A conjugate-gradient-type rational Krylov subspace method for ill-posed problems

Conjugated gradients on the normal equation (CGNE) is a popular method to regularise linear inverse problems. The idea of the method can be summarized as minimising the residuum over a suitable Krylov subspace. It is shown that using the same idea for the shift-and-invert rational Krylov subspace yields an order-optimal regularisation scheme

Data-scalable Hessian preconditioning for distributed parameter PDE-constrained inverse problems

Hessian preconditioners are the key to efficient numerical solution of large-scale distributed parameter PDE-constrained inverse problems with highly informative data. Such inverse problems arise in many applications, yet solving them remains computationally costly. With existing methods, the computational cost depends on spectral properties of the Hessian which worsen as more informative data are used to reconstruct the unknown parameter field. The best case scenario from a scientific standpoint (lots of high-quality data) is therefore the worst case scenario from a computational standpoint (large computational cost). In this dissertation, we argue that the best way to overcome this predicament is to build data-scalable Hessian/KKT preconditioners---preconditioners that perform well even if the data are highly informative about the parameter. We present a novel data-scalable KKT preconditioner for a diffusion inverse problem, a novel data-scalable Hessian preconditioner for an advection inverse problem, and a novel data-scalable domain decomposition preconditioner for an auxiliary operator that arises in connection with KKT preconditioning for a wave inverse problem. Our novel preconditioners outperform existing preconditioners in all three cases: they are robust to large numbers of observations in the diffusion inverse problem, large Peclet numbers in the advection inverse problem, and high wave frequencies in the wave inverse problem.Computational Science, Engineering, and Mathematic

SCEE 2008 book of abstracts : the 7th International Conference on Scientific Computing in Electrical Engineering (SCEE 2008), September 28 – October 3, 2008, Helsinki University of Technology, Espoo, Finland

This report contains abstracts of presentations given at the SCEE 2008 conference.reviewe

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

The application of Krylov subspace methods for the calculation of forward solutions and model sensitivities of 3D time domain marine controlled source electromagnetic problems

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

System- and Data-Driven Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques