## Schriften zur Funktionalanalysis und Geomathematik

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- Preprint (34)
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#### Keywords

- Wavelet (8)
- Inverses Problem (7)
- Mehrskalenanalyse (6)
- Approximation (4)
- Regularisierung (4)
- Galerkin-Methode (3)
- Gravimetrie (3)
- Gravitationsfeld (3)
- Kugelflächenfunktion (3)
- Spline (3)

- 30
- Locally Supported Approximate Identities on the Unit Ball (2006)
- We present a constructive theory for locally supported approximate identities on the unit ball in \(\mathbb{R}^3\). The uniform convergence of the convolutions of the derived kernels with an arbitrary continuous function \(f\) to \(f\), i.e. the defining property of an approximate identity, is proved. Moreover, an explicit representation for a class of such kernels is given. The original publication is available at www.springerlink.com

- 33
- Splines on the 3-dimensional Ball and their Application to Seismic Body Wave Tomography (2007)
- In this paper we construct spline functions based on a reproducing kernel Hilbert space to interpolate/approximate the velocity field of earthquake waves inside the Earth based on traveltime data for an inhomogeneous grid of sources (hypocenters) and receivers (seismic stations). Theoretical aspects including error estimates and convergence results as well as numerical results are demonstrated.

- 50
- Mathematical aspects of stress field simulations in deep geothermal reservoirs (2011)
- 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.

- 49
- Mathematische Methoden in der Geothermie (2011)
- 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.

- 17
- Split Operators for Oblique Boundary Value Problems (2005)
- In the field of gravity determination a special kind of boundary value problem respectively ill-posed satellite problem occurs; the data and hence side condition of our PDE are oblique second order derivatives of the gravitational potential. In mathematical terms this means that our gravitational potential \(v\) fulfills \(\Delta v = 0\) in the exterior space of the Earth and \(\mathscr D v = f\) on the discrete data location which is on the Earth's surface for terrestrial measurements and on a satellite track in the exterior for spaceborne measurement campaigns. \(\mathscr D\) is a first order derivative for methods like geometric astronomic levelling and satellite-to-satellite tracking (e.g. CHAMP); it is a second order derivative for other methods like terrestrial gradiometry and satellite gravity gradiometry (e.g. GOCE). Classically one can handle first order side conditions which are not tangential to the surface and second derivatives pointing in the radial direction employing integral and pseudo differential equation methods. We will present a different approach: We classify all first and purely second order operators \(\mathscr D\) which fulfill \(\Delta \mathscr D v = 0\) if \(\Delta v = 0\). This allows us to solve the problem with oblique side conditions as if we had ordinary i.e. non-derived side conditions. The only additional work which has to be done is an inversion of \(\mathscr D\), i.e. integration.

- 24
- Easy Differentiation and Integration of Homogeneous Harmonic Polynomials (2005)
- We will give explicit differentiation and integration rules for homogeneous harmonic polynomial polynomials and spherical harmonics in IR^3 with respect to the following differential operators: partial_1, partial_2, partial_3, x_3 partial_2 - x_2 partial_3, x_3 partial_1 - x_1 partial_3, x_2 partial_1 - x_1 partial_2 and x_1 partial_1 + x_2 partial_2 + x_3 partial_3. A numerical application to the problem of determining the geopotential field will be shown.

- 3
- A Tree Algorithm for Isotropic Finite Elements on the Sphere (2003)
- The Earth's surface is an almost perfect sphere. Deviations from its spherical shape are less than 0,4% of its radius and essentially arise from its rotation. All equipotential surfaces are nearly spherical, too. In consequence, multiscale modelling of geoscientifically relevant data on the sphere involving rotational symmetry of the trial functions used for the approximation plays an important role. In this paper we deal with isotropic kernel functions showing local support and (one-dimensional) polynomial structure (briefly called isotropic finite elements) for reconstructing square--integrable functions on the sphere. Essential tool is the concept of multiresolution analysis by virtue of the spherical up function. The main result is a tree algorithm in terms of (low--order) isotropic finite elements.

- 44
- Spherical Fast Multiscale Approximation by Locally Compact Orthogonal Wavelets (2009)
- Using a stereographical projection to the plane we construct an O(N log(N)) algorithm to approximate scattered data in N points by orthogonal, compactly supported wavelets on the surface of a 2-sphere or a local subset of it. In fact, the sphere is not treated all at once, but is split into subdomains whose results are combined afterwards. After choosing the center of the area of interest the scattered data points are mapped from the sphere to the tangential plane through that point. By combining a k-nearest neighbor search algorithm and the two dimensional fast wavelet transform a fast approximation of the data is computed and mapped back to the sphere. The algorithm is tested with nearly 1 million data points and yields an approximation with 0.35% relative errors in roughly 2 minutes on a standard computer using our MATLAB implementation. The method is very flexible and allows the application of the full range of two dimensional wavelets.

- 13
- Regularization without Preliminary Knowledge of Smoothness and Error Behavior (2004)
- The mathematical formulation of many physical problems results in the task of inverting a compact operator. The only known sensible solution technique is regularization which poses a severe problem in itself. Classically one dealt with deterministic noise models and required both the knowledge of smoothness of the solution function and the overall error behavior. We will show that we can guarantee an asymptotically optimal regularization for a physically motivated noise model under no assumptions for the smoothness and rather weak assumptions on the noise behavior which can mostly obtained out of two input data sets. An application to the determination of the gravitational field out of satellite data will be shown.

- 41
- On Mathematical Aspects of a Combined Inversion of Gravity and Normal Mode Variations by a Spline Method (2008)
- This paper provides a brief overview of two linear inverse problems concerned with the determination of the Earth’s interior: inverse gravimetry and normal mode tomography. Moreover, a vector spline method is proposed for a combined solution of both problems. This method uses localised basis functions, which are based on reproducing kernels, and is related to approaches which have been successfully applied to the inverse gravimetric problem and the seismic traveltime tomography separately.