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Tue, 17 Jul 2012 09:05:27 +0200Tue, 17 Jul 2012 09:05:27 +0200Annual Report
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3200
Annual Report, Jahrbuch AG Magnetismusperiodicalhttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/3200Tue, 17 Jul 2012 09:05:27 +0200Mathematical aspects of stress field simulations in deep geothermal reservoirs
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2671
This report gives an insight into basics of stress field simulations for geothermal reservoirs.
The quasistatic equations of poroelasticity are deduced from constitutive equations, balance
of mass and balance of momentum. Existence and uniqueness of a weak solution is shown.
In order of to find an approximate solution numerically, usage of the so–called method of
fundamental solutions is a promising way. The idea of this method as well as a sketch of
how convergence may be proven are given.Matthias Augustinreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2671Fri, 08 Jul 2011 07:18:26 +0200Mathematische Methoden in der Geothermie
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2669
Insbesondere bei der industriellen Nutzung tiefer geothermischer Systeme gibt es Risiken, die im Hinblick auf eine zukunftsträchtige Rolle der Ressource "Geothermie" innerhalb der Energiebranche eingeschätzt und minimiert werden müssen. Zur Förderung und Unterstützung dieses Prozesses kann die Mathematik einen entscheidenden Beitrag leisten. Um dies voranzutreiben haben wir zur Charakterisierung tiefer geothermischer Systeme ein Säulenmodell entwickelt, das die Bereiche Exploration, Bau und Produktion näher beleuchtet. Im Speziellen beinhalten die Säulen: Seismische Erkundung, Gravimetrie/Geomagnetik, Transportprozesse, Spannungsfeld.Matthias Augustin; Willi Freeden; Christian Gerhards; Sandra Möhringer; Isabel Ostermannpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2669Wed, 06 Jul 2011 03:44:25 +0200Modeling Deep Geothermal Reservoirs: Recent Advances and Future Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2155
Due to the increasing demand of renewable energy production facilities, modeling geothermal reservoirs is a central issue in today's engineering practice. After over 40 years of study, many models have been proposed and applied to hundreds of sites worldwide. Nevertheless, with increasing computational capabilities new efficient methods are becoming available. The aim of this paper is to present recent progress on seismic processing as well as fluid and thermal flow simulations for porous and fractured subsurface systems. The commonly used methods in industrial energy exploration and production such as forward modeling, seismic migration, and inversion methods together with continuum and discrete flow models for reservoir monitoring and management are reviewed. Furthermore, for two specific features numerical examples are presented. Finally, future fields of studies are described.Maxim Ilyasov; Isabel Ostermann; Alessandro Punzipreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2155Tue, 08 Dec 2009 10:09:54 +0100Regularized Multiresolution Recovery of the Mass Density Distribution From Satellite Data of the Earth´s Gravitational Field
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1586
The inverse problem of recovering the Earth's density distribution from data of the first or second derivative of the gravitational potential at satellite orbit height is discussed for a ball-shaped Earth. This problem is exponentially ill-posed. In this paper a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part the second radial derivative of the gravitational potential at 200 km orbitheight is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG (satellite gravity gradiometry) satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust. Moreover, the noise sensitivity of the regularization technique is analyzed numerically.Volker Michelreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1586Tue, 16 Nov 2004 14:25:33 +0100Algebraic Systems Theory
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1567
Control systems are usually described by differential equations, but their properties of interest are most naturally expressed in terms of the system trajectories, i.e., the set of all solutions to the equations. This is the central idea behind the so-called "behavioral approach" to systems and control theory. On the other hand, the manipulation of linear systems of differential equations can be formalized using algebra, more precisely, module theory and homological methods ("algebraic analysis"). The relationship between modules and systems is very rich, in fact, it is a categorical duality in many cases of practical interest. This leads to algebraic characterizations of structural systems properties such as autonomy, controllability, and observability. The aim of these lecture notes is to investigate this module-system correspondence. Particular emphasis is put on the application areas of one-dimensional rational systems (linear ODE with rational coefficients), and multi-dimensional constant systems (linear PDE with constant coefficients).Eva Zerzreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1567Wed, 25 Aug 2004 09:34:44 +0200Multiscale Modeling of CHAMP-Data
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1457
