## Schriften zur Funktionalanalysis und Geomathematik

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25

We analyze the regular oblique boundary problem for the Poisson equation on a C^1-domain with stochastic inhomogeneities. At first we investigate the deterministic problem. Since our assumptions on the inhomogeneities and coefficients are very weak, already in order to formulate the problem we have to work out properties of functions from Sobolev spaces on submanifolds. An further analysis of Sobolev spaces on submanifolds together with the Lax-Milgram lemma enables us to prove an existence and uniqueness result for weak solution to the oblique boundary problem under very weak assumptions on coefficients and inhomogeneities. Then we define the spaces of stochastic functions with help of the tensor product. These spaces enable us to extend the deterministic formulation to the stochastic setting. Under as weak assumptions as in the deterministic case we are able to prove the existence and uniqueness of a stochastic weak solution to the regular oblique boundary problem for the Poisson equation. Our studies are motivated by problems from geodesy and through concrete examples we show the applicability of our results. Finally a Ritz-Galerkin approximation is provided. This can be used to compute the stochastic weak solution numerically.

43

In this article we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients and inhomogeneities. Main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in [1]. Moreover we prove regularisation results for the weak solutions of both, the inner and the outer problem. We investigate the non-admissible direction for the oblique vector field, state results with stochastic inhomogeneities and provide a Ritz-Galerkinm approximation. The results are applicable to problems from Geomathematics, see e.g. [2] and [3].

46

In this article we combine the modern theory of Sobolev spaces with the classical theory of limit formulae and jump relations of potential theory. Also other authors proved the convergence in Lebesgue spaces for integrable functions. The achievement of this paper is the L2 convergence for the weak derivatives of higher orders. Also the layer functions F are elements of Sobolev spaces and a two dimensional suitable smooth submanifold in R3, called regular Cm-surface. We are considering the potential of the single layer, the potential of the double layer as well as their first order normal derivatives. Main tool is the convergence in Cm-norm which is proved with help of some results taken from [14]. Additionally, we need a result about the limit formulae in L2-norm, which can be found in [16], and a reduction result which we took from [19]. Moreover we prove the convergence in the Hölder spaces Cm,alpha. Finally, we give an application of the limit formulae and jump relations to Geomathematics. We generalize a density results, see e.g. [11], from L2 to Hm,2. For it we prove the limit formula for U1 in (Hm,2)' also.

1

We study a possiblity to use the structure of the regularization error for a posteriori choice of the regularization parameter. As a result, a rather general form of a selection criterion is proposed, and its relation to the heuristical quasi-optimality principle of Tikhonov and Glasko (1964), and to an adaptation scheme proposed in a statistical context by Lepskii (1990), is discussed. The advantages of the proposed criterion are illustrated by using such examples as self-regularization of the trapezoidal rule for noisy Abel-type integral equations, Lavrentiev regularization for non-linear ill-posed problems and an inverse problem of the two-dimensional profile reconstruction.

36

This paper presents a wavelet analysis of temporal and spatial variations of the Earth's gravitational potential based on tensor product wavelets. The time--space wavelet concept is realized by combining Legendre wavelets for the time domain and spherical wavelets for the space domain. In consequence, a multiresolution analysis for both, temporal and spatial resolution, is formulated within a unified concept. The method is then numerically realized by using first synthetically generated data and, finally, several real data sets.

28

We show the numerical applicability of a multiresolution method based on harmonic splines on the 3-dimensional ball which allows the regularized recovery of the harmonic part of the Earth's mass density distribution out of different types of gravity data, e.g. different radial derivatives of the potential, at various positions which need not be located on a common sphere. This approximated harmonic density can be combined with its orthogonal anharmonic complement, e.g. determined out of the splitting function of free oscillations, to an approximation of the whole mass density function. The applicability of the presented tool is demonstrated by several test calculations based on simulated gravity values derived from EGM96. The method yields a multiresolution in the sense that the localization of the constructed spline basis functions can be increased which yields in combination with more data a higher resolution of the resulting spline. Moreover, we show that a locally improved data situation allows a highly resolved recovery in this particular area in combination with a coarse approximation elsewhere which is an essential advantage of this method, e.g. compared to polynomial approximation.

8

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.

29

We introduce a method to construct approximate identities on the 2-sphere which have an optimal localization. This approach can be used to accelerate the calculations of approximations on the 2-sphere essentially with a comparably small increase of the error. The localization measure in the optimization problem includes a weight function which can be chosen under some constraints. For each choice of weight function existence and uniqueness of the optimal kernel are proved as well as the generation of an approximate identity in the bandlimited case. Moreover, the optimally localizing approximate identity for a certain weight function is calculated and numerically tested.

5

The article is concerned with the modelling of ionospheric current systems from induced magnetic fields measured by satellites in a multiscale framework. Scaling functions and wavelets are used to realize a multiscale analysis of the function spaces under consideration and to establish a multiscale regularization procedure for the inversion of the considered vectorial operator equation. Based on the knowledge of the singular system a regularization technique in terms of certain product kernels and corresponding convolutions can be formed. In order to reconstruct ionospheric current systems from satellite magnetic field data, an inversion of the Biot-Savart's law in terms of multiscale regularization is derived. The corresponding operator is formulated and the singular values are calculated. The method is tested on real magnetic field data of the satellite CHAMP and the proposed satellite mission SWARM.

7

A wavelet technique, the wavelet-Mie-representation, is introduced for the analysis and modelling of the Earth's magnetic field and corresponding electric current distributions from geomagnetic data obtained within the ionosphere. The considerations are essentially based on two well-known geomathematical keystones, (i) the Helmholtz-decomposition of spherical vector fields and (ii) the Mie-representation of solenoidal vector fields in terms of poloidal and toroidal parts. The wavelet-Mie-representation is shown to provide an adequate tool for geomagnetic modelling in the case of ionospheric magnetic contributions and currents which exhibit spatially localized features. An important example are ionospheric currents flowing radially onto or away from the Earth. To demonstrate the functionality of the approach, such radial currents are calculated from vectorial data of the MAGSAT and CHAMP satellite missions.