g., tumor cells in medical imaging, or oil deposits in seismic imaging). The prediction of the scattered fields from known incident waves and given material properties is called simulation or forward problem . In the frequency domain, the forward problem in computational wave propagation is governed by the Helmholtz equation. In contrast, estimating the material properties from measured scattered fields is generally called the inverse problem . In seismology, the inverse problem is often called seismic imaging problem ; the qualifier "full-waveform" is used, when the true Helmholtz equation without any approximation is used to model the propagating wave fields. Seismic imaging has experienced significant developments during the last decade. It can be either implemented in the time-domain or in the frequency-domain. Time-domain approaches require storing and accessing the whole time-history of the forward and the backward wave propagation, which can be prohibitively expensive in terms of computational time and storage. An alternative approach is to work in the frequency-domain formulation, i.e., the Helmholtz equation and its various generalizations. This approach is very attractive because it avoids storing the entire wave propagation history. However, the frequency approach is not widespread for three-dimensional seismic simulation or imaging applications due to the lack of efficient 3D Helmholtz solvers. Recent research on Helmholtz solvers and inexact interior-point optimization methods indicates that these algorithms can also be applied to large-scale frequency-domain full-waveform inversion. In particular, the imposition of a priori bounds to avoid false minima has proved very effective. Moreover, larger parallel architectures and new algorithms for wave propagation now provide the computa- tional ability to simulate three-dimensional waves in heterogeneous media with greater accuracy. To achieve accurate full-wave form inversion in three space dimensions, we propose to extend recent developments in computational methods for nonlinear optimization and wave propagation, such as inverse calculation of se- lected entries in the Helmholtz operator, sweeping or moving boundary PML preconditioners, and their application on highly-parallel architectures. More specifically, the scientific goals of our project are: to develop 3D Helmholtz preconditioning solvers for large-scale high-frequency 3D applications in strongly heterogeneous media, to develop new inexact-Newton full-waveform inversion methods that include a priori bounds, and to apply these computational methods to realistic seismic applications. Advanced Methods for Computational Electromagnetics Research Project | 3 Project Members1. Our project aims at the development of new computational methods for the simulation of electromagnetic phenomena. Both theoretical and computational aspects are investigated. We pursue the development of flexible and efficient numerical methods for acoustic or electromagnetic wave propagation, which combine modern developments in numerical analysis, such as discontinuous Galerkin finite element methods, local time stepping, and high-order local nonreflecting boundary conditions. 2. The accurate and reliable simulation of electromagnetic fields is of fundamental importance in a wide range of engineering applications such as fiber optics, wireless communication, radar technology, inverse scattering, non-invasive testing, and optical microscopy. Furthermore, the methods developed here can directly be applied to (the much simpler) acoustic wave phenomena, pervasive in medical applications, such as ultra-sound imaging and microscopy. 3. Computational electromagnetics (CEM) presents a number of challenges. First, electromagnetic fields tend to be highly oscillatory and wave-dominated; moreover, they can develop singularities at material interfaces and boundaries. Second, the phenomena of interest typically involve complicated geometries, multi-physics, inhomogeneous media, and even nonlinear materials. Third, the underlying partial differential equations often need to be solved in an unbounded domain, which needs to be truncated by an artificial boundary to confine the region of interest to a finite computational domain. To fully address this wide range of difficulties, great flexibility is needed from any computational approach. In continuation of our previous work, we shall pursue the development of novel discontinuous Galerkin (DG) techniques to achieve the required flexibility. These discretization techniques greatly facilitate the handling of material interfaces and non-matching grids; they also permit the coupling of different elements of arbitrary shapes and local spaces of different types. In particular, starting from our new symmetric, interior penalty DG finite element discretization we wish to develop the first explicit, energy conserving, local time-stepping DG-method for time dependent electromagnetic wave propagation. To handle problems in unbounded domains, we shall use either standard Perfectly Matched Layers or our new local high-order Nonreflecting Boundary Condition (NBC). In particular, we shall derive the first completely local NBC for time dependent multiple scattering problems. Multiscale analysis and simulation of waves in strongly heterogeneous media. Research Project | 2 Project MembersWhen a wave propagates through a homogeneous medium it retains its initial shape even over long distances. As it encounters an inhomogeneity, however, the wave interacts with the medium and complicated scattered wave patterns emerge. From that information, it is possible to infer characteristics of the inhomogeneity, such as its shape or density, even when buried deeply inside the medium. Many applications in science, engineering and medicine relie on the particular features of waves propagating through an inhomogeneous medium for remote sensing, such as geophysical imaging, ultrasound, non-destructive testing in material science, mine detection, or the design of meta-materials or photonic crystals. If the variations of the medium occur at a scale $e$ much smaller than the size of the domain or the wave length, standard numerical methods become prohibitively expensive due to their need to discretize the entire computational domain down to the smallest scales. Thus we seek heterogeneous multiscale methods (HMM) that permit the simulation of waves propagating through strongly varying heterogenous media, at a computational cost independent of $e$. At later time, as the wave propagates through a strongly heterogeous medium, it develops a secondary wave train of dispersive nature, which is not captured by classical homogenization theory. Therefore we shall devise an HMM scheme for the wave equation in strongly heterogeous media, which applies in more general situations without precomputing the homogenized limit problem. Clearly to explore and discover unknown inhomogeneities deeply buried inside a medium, an efficient forward solver is not sufficient. By comparing the