Prof. Dr. Marcus J. Grote Department of Mathematics and Computer Sciences Profiles & Affiliations OverviewResearch Publications Projects & Collaborations Projects & Collaborations OverviewResearch Publications Projects & Collaborations Profiles & Affiliations Projects & Collaborations 11 foundShow per page10 10 20 50 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. 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. Large-Scale Parallel Nonlinear Optimization for High Resolution 3D-Seismic Imaging Research Project | 5 Project MembersCurrent methods in global or local-scale seismic tomography rely on approximate descriptions of wave propagation with the result of severely limiting the resolution of tomographic images. However, to truly understand the dynamics of our planet, we need to be able to seismically map its deep structure at resolutions much higher than it is nowadays possible. Major geophysical questions that require high resolution 3D imaging at the planetary scale include a better understanding, e.g., of the nature of mantle plumes and sinking tectonic plates. At the regional scale, reliable seismic images are crucial for more accurate earthquake location and the compilation of seismic hazard maps. Recent advances in algorithms, software development, and high performance computing systems have resulted in PDE-based solvers that scale up to millions of variables, make use of thousands of processors, and accommodate complex multiple-physics. As partial differential equations (PDE) solvers also mature in the Earth Sciences, there is an increasing interest in solving nonlinear seismic inversion problems governed by PDE-based models. Larger computer architectures and new algorithms for optimization and wave propagation now provide the computational ability to address the geophysical issues mentioned above in a more rigorous way: namely, to abandon asymptotic ray-theory approximations in favor of time-dependent PDE-based models, and replace linearized inversions by truly nonlinear optimization. To achieve this goal, it will be necessary to combine recent developments in computational methods for nonlinear optimization and wave propagation, such as high-order finite element discretizations, local time-stepping, iterative methods, and inexact parallel interior-point methods. More specifically, the scientific goals of the project are: to develop parallel numerical methods for forward wave propagation and large-scale nonlinear optimization, to explore the performance of such methods on emerging petascale architectures (e.g. GPU, Cell BE) and to develop a new generation of a seismic inversion code for 3D Earth imaging. Numerical methods for stochastically driven wave equations Research Project | 3 Project MembersThe interest in stochastic partial differential equations is growing during the last decades. Unfortunately, there are very few equations for which an analytical solution is known. For those stochastic partial differential equations without a known solution, one hope is that numerical methods will help to better understand them. In this project, we will focus on numerical discretisations of stochastic wave equations. For problems such as wave propagation through the ocean, the properties of the water fluctuate randomly due to the presence of turbulence: a deterministic approach is thus not enough to describe the motion of the wave. This leads us to consider stochastic wave equations. As a model problem, we plan to study numerical discretisations of the one-dimensional stochastic wave equation, where the space-time noise is white in time and spatially correlated. This partial differential equation can be seen as a model of a nonlinear string submerged in a turbulent fluid. For the numerical discretisation of such a problem, we first discretise in space (method of lines) and then in time with a geometric integrator . After a pseudo-spectral semi-discretisation in space of the stochastic wave equation, we obtain a system of second-order stochastic differential equations which are oscillators-like equations. A deep understanding of this system of stochastic equations is primordial for a proper numerical treatment of the problem. To get more insight into the behaviour of such kind of system, we firstly concentrate on the study of efficient numerical methods for stochastic oscillators . We then adapt these numerical schemes to systems of second-order stochastic differential equations and finally we look at the full discretisation of the stochastically driven wave equation . Our new geometric integrators will permit a deeper understanding of the numerical discretisation of stochastic oscillators, which will lead to an optimal numerical treatments of stochastically driven wave equations. Advanced Methods for Computational Electromagnetics Research Project | 2 Project MembersComputational electromagnetics (CEM) presents a number of challenges. First, electromagnetic fields tend to be highly {em oscillatory} and {em wave-dominated}; moreover, they can develop {em singularities} at material interfaces and boundaries. Second, the phenomena of interest typically involve {em complicated} geometries, {em multi-physics}, {em inhomogeneous} media, and even {em nonlinear} materials. Third, the underlying partial differential equations often need to be solved in an unbounded domain, which needs to be truncated by an {em artificial boundary} to confine the region of interest to a finite computational domain. Fourth, standard preconditioners are typically ineffective for the highly indefinite ill-conditioned sparse linear systems of equations that appear in higher frequency regimes. 