The following three papers present recent developments in multiscale gravitational field modeling by the use of CHAMP or CHAMP-related data. Part A - The Model SWITCH-03: Observed orbit perturbations of the near-Earth orbiting satellite CHAMP are analyzed to recover the long-wavelength features of the Earth's gravitational potential. More precisely, by tracking the low-flying satellite CHAMP by the high-flying satellites of the Global Positioning System (GPS) a kinematic orbit of CHAMP is obtainable from GPS tracking observations, i.e. the ephemeris in cartesian coordinates in an Earth-fixed coordinate frame (WGS84) becomes available. In this study we are concerned with two tasks: First we present new methods for preprocessing, modelling and analyzing the emerging tracking data. Then, in a first step we demonstrate the strength of our approach by applying it to simulated CHAMP orbit data. In a second step we present results obtained by operating on a data set derived from real CHAMP data. The modelling is mainly based on a connection between non-bandlimited spherical splines and least square adjustment techniques to take into account the non-sphericity of the trajectory. Furthermore, harmonic regularization wavelets for solving the underlying Satellite-to-Satellite Tracking (SST) problem are used within the framework of multiscale recovery of the Earth's gravitational potential leading to SWITCH-03 (Spline and Wavelet Inverse Tikhonov regularized CHamp data). Further it is shown how regularization parameters can be adapted adequately to a specific region improving a globally resolved model. Finally we give a comparison of the developed model to the EGM96 model, the model UCPH2002_02_0.5 from the University of Copenhagen and the GFZ models EIGEN-1s and EIGEN-2. Part B - Multiscale Solutions from CHAMP: CHAMP orbits and accelerometer data are used to recover the long- to medium- wavelength features of the Earth's gravitational potential. In this study we are concerned with analyzing preprocessed data in a framework of multiscale recovery of the Earth's gravitational potential, allowing both global and regional solutions. The energy conservation approach has been used to convert orbits and accelerometer data into in-situ potential. Our modelling is spacewise, based on (1) non-bandlimited least square adjustment splines to take into account the true (non-spherical) shape of the trajectory (2) harmonic regularization wavelets for solving the underlying inverse problem of downward continuation. Furthermore we can show that by adapting regularization parameters to specific regions local solutions can improve considerably on global ones. We apply this concept to kinematic CHAMP orbits, and, for test purposes, to dynamic orbits. Finally we compare our recovered model to the EGM96 model, and the GFZ models EIGEN-2 and EIGEN-GRACE01s. Part C - Multiscale Modeling from EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, EGM96: Spherical wavelets have been developed by the Geomathematics Group Kaiserslautern for several years and have been successfully applied to georelevant problems. Wavelets can be considered as consecutive band-pass filters and allow local approximations. The wavelet transform can also be applied to spherical harmonic models of the Earth's gravitational field like the most up-to-date EIGEN-1S, EIGEN-2, EIGEN-GRACE01S, UCPH2002_0.5, and the well-known EGM96. Thereby, wavelet coefficients arise and these shall be made available to other interested groups. These wavelet coefficients allow the reconstruction of the wavelet approximations. Different types of wavelets are considered: bandlimited wavelets (here: Shannon and Cubic Polynomial (CP)) as well as non-bandlimited ones (in our case: Abel-Poisson). For these types wavelet coefficients are computed and wavelet variances are given. The data format of the wavelet coefficients is also included.W. Freeden; V. Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1457Mon, 08 Dec 2003 11:25:15 +0100Regularized Multiresolution Recovery of the Mass Density Distribution from Satellite Data of the Earth's Gravitational Field
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1413
The inverse problem of recovering the Earth's density distribution from satellite data of the first or second derivative of the gravitational potential at orbit height is discussed. This problem is exponentially ill-posed. In this paper a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part the second radial derivative of the gravitational potential at 200 km orbit height is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust.Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1413Tue, 15 Jul 2003 13:04:31 +0200Unde tales melancholici sunt, et optimi fiunt mathematici (H. v. Gent) Zur Interaktion von Mathematik und Melancholie
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1263
Knut Radbruchpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1263Tue, 20 Nov 2001 00:00:00 +0100Presentation of power-ordered sets
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1259
Power-ordered sets are not always lattices. In the case of distributive lattices we give a description by disjoint of chains. Finite power-ordered sets have a polarity. We introduct the leveled lattices and show examples with trivial tolerance. Finally we give a list of Hasse diagrams of power-ordered sets.Dietmar Schweigertpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1259Fri, 07 Sep 2001 00:00:00 +0200The finite-section approximation for ill-posed integral equations on the half-line
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1260
Integral equations on the half of line are commonly approximated by the finite-section approximation, in which the infinite upper limit is replaced by apositie number called finite-section parameter. In this paper we consider the finite-section approximation for first kind intgral equations which are typically ill-posed and call for regularization. For some classes of such equations corresponding to inverse problems from optics and astronomy we indicate the finite-section parameters that allows to apply standard regularization techniques. Two discretization schemes for the finite-section equations ar also proposed and their efficiency is studied.Sergei Pereverzev; Eberhard Schockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1260Fri, 07 Sep 2001 00:00:00 +0200Maximal Cohen-Macaulay Modules over the Cone of an Elliptic Curve
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1261
Radu Laza; Gerhard Pfister; Popescu Dorinpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1261Fri, 07 Sep 2001 00:00:00 +0200Derived Tameness of some Associative Algebras
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1262
In this article we propose a new method to deal with the derived categories of some semi-perfect k-algebras. This method gives us a unifying description of the derived categories of a branch class of algebras such as gentle algebras, some lannish algebras, of the Gelfand quiver etc.Igor Burban; Yuriy Drozdpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1262Fri, 07 Sep 2001 00:00:00 +0200Construction of Moduli Spaces for Space Curve Singularities of Multiplicity 3
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1257
Anne Frühbis-Krügerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1257Thu, 31 May 2001 00:00:00 +0200Kryptanalyse der Verschlüsselungsalgorithmen der Microsoft Word Versionen 2.0-7.0
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1258
Roland Fiat; Andreas Guthmann; Georg Kuxpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1258Thu, 31 May 2001 00:00:00 +0200Geophysical Field Modelling by Multiresolution Analysis
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/628
Wavelet transform originated in 1980's for the analysis of seismic signals has seen an explosion of applications in geophysics. However, almost all of the material is based on wavelets over Euclidean spaces. This paper deals with the generalization of the theory and algorithmic aspects of wavelets to a spherical earth's model and geophysically relevant vector fields such as the gravitational, magnetic, elastic field of the earth.A scale discrete wavelet approach is considered on the sphere thereby avoiding any type of tensor-valued 'basis (kernel) function'. The generators of the vector wavelets used for the fast evaluation are assumed to have compact supports. Thus the scale and detail spaces are finite-dimensional. As an important consequence, detail information of the vector field under consideration can be obtained only by a finite number of wavelet coefficients for each scale. Using integration formulas that are exact up to a prescribed polynomial degree, wavelet decomposition and reconstruction are investigated for bandlimited vector fields. A pyramid scheme for the recursive computation of the wavelet coefficients from level to level is described in detail. Finally, data compression is discussed for the EGM96 model of the earth's gravitational field.Michael Bayer; Stefan Beth; Willi Freedenpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/628Mon, 03 Apr 2000 00:00:00 +0200A: New Wavelet Methods for Approximating Harmonic Functions; B: Satellite Gradiometry - from Mathematical and Numerical Point of View
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/537
Some new approximation methods are described for harmonic functions corresponding to boundary values on the (unit) sphere. Starting from the usual Fourier (orthogonal) series approach, we propose here nonorthogonal expansions, i.e. series expansions in terms of overcomplete systems consisting of localizing functions. In detail, we are concerned with the so-called Gabor, Toeplitz, and wavelet expansions. Essential tools are modulations, rotations, and dilations of a mother wavelet. The Abel-Poisson kernel turns out to be the appropriate mother wavelet in approximation of harmonic functions from potential values on a spherical boundary.Willi Freeden; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/537Mon, 03 Apr 2000 00:00:00 +0200Spherical Wavelet Transform and its Discretization
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/555
A continuous version of spherical multiresolution is described, starting from continuous wavelet transform on the sphere. Scale discretization enables us to construct spherical counterparts to Daubechies wavelets and wavelet packets (known from Euclidean theory). Essential tool is the theory of singular integrals on the sphere. It is shown that singular integral operators forming a semigroup of contraction operators of class (Co) (like Abel-Poisson or Gauß-Weierstraß operators) lead in canonical way to (pyramidal) algorithms.Willi Freeden; U. Windheuserpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/555Mon, 03 Apr 2000 00:00:00 +0200An Adaptive Hierarchical Approximation Method on the Sphere Using Axisymmetric Locally Supported Basis Functions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/561
The paper discusses the approximation of scattered data on the sphere which is one of the major tasks in geomathematics. Starting from the discretization of singular integrals on the sphere the authors devise a simple approximation method that employs locally supported spherical polynomials and does not require equidistributed grids. It is the basis for a hierarchical approximation algorithm using differently scaled basis functions, adaptivity and error control. The method is applied to two examples one of which is a digital terrain model of Australia.Willi Freeden; J. Fröhhlich; R. Brandpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/561Mon, 03 Apr 2000 00:00:00 +0200Deformation Analysis Using Navier Spline Interpolation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/567
The static deformation of the surface of the earth caused by surface pressure like the water load of an ocean or an artificial lake is discussed. First a brief mention is made on the solution of the Boussenesq problem for an infinite halfspace with the elastic medium to be assumed as homogeneous and isotropic. Then the elastic response for realistic earth models is determinied by spline interpolation using Navier splines. Major emphasis is on the derteminination of the elastic field caused by water loads from surface tractions on the (real) earth" s surface. Finally the elastic deflection of an artificial lake assuming a homogeneous isotropic crust is compared for both evaluation methods.Willi Freeden; E. Groten; Michael Schreiner; W. Söhhne; M. Tücckspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/567Mon, 03 Apr 2000 00:00:00 +0200Equidistribution on the Sphere
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/574