response from the simulation with true measurements, it is possible to iteratively improve upon the initial guess of the medium characteristics and determine the hidden scatterer. Such an inverse medium problem is probably best formulated as a PDE-constrained optimization problem. It is generally ill-posed, contains many (false) local solutions and it is usually significantly more difficult to solve than the forward problem. To overcome these difficulties, we shall devise numerical methods that guarantee superlinear global convergence, include inequality constraints to exclude unphysical false solutions, and handle large ill-conditioned indefinite linear systems. When a wave propagates through a homogeneous medium it retains its initial shape even over longdistances. As it encounters an inhomogeneity, however, the wave interacts with the mediumand complicated scattered wave patterns emerge. From that information, it is possible to infer characteristics of the inhomogeneity, such as its shape or density, even when buried deeply inside the medium. Many applications in science, engineering and medicine relie on the particular features of waves propagating through an inhomogeneous medium for remote sensing, such as geophysical imaging, ultrasound, non-destructive testing in material science,mine detection, or the design of meta-materials or photonic crystals. If the variations of the medium occur at a scale much smaller than the size of the domainor the wave length, standard numerical methods become prohibitively expensive due to theirneed to discretize the entire computational domain down to the smallest scales. Thus we seek heterogeneous multiscale methods (HMM)that permit the simulation of waves propagating through strongly varying heterogenous media,at a computational cost independent of smaller scales. At later time, as the wave propagates through a stronglyheterogeous medium, it develops a secondary wave train of dispersive nature, which is notcaptured by classical homogenization theory. Therefore we shall devise an HMM scheme for thewave equation in strongly heterogeous media, which applies in more general situations withoutprecomputing the homogenized limit problem. Clearly to explore and discover unknown inhomogeneities deeply buried inside a medium, an efficient forward solver is not sufficient. By comparing the response from the simulationwith true measurements, it is possible to iteratively improve upon the initial guess ofthe medium characteristics and determine the hidden scatterer. Such an inverse medium problemis probably best formulated as a PDE-constrained optimization problem. It is generallyill-posed, contains many (false) local solutions and it is usually significantly more difficult to solve than the forward problem. To overcome these difficulties,we shall devise numerical methods that guarantee superlinear global convergence, include inequalityconstraints to exclude unphysical false solutions, and handle large ill-conditioned indefinitelinear systems. 12 12 OverviewMembersPublicationsProjects & Collaborations
Projects & Collaborations 14 foundShow per page10 10 20 50 SPACE-TIME ADAPTIVE EXPLICIT TIME-STEPPING FOR WAVE PROPAGATION Research Project | 1 Project MembersNo Description available Local Time-Stepping Methods: Stability, Convergence and Advanced Applications Research Project | 4 Project MembersDie effiziente und detailtreue Computersimulation von Wellenphänomenen ist zentral bei vielen Anwendungen wie z.B. der medizinischen Bildgebung, der Seismik, dem Mobilfunk oder auch der zerstörungsfreien Materialprüfung. Numerische Verfahren zur Simulation von Wellenphänomenen basieren auf einer Raum- und Zeitdiskretisierung der Wellengleichung. Für die räumliche Diskretisierung sind finite Elemente Verfahren bestens geeignet, besonders weil sie auch in beliebig komplizierten Geometrien einsetzbar sind. Für die zeitliche Diskretisierung sind explizite Zeitschrittverfahren besonders effizient und lassen sich auch gut auf parallelen Hochleistungsrechnern einsetzen. Dabei benutzen Standardverfahren den gleichen Zeitschritt im gesamten Rechengebiet, der wegen der CFL Stabilitätsbedingung durch die kleinste Maschenweite bestimmt ist. Dies ist bei lokal verfeinerten Gittern höchst ineffizient. Um diese CFL Stabilitätsschranke zu umgehen, wurden in den letzten Jahren verschiedene lokale Zeitschrittverfahren (Local Time Stepping = LTS) entwickelt, die in den kleineren Elementen kleinere Zeitschritte und in den grösseren Elementen grössere Zeitschritte erlauben. Auch wenn lokale Zeitschrittverfahren sich in der Praxis schon bewährt haben, gibt es bisher noch keine allgemeine Konvergenztheorie. Bei der Zeitintegration der Wellengleichung unterscheidet man i.A. zwei Klassen von expliziten Zeitschrittverfahren: Leap-Frog (LF) Verfahren, die auf der klassischen Formulierung 2. Ordnung basieren, und Runge-Kutta (RK) Verfahren, die auf der Umformulierung als System 1. Ordnung basieren. Für LF Verfahren konnten wir 2019 zum ersten Mal optimale Konvergenzraten für ein auf LF basierendes LTS Verfahren beweisen, jedoch nicht unter einer optimalen (CFL) Stabilitätsschranke. Es geht nun darum, das LTS-LF Verfahren leicht zu modifizieren, damit optimale Konvergenzraten auch unter einer optimalen CFL Bedingung bewiesen werden können. Ausserdem möchten wir versuchen, diese Konvergenztheorie auch auf RK basierende LTS Verfahren zu erweitern. Ziel dieses Projekts ist es deshalb, einerseits eine rigorose Konvergenztheorie für explizite lokale Zeitschritte zu etablieren und andererseits deren Nutzen sowohl für Simulationen bei Unsicherheiten in der Wellengeschwindigkeit als auch für zeitharmonische Simulationen zu demonstrieren. Adaptive Spectral Decompositions for Inverse Medium Problems Research Project | 3 Project MembersInverse medium problems involve the reconstruction of a spatially varying unknown medium from available observations by exploring a restricted search space of possible solutions. Standard grid-based representations are very general but all too often computationally prohibitive due to the high dimension of the search space. Adaptive spectral decompositions instead expand the unknown medium in a basis of eigenfunctions of a judicious elliptic operator, which depends itself on the medium. We combine the AS decomposition with standard inexact Newton-type methods for the solution of time-harmonic and time-dependent wave scattering problems. By repeatedly adapting both the eigenfunction basis and its dimension, the resulting adaptive spec- tral inversion (ASI) method substantially reduces the dimension of the search space during the nonlinear optimization. Rigorous estimates of the AS decomposition are proved for a general piecewise constant medium. Numerical results illustrate the accuracy and efficiency of the ASI method for time-harmonic inverse scattering problems, including realistic subsurface models from geophysics. A Gauss-Jacobi Kernel Compression Scheme for Fractional Differential Equations Research Project | 1 Project MembersA scheme for