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Inversion methods and resolution analysis for the 2D/3D reconstruction of resistivity structures from DC measurementsGünther, Thomas 25 November 2009 (has links) (PDF)
The presented thesis deals with the multi-dimensional reconstruction of the earth's conductivity distribution based on DC resistivity data. This task represents a nonlinear and ill-posed minimization problem with many degrees of freedom. In this work, techniques for regularization and controlling of this problem are depicted and classified. Particularly, it is concentrated on explicit regularization types, which impose constraints onto the model. The system of equations as resulting from the application of the Gauss-Newton minimization can be solved efficiently. Furthermore, it is shown how the regularization strength can be controlled. The method of non-linear resolution analysis plays a central role in the thesis. It represents a powerful tool to estimate the quality of inversion results. Furthermore, the derived resolution measures provide the basis for the optimization of experimental design concerning information content and efficiency. Methods of error estimation, forward modeling and the calculation of the Jacobian matrix for DC resistivity data are developed. Procedures for appropriate parameterization and inversion control are pointed out by studies of synthetic models. Different inversion and regularization methods are examined in detail. A linearized study is used to compare different data sets considering their efficiency. Moreover, a triplegrid-technique for the incorporation of topography into three-dimensional inversion is presented. Finally the inversion methods are applied to field data. The depicted optimization strategies are realized in practice, which increases the economic relevance of threedimensional data acquisition. The structure of the subsurface is imaged in detail for several applications in the fields of cavity detection, archaeology and the investigation of ground falls. The resolution analysis is successfully established to appraise the obtained results.
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Multi-level solver for degenerated problems with applications to p-versions of the femBeuchler, Sven 18 July 2003 (has links) (PDF)
Dissertation ueber die effektive Vorkonditionierung linearer Gleichungssysteme
resultierend aus der Diskretisierung eines elliptischen Randwertproblems 2. Ordnung mittels
der Methode der Finiten Elementen.
Als Vorkonditionierer werden multi-level artige Vorkonditionierer (BPX, Multi-grid, Wavelets) benutzt.
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Multiresolution weighted norm equivalences and applicationsBeuchler, Sven, Schneider, Reinhold, Schwab, Christoph 05 April 2006 (has links) (PDF)
We establish multiresolution norm equivalences in
weighted spaces <i>L<sup>2</sup><sub>w</sub></i>((0,1))
with possibly singular weight functions <i>w(x)</i>≥0
in (0,1).
Our analysis exploits the locality of the
biorthogonal wavelet basis and its dual basis
functions. The discrete norms are sums of wavelet
coefficients which are weighted with respect to the
collocated weight function <i>w(x)</i> within each scale.
Since norm equivalences for Sobolev norms are by now
well-known, our result can also be applied to
weighted Sobolev norms. We apply our theory to
the problem of preconditioning <i>p</i>-Version FEM
and wavelet discretizations of degenerate
elliptic problems.
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Fast solvers for degenerated problemsBeuchler, Sven 11 April 2006 (has links) (PDF)
In this paper, finite element discretizations of the
degenerated operator
-ω<sup>2</sup>(y) u<sub>xx</sub>-ω<sup>2</sup>(x)u<sub>yy</sub>=g
in the unit square are investigated, where the
weight function satisfies ω(ξ)=ξ<sup>α</sup>
with α ≥ 0.
We propose two multi-level methods in order to
solve the resulting system of linear algebraic
equations. The first method is a multi-grid
algorithm with line-smoother.
A proof of the smoothing property is given.
The second method is a BPX-like preconditioner
which we call MTS-BPX preconditioner.
We show that the upper eigenvalue bound of the
MTS-BPX preconditioned system matrix grows
proportionally to the level number.
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Contributions to the Minimal Realization Problem for Descriptor SystemsSokolov, Viatcheslav 15 June 2006 (has links) (PDF)
In this thesis we have studied several aspects of the minimal realization problem
for descriptor systems. These aspects include purely theoretical questions
such as that about the order of a minimal realization of a general improper
rational matrix and problems of a numerical nature, like rounding error analysis
of the computing a minimal realization from a nonminimal one. We have
also treated the minimal partial realization problem for general descriptor
systems with application to model reduction and to generalised eigenvalue
problems.
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Efficient Numerical Solution of Large Scale Algebraic Matrix Equations in PDE Control and Model Order ReductionSaak, Jens 21 October 2009 (has links) (PDF)
Matrix Lyapunov and Riccati equations are an important tool in mathematical systems theory. They are the key ingredients in balancing based model order reduction techniques and linear quadratic regulator problems. For small and moderately sized problems these equations are solved by techniques with at least cubic complexity which prohibits their usage in large scale applications.
Around the year 2000 solvers for large scale problems have been introduced. The basic idea there is to compute a low rank decomposition of the quadratic and dense solution matrix and in turn reduce the memory and computational complexity of the algorithms. In this thesis efficiency enhancing techniques for the low rank alternating directions implicit iteration based solution of large scale matrix equations are introduced and discussed. Also the applicability in the context of real world systems is demonstrated.
The thesis is structured in seven central chapters. After the introduction chapter 2 introduces the basic concepts and notations needed as fundamental tools for the remainder of the thesis. The next chapter then introduces a collection of test examples spanning from easily scalable academic test systems to badly conditioned technical applications which are used to demonstrate the features of the solvers. Chapter four and five describe the basic solvers and the modifications taken to make them applicable to an even larger class of problems. The following two chapters treat the application of the solvers in the context of model order reduction and linear quadratic optimal control of PDEs. The final chapter then presents the extensive numerical testing undertaken with the solvers proposed in the prior chapters.
Some conclusions and an appendix complete the thesis.
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Nichtlineare Effekte in chemischen Reaktoren und Trennapparaten /Zeyer, Klaus Peter. January 2009 (has links)
Zugl.: Magdeburg, Universiẗat, Habil.-Schr., 2009.
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Grundgleichungen und adaptive Finite-Elemente-Simulation bei "Großen Deformationen"Meyer, Arnd 27 November 2007 (has links) (PDF)
Eine einfache Darstellung der Grundgleichungen für
'Große Deformationen' und Herleitung eines geeigneten
Fehlerschätzers für die adaptive FEM.
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Efficiency improving implementation techniques for large scale matrix equation solversKöhler, Martin, Saak, Jens 11 June 2010 (has links) (PDF)
We address the important field of large scale matrix based algorithms in control and model order reduction. Many important tools from theory and applications in systems theory have been widely ignored during the recent decades in the context of PDE constraint optimal control problems and simulation of electric circuits. Often this is due to the fact that large scale matrices are suspected to be unsolvable in large scale applications. Since around 2000 efficient low rank theory for matrix equation solvers exists for sparse and also data sparse systems. Unfortunately upto now only incomplete or experimental Matlab implementations of most of these solvers have existed. Here we aim on the implementation of these algorithms in a higher programming language (in our case C) that allows for a high performance solver for many matrix equations arising in the context of large scale standard and generalized state space systems. We especially focus on efficient memory saving data structures and implementation techniques as well as the shared memory parallelization of the underlying algorithms.
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On the resolvent of the Laplacian on functions for degenerating surfaces of finite geometry / Über die Resolvente des Laplace-Operators auf Funktionen für degenerierende Flächen endlicher GeometrieSchulze, Michael 13 October 2004 (has links)
No description available.
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