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 theregion of interest to a finite computational domain. Multiscale analysis and simulation of waves in strongly heterogeous media Research Project | 4 Project MembersWe plan to study wave propagation in heterogeneous media both in the frequency and the time dependent regime. In particular, our aims are - to study via numerical simulation the limit problem, first in one and then in higher dimensions, and thus establish a deeper understanding of the dispersive response from the heterogeneous medium, and - to devise a heterogeneous multiscale method (HMM) scheme for wave propagation in strongly heterogeneous media, which should apply in more general situations without precomputing the analytical (homogenized) limit problem. We plan to study wave propagation in heterogeneous media both in thefrequency and the time dependent regime. In particular, our aims are - to study via numerical simulation the limit problem, first in one and then in higher dimensions, and thus establish a deeper understanding of the dispersive response from the heterogeneous medium, and - to devise a heterogeneous multiscale method (HMM) scheme for wave propagation in strongly heterogeneous media, which should apply in more general situations without precomputing the analytical (homogenized) limit problem. 12 12 OverviewResearch Publications Projects & Collaborations
Projects & Collaborations 11 foundShow per page10 10 20 50 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. 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. Large-Scale Parallel Nonlinear Optimization for High Resolution 3D-Seismic Imaging Research Project | 5 Project MembersCurrent methods in global or local-scale seismic tomography rely on approximate descriptions of wave propagation with the result of severely limiting the resolution of tomographic images. However, to truly understand the dynamics of our planet, we need to be able to seismically map its deep structure at resolutions much higher than it is nowadays possible. Major geophysical questions that require high resolution 3D imaging at the planetary scale include a better understanding, e.g., of the nature of mantle plumes and sinking tectonic plates. At the regional scale, reliable seismic images are crucial for more accurate earthquake location and the compilation of seismic hazard maps. Recent advances in algorithms, software development, and high performance computing systems have resulted in PDE-based solvers that scale up to millions of variables, make use of thousands of processors, and accommodate complex multiple-physics. As partial differential equations (PDE) solvers also mature in the Earth Sciences, there is an increasing interest in solving nonlinear seismic inversion problems governed by PDE-based models. Larger computer architectures and new algorithms for optimization and wave propagation now provide the computational ability to address the geophysical issues mentioned above in a more rigorous way: namely, to abandon asymptotic ray-theory approximations in favor of time-dependent PDE-based models, and replace linearized inversions by truly nonlinear optimization. To achieve this goal, it will be necessary to combine recent developments in computational methods for nonlinear optimization and wave propagation, such as high-order finite element discretizations, local time-stepping, iterative methods, and inexact parallel interior-point methods. More specifically, the scientific goals of the project are: to develop parallel numerical methods for forward wave propagation and large-scale nonlinear optimization, to explore the performance of such methods on emerging petascale architectures (e.g. GPU, Cell BE) and to develop a new generation of a seismic inversion code for 3D Earth imaging. Numerical methods for stochastically driven wave equations Research Project | 3 Project MembersThe interest in stochastic partial differential equations is growing during the last decades. Unfortunately, there are very few equations for which an analytical solution is known. For those stochastic partial differential equations without a known solution, one hope is that numerical methods will help to better understand them. In this project, we will focus on numerical discretisations of stochastic wave equations. For problems such as wave propagation through the ocean, the properties of the water fluctuate randomly due to the presence of turbulence: a deterministic approach is thus not enough to describe the motion of the wave. This leads us to consider stochastic wave equations. As a model problem, we plan to study numerical discretisations of the one-dimensional stochastic wave equation, where the space-time noise is white in time and spatially correlated. This partial differential equation can be seen as a model of a nonlinear string submerged in a turbulent fluid. For the numerical discretisation of such a problem, we first discretise in space (method of lines) and then in time with a geometric integrator . After a pseudo-spectral semi-discretisation in space of the stochastic wave