approximating the kernel $w$ of the fractional $a$-integral by a linear combination of exponentials is proposed and studied. The scheme is based on the application of a composite Gauss-Jacobi quadrature rule to an integral representation of $w$. This results in an approximation of $w$ in an interval $[del,T]$, with $00$, and $delin(0,T)$. Local Time-Stepping Methods for Wave Propagation Research Project | 3 Project MembersWave phenomena are ubiquitous across science, technology and medicine. Typical applications in- clude ultrasound imaging, wireless communications and seismic tomography. In this proposal we shall analyze and further develop time integration methods for the numerical simulation of acoustic, electromagnetic or elastic wave phenomena. For the spatial discretization, we use either conforming finite elements or discontinuous Galerkin methods, which accomodate arbitrary meshes and geometry. For the time discretization, we consider recently derived local time-stepping (LTS) methods, which overcome the bottleneck due to local mesh refinement by taking smaller time-steps precisely where the smallest elements are located. Explicit LTS methods have already proved useful in many applications and shown nearly optimal speed-up on HPC architectures. Convergence (in the ODE sense) to the semi-discrete solution on a fixed mesh is fairly standard. However, a general convergence theory in the PDE sense, which establishes convergence to the (true) continuous solution as both the time-step and the mesh-size tend to zero, is still lacking. This proposal therefore aims at establishing a rigourous convergence theory for explicit LTS meth- ods, which fall into two separate categories. Hence this proposal consists of two separate projects. In the first project, we shall prove optimal space-time convergence of LTS methods based on energy conserving leap-frog (LF) methods. Moreover, we shall compare the accuracy of different fourth-order LTS methods in particular for long-time simulations. In the second project, we shall derive a complete space-time convergence theory for Runge-Kutta (RK) based LTS methods. Moreover, we shall demonstrate the usefulness of LTS methods also in time-harmonic regimes, when the controllability method is used for the solution of the Helmholtz equation. Advanced Methods in Computational Electromagnetics Research Project | 3 Project MembersMathematisch werden die verschiedensten Wellenphänomene im Grunde durch die klassische Wellengleichung und die Maxwell-Gleichungen beschrieben. Da sich deren Lösung nur in den einfachsten "akademischen" Fällen mit Bleistift und Papier bestimmen lässt, greift man in der Praxis auf Methoden der Numerischen Mathematik zurück, mit denen sich näherungsweise Wellenphänomene auch in kompliziertesten Situationen auf dem Computer simulieren lassen. Um eine möglichst detailtreue Darstellung des Phänomens zu erreichen, sind eine hohe Auflösung und ein riesiger Lösungsraum erforderlich, die nur mit modernsten numerischen Methoden und den neusten Hochleistungsrechnern zu bewältigen sind. In diesem Projekt werden wir neuartige numerische Verfahren zur Ort- und Zeitdiskretisierung obiger Gleichungen entwickeln. Für die Ortdiskretisierung werden wir auf diskontinuierliche Galerkin (DG) Finite Elemente (FE) Methoden zurückgreifen, die besonders flexibel und effizient sind und teils in früheren Projekten entwickelt wurden. Für die Zeitdiskretisierung werden wir neue explizite lokale Zeitschrittverfahren entwickeln, die auf Runge-Kutta Methoden basieren und es ermöglichen, die CFL-Stabilitätsgrenze zu umgehen. Dadurch werden bedeutend grössere Zeitschritte möglich, die wiederum zu schnelleren Computersimulationen führen. Meistens ist nicht nur die direkte Simulation (die Lösung des Vorwärtsproblems) gewünscht, sondern auch deren Anwendung bei der Lösung des inversen Problems, um z.B. anhand von Messdaten ein unbekanntes Medium zerstörungsfrei sichtbar zu machen (z.B. Ultraschall in der Medizin). Falls mehrere Hindernisse zur Überlagerung der Streufelder führen, wird es notwendig, die verschieden gestreuten Wellen von einander zu trennen. Dafür werden wir ein neues Verfahren entwickeln und erproben, das auf der Time Reversed Absorbing Conditions (TRAC) Methode basiert. Advanced Methods for Computational Electromagnetics Research Project | 1 Project MembersComputational electromagnetics (CEM) presents a number of challenges. First, electromagnetic fields tend to be highly oscillatory and wave-dominated ; moreover, they can develop singularities at material interfaces and boundaries. Second, the phenomena of interest typically involve complicated geometries, inhomogeneous media, and even nonlinear materials. Third, the underlying partial differential equations often need to be solved in an unbounded domain, which needs to be truncated by an artificial boundary to confine the region of interest to a finite computational domain. Thus, we seek efficient, reliable and inherently parallel computational methods which handle - complex geometry, corner singularities and material interfaces, - highly varying material properties, - wave-dominated behavior, - unbounded domains. To fully address the wide range of difficulties inherent to Maxwell's equations, great flexibility is needed from any computational approach. In continuation of our previous work, we shall pursue the development of novel explicit local time-stepping (LTS) methods and combine them with discon- tinuous Galerkin (DG) discretizations to achieve the required flexibility. These DG discretization techniques greatly facilitate the handling of material interfaces and non-matching grids; they also permit the coupling of different elements of arbitrary shapes and polynomial order. Starting from our new symmetric , interior penalty DG finite element discretization we have recently devised first explicit, energy conserving, high-order local time-stepping methods for electromagnetic wave propagation in non-conducting media. In our current work, we are studying new explicit LTS methods of arbitrarily high accuracy, which also handle lossy media or applied currents. While these LTS methods are based on classical multistep methods, we now wish to derive high-order explicit LTS methods starting from classical Runge-Kutta and low-storage Runge-Kutta methods for even greater flexibility and efficiency. Since wave phenomena typically occur in very large or truly unbounded regions, perfectly matched layers (PML) or nonreflecting boundary conditions (NBC) must be used to truncate the computational domain, yet without introducing spurious reflections. In our recent work we have derived a new PML for the wave equation in second-order form, which requires fewer auxiliary variables than previous fomulations and hence is more efficient. Currently we are extending that approach to Maxwell's equations and shall combine it with a DG discretization and our explicit local time-stepping methods. To handle situations of multiple scattering, where PML cannot be used, we have also derived the first local NBC for time-dependent multiple scattering problems. By using these new NBC, we now wish to include multiple-scattering effects into time-reversed absorbing conditions (TRAC), a completely new approach to recreate the past of a scattered field, but also to rapidly locate an unknown buried scatterer: both steps are crucial prior to any subsequent accurate numerical inversion procedure. Fast Methods for Frequency-Domain Full-Waveform Inversion in Strongly Heterogeneous Media Research Project | 4 Project MembersMany scientific and engineering problems - in such diverse areas as wave propagation in ultrasound to- mography, wireless communication, geophysical seismic imaging, and other areas such as atmospheric sciences, image registration, medicine, structural-fluid interactions, and chemical process industry - can be expressed in the form of a PDE-constrained optimization problem. For instance, the difficult task of non-destructively investigating a solid body, such as the Earth's interior, a piece of steel, or the human body only from surface measurements of propagating wave fields, can be formulated as a PDE-constrained optimization problem. Indeed, a distinct feature of waves propagating through a homogeneous medium is their ability to travel over long distances while retaining much of their shape and initial energy. Thus waves are ubiquitous for remote-sensing of well-defined bodies (e.g., micro-cracks, land mines) or more general inhomogeneities (e.g., tumor cells in medical imaging, or oil deposits in seismic imaging). The prediction of the scattered fields from known incident waves and given material properties is called simulation or forward problem . In the frequency domain, the forward problem in computational wave propagation is governed by the Helmholtz equation. In contrast, estimating the material properties from measured scattered fields is generally called the inverse problem . In seismology, the inverse problem is often called seismic imaging problem ; the qualifier "full-waveform" is used, when the true Helmholtz equation without any approximation is used to model the propagating wave fields. Seismic imaging has experienced significant developments during the last decade. It can be either implemented in the time-domain or in the frequency-domain. Time-domain approaches require storing and accessing the whole time-history of the forward and the backward wave propagation, which can be prohibitively expensive in terms of computational time and storage. An alternative approach is to work in the frequency-domain formulation, i.e., the Helmholtz equation and its various generalizations. This approach is very attractive because it avoids storing the entire wave propagation history. However, the frequency approach is not widespread for three-dimensional seismic simulation or imaging applications due to the lack of efficient 3D Helmholtz solvers. Recent research on Helmholtz solvers and inexact interior-point optimization methods indicates that these algorithms can also be applied to large-scale frequency-domain full-waveform inversion. In particular, the imposition of a priori bounds to avoid false minima has proved very effective. Moreover, larger parallel architectures and new algorithms for wave propagation now provide the computa- tional ability to simulate three-dimensional waves in heterogeneous media with greater accuracy. To achieve accurate full-wave form inversion in three space dimensions, we propose to extend recent developments in computational methods for nonlinear optimization and wave propagation, such as inverse calculation of se- lected entries in the Helmholtz operator, sweeping or moving boundary PML preconditioners, and their application on highly-parallel architectures. More specifically, the scientific goals of our project are: to develop 3D Helmholtz preconditioning solvers for large-scale high-frequency 3D applications in strongly heterogeneous media, to develop new inexact-Newton full-waveform inversion methods that include a priori bounds, and to apply these computational methods to realistic seismic applications. Advanced Methods for Computational Electromagnetics Research Project | 3 Project Members1. Our project aims at the development of new computational methods for the simulation of electromagnetic phenomena. Both theoretical and computational aspects are investigated. We pursue the development of flexible and efficient numerical methods for acoustic or electromagnetic wave propagation, which combine modern developments in numerical analysis, such as discontinuous Galerkin finite element methods, local time stepping, and high-order local nonreflecting boundary conditions. 2. The accurate and reliable simulation of electromagnetic fields is of fundamental importance in a wide range of engineering applications such as fiber optics, wireless communication, radar technology, inverse scattering, non-invasive testing, and optical microscopy. Furthermore, the methods developed here can directly be applied to (the much simpler) acoustic wave phenomena, pervasive in medical applications, such as ultra-sound imaging and microscopy. 3. Computational electromagnetics (CEM) presents a number of challenges. First, electromagnetic fields tend to be highly oscillatory and wave-dominated; moreover, they can develop singularities at material interfaces and boundaries. Second, the phenomena of interest typically involve complicated geometries, multi-physics, inhomogeneous media, and even nonlinear materials. Third, the underlying partial differential equations often need to be solved in an unbounded domain, which needs to be truncated by an artificial boundary to confine the region of interest to a finite computational domain. To fully address this wide range of difficulties, great flexibility is needed from any computational approach. In continuation of our previous work, we shall pursue the development of novel discontinuous Galerkin (DG) techniques to achieve the required flexibility. These discretization techniques greatly facilitate the handling of material interfaces and non-matching grids; they also permit the coupling of different elements of arbitrary shapes and local spaces of different types. In particular, starting from our new symmetric, interior penalty DG finite element discretization we wish to develop the first explicit, energy conserving, local time-stepping DG-method for time dependent electromagnetic wave propagation. To handle problems in unbounded domains, we shall use either standard Perfectly Matched Layers or our new local high-order Nonreflecting Boundary Condition (NBC). In particular, we shall derive the first completely local NBC for time dependent multiple scattering problems. Multiscale analysis and simulation of waves in strongly heterogeneous media. Research Project | 2 Project MembersWhen a wave propagates through a homogeneous medium it retains its initial shape even over long distances. As it encounters an inhomogeneity, however, the wave interacts with the medium and complicated scattered wave