A concept of generalized discrepancy, which involves pseudodifferential operators to give a criterion of equidistributed pointsets, is developed on the sphere. A simply structured formula in terms of elementary functions is established for the computation of the generalized discrepancy. With the help of this formula five kinds of point systems on the sphere, namely lattices in polar coordinates, transformed 2-dimensional sequences, rotations on the sphere, triangulation, and sum of three squares sequence, are investigated. Quantitative tests are done, and the results are compared with each other. Our calculations exhibit different orders of convergence of the generalized discrepancy for different types of point systems.Willi Freeden; J. Cuipreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/574Mon, 03 Apr 2000 00:00:00 +0200Combined Spherical Harmonic and Wavelet Expansion - a Future Concepts in Earth" s Gravitational Determination
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/576
The basic theory of spherical singular integrals is recapitulated. Criteria are given for measuring the space-frequency localization of functions on the sphere. The trade off between space localization on the sphere and frequency localization in terms of spherical harmonics is described in form of an uncertainty principle. A continuous version of spherical multiresolution is introduced, starting from continuous wavelet transform corresponding to spherical wavelets with vanishing moments up to a certain order. The wavelet transform is characterized by least-squares properties. Scale discretization enables us to construct spherical counterparts of wavelet packets and scale discrete Daubechies" wavelets. It is shown that singular integral operators forming a semigroup of contraction operators of class (Co) (like Abel-Poisson or Gauß-Weierstraß operators) lead in canonical way to pyramyd algorithms. Fully discretized wavelet transforms are obtained via approximate integration rules on the sphere. Finally applications to (geo-)physical reality are discussed in more detail. A combined method is proposed for approximating the low frequency parts of a physical quantity by spherical harmonics and the high frequency parts by spherical wavelets. The particular significance of this combined concept is motivated for the situation of today" s physical geodesy, viz. the determination of the high frequency parts of the earth" s gravitational potential under explicit knowledge of the lower order part in terms of a spherical harmonic expansion.Willi Freeden; U. Windheuserpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/576Mon, 03 Apr 2000 00:00:00 +0200A Survey on Spherical Spline Approximation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/590
Spline functions that approximate data given on the sphere are developed in a weighted Sobolev space setting. The flexibility of the weights makes possible the choice of the approximating function in a way which emphasizes attributes desirable for the particular application area. Examples show that certain choices of the weight sequences yield known methods. A convergence theorem containing explicit constants yields a usable error bound. Our survey ends with the discussion of spherical splines in geodetically relevant pseudodifferential equations.Willi Freeden; Michael Schreiner; R. Frankepreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/590Mon, 03 Apr 2000 00:00:00 +0200Gradiometry - an Inverse Problem in Modern Satellite Geodesy
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/595
Satellite gradiometry and its instrumentation is an ultra-sensitive detection technique of the space gravitational gradient (i.e. the Hesse tensor of the gravitational potential). Gradeometry will be of great significance in inertial navigation, gravity survey, geodynamics and earthquake prediction research. In this paper, satellite gradiometry formulated as an inverse problem of satellite geodesy is discussed from two mathematical aspects: Firstly, satellite gradiometry is considered as a continuous problem of harmonic downward continuation. The space-borne gravity gradients are assumed to be known continuously over the satellite (orbit) surface. Our purpose is to specify sufficient conditions under which uniqueness and existence can be guaranteed. It is shown that, in a spherical context, uniqueness results are obtainable by decomposition of the Hesse matrix in terms of tensor spherical harmonics. In particular, the gravitational potential is proved to be uniquely determined if second order radial derivatives are prescribed at satellite height. This information leads us to a reformulation of satellite gradiometry as a (Fredholm) pseudodifferential equation of first kind. Secondly, for a numerical realization, we assume the gravitational gradients to be known for a finite number of discrete points. The discrete problem is dealt with classical regularization methods, based on filtering techniques by means of spherical wavelets. A spherical singular integral-like approach to regularization methods is established, regularization wavelets are developed which allow the regularization in form of a multiresolution analysis. Moreover, a combined spherical harmonic and spherical regularization wavelet solution is derived as an appropriate tool in future (global and local) high-presision resolution of the earth" s gravitational potential.Willi Freeden; F. Schneider; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/595Mon, 03 Apr 2000 00:00:00 +0200Orthogonal and non-orthogonal multiresolution analysis, scale discrete and exact fully discrete wavelet transform on the sphere
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/597
Based on a new definition of delation a scale discrete version of spherical multiresolution is described, starting from a scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen. In particular, spherical Shannon wavelets are constructed that form an orthogonal multiresolution analysis. Finally fully discrete wavelet approximation is discussed in case of band-limited wavelets.Willi Freeden; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/597Mon, 03 Apr 2000 00:00:00 +0200