equation, we obtain a system of second-order stochastic differential equations which are oscillators-like equations. A deep understanding of this system of stochastic equations is primordial for a proper numerical treatment of the problem. To get more insight into the behaviour of such kind of system, we firstly concentrate on the study of efficient numerical methods for stochastic oscillators . We then adapt these numerical schemes to systems of second-order stochastic differential equations and finally we look at the full discretisation of the stochastically driven wave equation . Our new geometric integrators will permit a deeper understanding of the numerical discretisation of stochastic oscillators, which will lead to an optimal numerical treatments of stochastically driven wave equations. Advanced Methods for Computational Electromagnetics Research Project | 2 Project MembersComputational electromagnetics (CEM) presents a number of challenges. First, electromagnetic fields tend to be highly {em oscillatory} and {em wave-dominated}; moreover, they can develop {em singularities} at material interfaces and boundaries. Second, the phenomena of interest typically involve {em complicated} geometries, {em multi-physics}, {em inhomogeneous} media, and even {em nonlinear} materials. Third, the underlying partial differential equations often need to be solved in an unbounded domain, which needs to be truncated by an {em artificial boundary} to confine the region of interest to a finite computational domain. Fourth, standard preconditioners are typically ineffective for the highly indefinite ill-conditioned sparse linear systems of equations that appear in higher frequency regimes. 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 theregion of interest to a finite computational domain. Multiscale analysis and simulation of waves in strongly heterogeous media Research Project | 4 Project MembersWe plan to study wave propagation in heterogeneous media both in the frequency and the time dependent regime. In particular, our aims are - to study via numerical simulation the limit problem, first in one and then in higher dimensions, and thus establish a deeper understanding of the dispersive response from the heterogeneous medium, and - to devise a heterogeneous multiscale method (HMM) scheme for wave propagation in strongly heterogeneous media, which should apply in more general situations without precomputing the analytical (homogenized) limit problem. We plan to study wave propagation in heterogeneous media both in thefrequency and the time dependent regime. In particular, our aims are - to study via numerical simulation the limit problem, first in one and then in higher dimensions, and thus establish a deeper understanding of the dispersive response from the heterogeneous medium, and - to devise a heterogeneous multiscale method (HMM) scheme for wave propagation in strongly heterogeneous media, which should apply in more general situations without precomputing the analytical (homogenized) limit problem. 12 12
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.
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.
Large-Scale Parallel Nonlinear Optimization for High Resolution 3D-Seismic Imaging Research Project | 5 Project MembersCurrent methods in global or local-scale seismic tomography rely on approximate descriptions of wave propagation with the result of severely limiting the resolution of tomographic images. However, to truly understand the dynamics of our planet, we need to be able to seismically map its deep structure at resolutions much higher than it is nowadays possible. Major geophysical questions that require high resolution 3D imaging at the planetary scale include a better understanding, e.g., of the nature of mantle plumes and sinking tectonic plates. At the regional scale, reliable seismic images are crucial for more accurate earthquake location and the compilation of seismic hazard maps. Recent advances in algorithms, software development, and high performance computing systems have resulted in PDE-based solvers that scale up to millions of variables, make use of thousands of processors, and accommodate complex multiple-physics. As partial differential equations (PDE) solvers also mature in the Earth Sciences, there is an increasing interest in solving nonlinear seismic inversion problems governed by PDE-based models. Larger computer architectures and new algorithms for optimization and wave propagation now provide the computational ability to address the geophysical issues mentioned above in a more rigorous way: namely, to abandon asymptotic ray-theory approximations in favor of time-dependent PDE-based models, and replace linearized inversions by truly nonlinear optimization. To achieve this goal, it will be necessary to combine recent developments in computational methods for nonlinear optimization and wave propagation, such as high-order finite element discretizations, local time-stepping, iterative methods, and inexact parallel interior-point methods. More specifically, the scientific goals of the project are: to develop parallel numerical methods for forward wave propagation and large-scale nonlinear optimization, to explore the performance of such methods on emerging petascale architectures (e.g. GPU, Cell BE) and to develop a new generation of a seismic inversion code for 3D Earth imaging.