patterns emerge. From that information, it is possible to infer characteristics of the inhomogeneity, such as its shape or density, even when buried deeply inside the medium. Many applications in science, engineering and medicine relie on the particular features of waves propagating through an inhomogeneous medium for remote sensing, such as geophysical imaging, ultrasound, non-destructive testing in material science, mine detection, or the design of meta-materials or photonic crystals. If the variations of the medium occur at a scale $e$ much smaller than the size of the domain or the wave length, standard numerical methods become prohibitively expensive due to their need to discretize the entire computational domain down to the smallest scales. Thus we seek heterogeneous multiscale methods (HMM) that permit the simulation of waves propagating through strongly varying heterogenous media, at a computational cost independent of $e$. At later time, as the wave propagates through a strongly heterogeous medium, it develops a secondary wave train of dispersive nature, which is not captured by classical homogenization theory. Therefore we shall devise an HMM scheme for the wave equation in strongly heterogeous media, which applies in more general situations without precomputing the homogenized limit problem. Clearly to explore and discover unknown inhomogeneities deeply buried inside a medium, an efficient forward solver is not sufficient. By comparing the response from the simulation with true measurements, it is possible to iteratively improve upon the initial guess of the medium characteristics and determine the hidden scatterer. Such an inverse medium problem is probably best formulated as a PDE-constrained optimization problem. It is generally ill-posed, contains many (false) local solutions and it is usually significantly more difficult to solve than the forward problem. To overcome these difficulties, we shall devise numerical methods that guarantee superlinear global convergence, include inequality constraints to exclude unphysical false solutions, and handle large ill-conditioned indefinite linear systems. When a wave propagates through a homogeneous medium it retains its initial shape even over longdistances. As it encounters an inhomogeneity, however, the wave interacts with the mediumand complicated scattered wave patterns emerge. From that information, it is possible to infer characteristics of the inhomogeneity, such as its shape or density, even when buried deeply inside the medium. Many applications in science, engineering and medicine relie on the particular features of waves propagating through an inhomogeneous medium for remote sensing, such as geophysical imaging, ultrasound, non-destructive testing in material science,mine detection, or the design of meta-materials or photonic crystals. If the variations of the medium occur at a scale much smaller than the size of the domainor the wave length, standard numerical methods become prohibitively expensive due to theirneed to discretize the entire computational domain down to the smallest scales. Thus we seek heterogeneous multiscale methods (HMM)that permit the simulation of waves propagating through strongly varying heterogenous media,at a computational cost independent of smaller scales. At later time, as the wave propagates through a stronglyheterogeous medium, it develops a secondary wave train of dispersive nature, which is notcaptured by classical homogenization theory. Therefore we shall devise an HMM scheme for thewave equation in strongly heterogeous media, which applies in more general situations withoutprecomputing the homogenized limit problem. Clearly to explore and discover unknown inhomogeneities deeply buried inside a medium, an efficient forward solver is not sufficient. By comparing the response from the simulationwith true measurements, it is possible to iteratively improve upon the initial guess ofthe medium characteristics and determine the hidden scatterer. Such an inverse medium problemis probably best formulated as a PDE-constrained optimization problem. It is generallyill-posed, contains many (false) local solutions and it is usually significantly more difficult to solve than the forward problem. To overcome these difficulties,we shall devise numerical methods that guarantee superlinear global convergence, include inequalityconstraints to exclude unphysical false solutions, and handle large ill-conditioned indefinitelinear systems. 12 12
SPACE-TIME ADAPTIVE EXPLICIT TIME-STEPPING FOR WAVE PROPAGATION Research Project | 1 Project MembersNo Description available
Local Time-Stepping Methods: Stability, Convergence and Advanced Applications Research Project | 4 Project MembersDie effiziente und detailtreue Computersimulation von Wellenphänomenen ist zentral bei vielen Anwendungen wie z.B. der medizinischen Bildgebung, der Seismik, dem Mobilfunk oder auch der zerstörungsfreien Materialprüfung. Numerische Verfahren zur Simulation von Wellenphänomenen basieren auf einer Raum- und Zeitdiskretisierung der Wellengleichung. Für die räumliche Diskretisierung sind finite Elemente Verfahren bestens geeignet, besonders weil sie auch in beliebig komplizierten Geometrien einsetzbar sind. Für die zeitliche Diskretisierung sind explizite Zeitschrittverfahren besonders effizient und lassen sich auch gut auf parallelen Hochleistungsrechnern einsetzen. Dabei benutzen Standardverfahren den gleichen Zeitschritt im gesamten Rechengebiet, der wegen der CFL Stabilitätsbedingung durch die kleinste Maschenweite bestimmt ist. Dies ist bei lokal verfeinerten Gittern höchst ineffizient. Um diese CFL Stabilitätsschranke zu umgehen, wurden in den letzten Jahren verschiedene lokale Zeitschrittverfahren (Local Time Stepping = LTS) entwickelt, die in den kleineren Elementen kleinere Zeitschritte und in den grösseren Elementen grössere Zeitschritte erlauben. Auch wenn lokale Zeitschrittverfahren sich in der Praxis schon bewährt haben, gibt es bisher noch keine allgemeine Konvergenztheorie. Bei der Zeitintegration der Wellengleichung unterscheidet man i.A. zwei Klassen von expliziten Zeitschrittverfahren: Leap-Frog (LF) Verfahren, die auf der klassischen Formulierung 2. Ordnung basieren, und Runge-Kutta (RK) Verfahren, die auf der Umformulierung als System 1. Ordnung basieren. Für LF Verfahren konnten wir 2019 zum ersten Mal optimale Konvergenzraten für ein auf LF basierendes LTS Verfahren beweisen, jedoch nicht unter einer optimalen (CFL) Stabilitätsschranke. Es geht nun darum, das LTS-LF Verfahren leicht zu modifizieren, damit optimale Konvergenzraten auch unter einer optimalen CFL Bedingung bewiesen werden können. Ausserdem möchten wir versuchen, diese Konvergenztheorie auch auf RK basierende LTS Verfahren zu erweitern. Ziel dieses Projekts ist es deshalb, einerseits eine rigorose Konvergenztheorie für explizite lokale Zeitschritte zu etablieren und andererseits deren Nutzen sowohl für Simulationen bei Unsicherheiten in der Wellengeschwindigkeit als auch für zeitharmonische Simulationen zu demonstrieren.
