Near-optimal perfectly matched layers for indefinite Helmholtz problems

A new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, designed with the help of a classical result by Zolotarev. Using Krein's interpretation of a Stieltjes continued fraction, this interpolant can be converted into a three-term finite difference discretization of a perfectly matched layer (PML) which converges exponentially fast in the number of grid points. The convergence rate is asymptotically optimal for both propagative and evanescent wave modes. Several numerical experiments and illustrations are included.Comment: Accepted for publication in SIAM Review. To appear 201

Efficient domain decomposition algorithms for the solution of the helmholtz equation

The purpose of this thesis is to formulate and investigate new iterative methods for the solution of scattering problems based on the domain decomposition approach. This work is divided into three parts. In the first part, a new domain decomposition method for the perfectly matched layer system of equations is presented. Analysis of a simple model problem shows that the convergence of the new algorithm is guaranteed provided that a non-local, square-root transmission operator is used. For efficiency, in practical simulations such operators need to be localized. Current, state of the art domain decomposition algorithms use the localization technique based on rational approximation of the symbol of the transmission operator. However, the original formulation of the procedure assumed decompositions that contain no cross-points and consequently could not be used in the cross-point algorithm. In the context of the perfectly matched layer problem, we adapt the cross-point technique and combined with the rational approximation of the square root transmission operator to yield an effective algorithm. Furthermore, to reduce Krylov subspace iterations, we present a new, adequate and efficient preconditioner for the perfectly matched layer problem. The new, zero frequency limit preconditioner shows great reduction in the required number of iterations while being extremely easy to construct. In the second part of the thesis, a new domain decomposition algorithm is considered. From theoretical point of view, its formulation guarantees well-posedness of local problems. Its practicality on the other hand is evident from its efficiency and ease of implementations as compared with other, state of the art domain decomposition approaches. Moreover, the method exhibits robustness with respect to the problem frequency and is suitable for large scale simulations on a parallel computer. Finally, the third part of the thesis presents an extensible, object oriented architecture that supports development of parallel domain decomposition algorithms where local problems are solved by the finite element method. The design hides mesh implementation details and is capable of supporting various families of finite elements together with quadrature formulas of suitable degree of precision

An introduction to operator preconditioning for the fast iterative integral equation solution of time-harmonic scattering problems

International audienceThe aim of this paper is to provide an introduction to the improved iterative Krylov solution of boundary integral equations for time-harmonic scattering problems arising in acoustics, electromagnetism and elasticity. From the point of view of computational methods, considering large frequencies is a challenging issue in engineering since it leads to solving highly indefinite large scale complex linear systems which generally implies a convergence breakdown of iterative methods. More specifically, we explain the problematic and some partial solutions through analytical preconditioning for high-frequency acoustic scattering problems and the introduction of new combined field integral equations. We complete the paper with some recent extensions to the case of electromagnetic and elastic waves equations

Die Anwendung von Krylov Unterraum Methoden zur Berechnung von Forwärts Lösungen und Model Sensitivitäten von 3D mariner, aktiver elektromagnetischer Probleme im Zeitbereich

To reduce the run-times of 3D modeling and inversion software for the interpretation of marine controlled source electromagnetics (CSEM) in time domain, the implementation of efficient algorithms on massive parallel hardware is presented. Two forward modeling implementations as well as an implementation for sensitivity calculation are illustrated. The first forward code is an implementation of the spectral Lanczos decomposition method on a graphics processing unit (GPU). The applicability of the code for a CSEM system, how it is used at GEOMAR, is demonstrated. In the second forward code, the SLDM is replaced by the more efficient Rational Krylov Subspace Method (RKSM). This reduces the dimension and run-time of the problem drastically. The accuracy of the code is investigated for different models and conductivity contrasts. The run-times of SLDM and RKSM are compared on different architectures. The sensitivities are computed with the MOR-method (Model Order Reduction). It is shown that the method works and the applicability to a real data set is shown.Zur Reduzierung der Laufzeiten von 3D Modellierungs- und Inversions-Software für die Interpretation von mariner, aktiver Elektromagnetik (engl. CSEM, controlled source electro magnetics) im Zeitbereich, werden effiziente Algorithmen und Implementierungen auf massiv-paralleler Hardware vorgestellt. Zwei Implementierungen zur Berechnung der Vorwärts Modellierung, sowie eine Implementierung zur Berechnung der Sensitivitäten werden dargestellt. Bei dem ersten Vorwärts Code handelt es sich um eine Implementierung der Spektralen Lanczos Zerlegung (engl. SLDM, Spectral Lanczos Decomposition Method) auf dem Prozessor von Graphik Karten (engl. GPU, Graphics Processing Unit). Die Anwendbarkeit des Codes wird für ein CSEM System demonstriert, wie es am GEOMAR im Einsatz ist. Bei dem Zweiten Vorwärts Code wird die SLDM durch das effektivere Rationale Krylov Unterraum Verfahren (engl. RKSM, Rational Krylov Subspace Method) ersetzt. Die Genauigkeit des Codes wird für verschiedene Modelle und Kontraste des elektrischen Leitwertes untersucht. Ein Laufzeitvergleich von SLDM und RKSM wird gegeben.Die Sensitivitäten werden mit dem MOR-Verfahren (engl. Model Order Reduction) berechnet. Es wird gezeigt, dass die Methode funktioniert und seine Anwendbarkeit auf einen echten Datensatz demonstriert