Numerical methods for stochastically driven wave equations Research Project | 3 Project MembersThe interest in stochastic partial differential equations is growing during the last decades. Unfortunately, there are very few equations for which an analytical solution is known. For those stochastic partial differential equations without a known solution, one hope is that numerical methods will help to better understand them. In this project, we will focus on numerical discretisations of stochastic wave equations. For problems such as wave propagation through the ocean, the properties of the water fluctuate randomly due to the presence of turbulence: a deterministic approach is thus not enough to describe the motion of the wave. This leads us to consider stochastic wave equations. As a model problem, we plan to study numerical discretisations of the one-dimensional stochastic wave equation, where the space-time noise is white in time and spatially correlated. This partial differential equation can be seen as a model of a nonlinear string submerged in a turbulent fluid. For the numerical discretisation of such a problem, we first discretise in space (method of lines) and then in time with a geometric integrator . After a pseudo-spectral semi-discretisation in space of the stochastic wave equation, we obtain a system of second-order stochastic differential equations which are oscillators-like equations. A deep understanding of this system of stochastic equations is primordial for a proper numerical treatment of the problem. To get more insight into the behaviour of such kind of system, we firstly concentrate on the study of efficient numerical methods for stochastic oscillators . We then adapt these numerical schemes to systems of second-order stochastic differential equations and finally we look at the full discretisation of the stochastically driven wave equation . Our new geometric integrators will permit a deeper understanding of the numerical discretisation of stochastic oscillators, which will lead to an optimal numerical treatments of stochastically driven wave equations.
Advanced Methods for Computational Electromagnetics Research Project | 2 Project MembersComputational electromagnetics (CEM) presents a number of challenges. First, electromagnetic fields tend to be highly {em oscillatory} and {em wave-dominated}; moreover, they can develop {em singularities} at material interfaces and boundaries. Second, the phenomena of interest typically involve {em complicated} geometries, {em multi-physics}, {em inhomogeneous} media, and even {em nonlinear} materials. Third, the underlying partial differential equations often need to be solved in an unbounded domain, which needs to be truncated by an {em artificial boundary} to confine the region of interest to a finite computational domain. Fourth, standard preconditioners are typically ineffective for the highly indefinite ill-conditioned sparse linear systems of equations that appear in higher frequency regimes. 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 theregion of interest to a finite computational domain.
Multiscale analysis and simulation of waves in strongly heterogeous media Research Project | 4 Project MembersWe plan to study wave propagation in heterogeneous media both in the frequency and the time dependent regime. In particular, our aims are - to study via numerical simulation the limit problem, first in one and then in higher dimensions, and thus establish a deeper understanding of the dispersive response from the heterogeneous medium, and - to devise a heterogeneous multiscale method (HMM) scheme for wave propagation in strongly heterogeneous media, which should apply in more general situations without precomputing the analytical (homogenized) limit problem. We plan to study wave propagation in heterogeneous media both in thefrequency and the time dependent regime. In particular, our aims are - to study via numerical simulation the limit problem, first in one and then in higher dimensions, and thus establish a deeper understanding of the dispersive response from the heterogeneous medium, and - to devise a heterogeneous multiscale method (HMM) scheme for wave propagation in strongly heterogeneous media, which should apply in more general situations without precomputing the analytical (homogenized) limit problem.