Adaptive Spectral Decompositions for Inverse Medium Problems Research Project | 3 Project MembersInverse medium problems involve the reconstruction of a spatially varying unknown medium from available observations by exploring a restricted search space of possible solutions. Standard grid-based representations are very general but all too often computationally prohibitive due to the high dimension of the search space. Adaptive spectral decompositions instead expand the unknown medium in a basis of eigenfunctions of a judicious elliptic operator, which depends itself on the medium. We combine the AS decomposition with standard inexact Newton-type methods for the solution of time-harmonic and time-dependent wave scattering problems. By repeatedly adapting both the eigenfunction basis and its dimension, the resulting adaptive spec- tral inversion (ASI) method substantially reduces the dimension of the search space during the nonlinear optimization. Rigorous estimates of the AS decomposition are proved for a general piecewise constant medium. Numerical results illustrate the accuracy and efficiency of the ASI method for time-harmonic inverse scattering problems, including realistic subsurface models from geophysics.
A Gauss-Jacobi Kernel Compression Scheme for Fractional Differential Equations Research Project | 1 Project MembersA scheme for approximating the kernel $w$ of the fractional $a$-integral by a linear combination of exponentials is proposed and studied. The scheme is based on the application of a composite Gauss-Jacobi quadrature rule to an integral representation of $w$. This results in an approximation of $w$ in an interval $[del,T]$, with $00$, and $delin(0,T)$.
Local Time-Stepping Methods for Wave Propagation Research Project | 3 Project MembersWave phenomena are ubiquitous across science, technology and medicine. Typical applications in- clude ultrasound imaging, wireless communications and seismic tomography. In this proposal we shall analyze and further develop time integration methods for the numerical simulation of acoustic, electromagnetic or elastic wave phenomena. For the spatial discretization, we use either conforming finite elements or discontinuous Galerkin methods, which accomodate arbitrary meshes and geometry. For the time discretization, we consider recently derived local time-stepping (LTS) methods, which overcome the bottleneck due to local mesh refinement by taking smaller time-steps precisely where the smallest elements are located. Explicit LTS methods have already proved useful in many applications and shown nearly optimal speed-up on HPC architectures. Convergence (in the ODE sense) to the semi-discrete solution on a fixed mesh is fairly standard. However, a general convergence theory in the PDE sense, which establishes convergence to the (true) continuous solution as both the time-step and the mesh-size tend to zero, is still lacking. This proposal therefore aims at establishing a rigourous convergence theory for explicit LTS meth- ods, which fall into two separate categories. Hence this proposal consists of two separate projects. In the first project, we shall prove optimal space-time convergence of LTS methods based on energy conserving leap-frog (LF) methods. Moreover, we shall compare the accuracy of different fourth-order LTS methods in particular for long-time simulations. In the second project, we shall derive a complete space-time convergence theory for Runge-Kutta (RK) based LTS methods. Moreover, we shall demonstrate the usefulness of LTS methods also in time-harmonic regimes, when the controllability method is used for the solution of the Helmholtz equation.
Advanced Methods in Computational Electromagnetics Research Project | 3 Project MembersMathematisch werden die verschiedensten Wellenphänomene im Grunde durch die klassische Wellengleichung und die Maxwell-Gleichungen beschrieben. Da sich deren Lösung nur in den einfachsten "akademischen" Fällen mit Bleistift und Papier bestimmen lässt, greift man in der Praxis auf Methoden der Numerischen Mathematik zurück, mit denen sich näherungsweise Wellenphänomene auch in kompliziertesten Situationen auf dem Computer simulieren lassen. Um eine möglichst detailtreue Darstellung des Phänomens zu erreichen, sind eine hohe Auflösung und ein riesiger Lösungsraum erforderlich, die nur mit modernsten numerischen Methoden und den neusten Hochleistungsrechnern zu bewältigen sind. In diesem Projekt werden wir neuartige numerische Verfahren zur Ort- und Zeitdiskretisierung obiger Gleichungen entwickeln. Für die Ortdiskretisierung werden wir auf diskontinuierliche Galerkin (DG) Finite Elemente (FE) Methoden zurückgreifen, die besonders flexibel und effizient sind und teils in früheren Projekten entwickelt wurden. Für die Zeitdiskretisierung werden wir neue explizite lokale Zeitschrittverfahren entwickeln, die auf Runge-Kutta Methoden basieren und es ermöglichen, die CFL-Stabilitätsgrenze zu umgehen. Dadurch werden bedeutend grössere Zeitschritte möglich, die wiederum zu schnelleren Computersimulationen führen. Meistens ist nicht nur die direkte Simulation (die Lösung des Vorwärtsproblems) gewünscht, sondern auch deren Anwendung bei der Lösung des inversen Problems, um z.B. anhand von Messdaten ein unbekanntes Medium zerstörungsfrei sichtbar zu machen (z.B. Ultraschall in der Medizin). Falls mehrere Hindernisse zur Überlagerung der Streufelder führen, wird es notwendig, die verschieden gestreuten Wellen von einander zu trennen. Dafür werden wir ein neues Verfahren entwickeln und erproben, das auf der Time Reversed Absorbing Conditions (TRAC) Methode basiert.
Advanced Methods for Computational Electromagnetics Research Project | 1 Project MembersComputational electromagnetics (CEM) presents a number of challenges. First, electromagnetic fields tend to be highly oscillatory and wave-dominated ; moreover, they can develop singularities at material interfaces and boundaries. Second, the phenomena of interest typically involve complicated geometries, inhomogeneous media, and even nonlinear materials. Third, the underlying partial differential equations often need to be solved in an unbounded domain, which needs to be truncated by an artificial boundary to confine the region of interest to a finite computational domain. Thus, we seek efficient, reliable and inherently parallel computational methods which handle - complex geometry, corner singularities and material interfaces, - highly varying material properties, - wave-dominated behavior, - unbounded domains. To fully address the wide range of difficulties inherent to Maxwell's equations, great flexibility is needed from any computational approach. In continuation of our previous work, we shall pursue the development of novel explicit local time-stepping (LTS) methods and combine them with discon- tinuous Galerkin (DG) discretizations to achieve the required flexibility. These DG discretization techniques greatly facilitate the handling of material interfaces and non-matching grids; they also permit the coupling of different elements of arbitrary shapes and polynomial order. Starting from our new symmetric , interior penalty DG finite element discretization we have recently devised first explicit, energy conserving, high-order local time-stepping methods for electromagnetic wave propagation in non-conducting media. In our current work, we are studying new explicit LTS methods of arbitrarily high accuracy, which also handle lossy media or applied currents. While these LTS methods are based on classical multistep methods, we now wish to derive high-order explicit LTS methods starting from classical Runge-Kutta and low-storage Runge-Kutta methods for even greater flexibility and efficiency. Since wave phenomena typically occur in very large or truly unbounded regions, perfectly matched layers (PML) or nonreflecting boundary conditions (NBC) must be used to truncate the computational domain, yet without introducing spurious reflections. In our recent work we have derived a new PML for the wave equation in second-order form, which requires fewer auxiliary variables than previous fomulations and hence is more efficient. Currently we are extending that approach to Maxwell's equations and shall combine it with a DG discretization and our explicit local time-stepping methods. To handle situations of multiple scattering, where PML cannot be used, we have also derived the first local NBC for time-dependent multiple scattering problems. By using these new NBC, we now wish to include multiple-scattering effects into time-reversed absorbing conditions (TRAC), a completely new approach to recreate the past of a scattered field, but also to rapidly locate an unknown buried scatterer: both steps are crucial prior to any subsequent accurate numerical inversion procedure.
Fast Methods for Frequency-Domain Full-Waveform Inversion in Strongly Heterogeneous Media Research Project | 4 Project MembersMany scientific and engineering problems - in such diverse areas as wave propagation in ultrasound to- mography, wireless communication, geophysical seismic imaging, and other areas such as atmospheric sciences, image registration, medicine, structural-fluid interactions, and chemical process industry - can be expressed in the form of a PDE-constrained optimization problem. For instance, the difficult task of non-destructively investigating a solid body, such as the Earth's interior, a piece of steel, or the human body only from surface measurements of propagating wave fields, can be formulated as a PDE-constrained optimization problem. Indeed, a distinct feature of waves propagating through a homogeneous medium is their ability to travel over long distances while retaining much of their shape and initial energy. Thus waves are ubiquitous for remote-sensing of well-defined bodies (e.g., micro-cracks, land mines) or more general inhomogeneities (e.g., tumor cells in medical imaging, or oil deposits in seismic imaging). The prediction of the scattered fields from known incident waves and given material properties is called simulation or forward problem . In the frequency domain, the forward problem in computational wave propagation is governed by the Helmholtz equation. In contrast, estimating the material properties from measured scattered fields is generally called the inverse problem . In seismology, the inverse problem is often called seismic imaging problem ; the qualifier "full-waveform" is used, when the true Helmholtz equation without any approximation is used to model the propagating wave fields. Seismic imaging has experienced significant developments during the last decade. It can be either implemented in the time-domain or in the frequency-domain. Time-domain approaches require storing and accessing the whole time-history of the forward and the backward wave propagation, which can be prohibitively expensive in terms of computational time and storage. An alternative approach is to work in the frequency-domain formulation, i.e., the Helmholtz equation and its various generalizations. This approach is very attractive because it avoids storing the entire wave propagation history. However, the frequency approach is not widespread for three-dimensional seismic simulation or imaging applications due to the lack of efficient 3D Helmholtz solvers. Recent research on Helmholtz solvers and inexact interior-point optimization methods indicates that these algorithms can also be applied to large-scale frequency-domain full-waveform inversion. In particular, the imposition of a priori bounds to avoid false minima has proved very effective. Moreover, larger parallel architectures and new algorithms for wave propagation now provide the computa- tional ability to simulate three-dimensional waves in heterogeneous media with greater accuracy. To achieve accurate full-wave form inversion in three space dimensions, we propose to extend recent developments in computational methods for nonlinear optimization and wave propagation, such as inverse calculation of se- lected entries in the Helmholtz operator, sweeping or moving boundary PML preconditioners, and their application on highly-parallel architectures. More specifically, the scientific goals of our project are: to develop 3D Helmholtz preconditioning solvers for large-scale high-frequency 3D applications in strongly heterogeneous media, to develop new inexact-Newton full-waveform inversion methods that include a priori bounds, and to apply these computational methods to realistic seismic applications.
Advanced Methods for Computational Electromagnetics Research Project | 3 Project Members1. Our project aims at the development of new computational methods for the simulation of electromagnetic phenomena. Both theoretical and computational aspects are investigated. We pursue the development of flexible and efficient numerical methods for acoustic or electromagnetic wave propagation, which combine modern developments in numerical analysis, such as discontinuous Galerkin finite element methods, local time stepping, and high-order local nonreflecting boundary conditions. 2. The accurate and reliable simulation of electromagnetic fields is of fundamental importance in a wide range of engineering applications such as fiber optics, wireless communication, radar technology, inverse scattering, non-invasive testing, and optical microscopy. Furthermore, the methods developed here can directly be applied to (the much simpler) acoustic wave phenomena, pervasive in medical applications, such as ultra-sound imaging and microscopy. 3. Computational electromagnetics (CEM) presents a number of challenges. First, electromagnetic fields tend to be highly oscillatory and wave-dominated; moreover, they can develop singularities at material interfaces and boundaries. Second, the phenomena of interest typically involve complicated geometries, multi-physics, inhomogeneous media, and even nonlinear materials. Third, the underlying partial differential equations often need to be solved in an unbounded domain, which needs to be truncated by an artificial boundary to confine the region of interest to a finite computational domain. To fully address this wide range of difficulties, great flexibility is needed from any computational approach. In continuation of our previous work, we shall pursue the development of novel discontinuous Galerkin (DG) techniques to achieve the required flexibility. These discretization techniques greatly facilitate the handling of material interfaces and non-matching grids; they also permit the coupling of different elements of arbitrary shapes and local spaces of different types. In particular, starting from our new symmetric, interior penalty DG finite element discretization we wish to develop the first explicit, energy conserving, local time-stepping DG-method for time dependent electromagnetic wave propagation. To handle problems in unbounded domains, we shall use either standard Perfectly Matched Layers or our new local high-order Nonreflecting Boundary Condition (NBC). In particular, we shall derive the first completely local NBC for time dependent multiple scattering problems.
Multiscale analysis and simulation of waves in strongly heterogeneous media. Research Project | 2 Project MembersWhen a wave propagates through a homogeneous medium it retains its initial shape even over long distances. As it encounters an inhomogeneity, however, the wave interacts with the medium and complicated scattered wave patterns emerge. From that information, it is possible to infer characteristics of the inhomogeneity, such as its shape or density, even when buried deeply inside the medium. Many applications in science, engineering and medicine relie on the particular features of waves propagating through an inhomogeneous medium for remote sensing, such as geophysical imaging, ultrasound, non-destructive testing in material science, mine detection, or the design of meta-materials or photonic crystals. If the variations of the medium occur at a scale $e$ much smaller than the size of the domain or the wave length, standard numerical methods become prohibitively expensive due to their need to discretize the entire computational domain down to the smallest scales. Thus we seek heterogeneous multiscale methods (HMM) that permit the simulation of waves propagating through strongly varying heterogenous media, at a computational cost independent of $e$. At later time, as the wave propagates through a strongly heterogeous medium, it develops a secondary wave train of dispersive nature, which is not captured by classical homogenization theory. Therefore we shall devise an HMM scheme for the wave equation in strongly heterogeous media, which applies in more general situations without precomputing the homogenized limit problem. Clearly to explore and discover unknown inhomogeneities deeply buried inside a medium, an efficient forward solver is not sufficient. By comparing the response from the simulation with true measurements, it is possible to iteratively improve upon the initial guess of the medium characteristics and determine the hidden scatterer. Such an inverse medium problem is probably best formulated as a PDE-constrained optimization problem. It is generally ill-posed, contains many (false) local solutions and it is usually significantly more difficult to solve than the forward problem. To overcome these difficulties, we shall devise numerical methods that guarantee superlinear global convergence, include inequality constraints to exclude unphysical false solutions, and handle large ill-conditioned indefinite linear systems. When a wave propagates through a homogeneous medium it retains its initial shape even over longdistances. As it encounters an inhomogeneity, however, the wave interacts with the mediumand complicated scattered wave patterns emerge. From that information, it is possible to infer characteristics of the inhomogeneity, such as its shape or density, even when buried deeply inside the medium. Many applications in science, engineering and medicine relie on the particular features of waves propagating through an inhomogeneous medium for remote sensing, such as geophysical imaging, ultrasound, non-destructive testing in material science,mine detection, or the design of meta-materials or photonic crystals. If the variations of the medium occur at a scale much smaller than the size of the domainor the wave length, standard numerical methods become prohibitively expensive due to theirneed to discretize the entire computational domain down to the smallest scales. Thus we seek heterogeneous multiscale methods (HMM)that permit the simulation of waves propagating through strongly varying heterogenous media,at a computational cost independent of smaller scales. At later time, as the wave propagates through a stronglyheterogeous medium, it develops a secondary wave train of dispersive nature, which is notcaptured by classical homogenization theory. Therefore we shall devise an HMM scheme for thewave equation in strongly heterogeous media, which applies in more general situations withoutprecomputing the homogenized limit problem. Clearly to explore and discover unknown inhomogeneities deeply buried inside a medium, an efficient forward solver is not sufficient. By comparing the response from the simulationwith true measurements, it is possible to iteratively improve upon the initial guess ofthe medium characteristics and determine the hidden scatterer. Such an inverse medium problemis probably best formulated as a PDE-constrained optimization problem. It is generallyill-posed, contains many (false) local solutions and it is usually significantly more difficult to solve than the forward problem. To overcome these difficulties,we shall devise numerical methods that guarantee superlinear global convergence, include inequalityconstraints to exclude unphysical false solutions, and handle large ill-conditioned indefinitelinear systems.
