Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

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DMV Tagung 2011 - Köln, 19. - 22. September Zdeněk Martinec Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin The adjoint sensitivity method of global electromagnetic induction for CHAMP magnetic data Martinec and McCreadie (2004) developed a time-domain spectral-finite element approach for the forward modelling of electromagnetic induction vector data as measured by the CHAMP satellite. Here, we present a new method of computing the sensitivity of the CHAMP electromagnetic induction data on the Earth’s mantle electrical conductivity, which we term the adjoint sensitivity method. The forward and adjoint initial boundary-value problems, both solved in the time domain, are identical, except for the specification of prescribed boundary conditions. The respective boundary-value data at the satellite’s altitude are the X magnetic component measured by the CHAMP vector magnetometer along satellite tracks for the forward method and the difference between the measured and predicted Z magnetic component for the adjoint method. The squares of these differences summed up over all CHAMP tracks determine the misfit. The sensitivity of the CHAMP data, that is the partial derivatives of the misfit function with respect to mantle conductivity parameters, are then determined by the scalar product of the forward and adjoint solutions, multiplied by the gradient of the conductivity and integrated over all CHAMP tracks. Such exactly determined sensitivities are checked against numerical differentiation of the misfit, and very good agreement is obtained. Literatur Martinec, Z. and McCreadie, H. (2004). Electromagnetic induction modelling based on satellite magnetic vector data. Geophys. J. Int., 157, 1045 - 1060. Isabel Ostermann Fraunhofer ITWM Kaiserslautern Modeling Heat Transport in Deep Geothermal Systems by Radial Basis Functions Geothermal power uses the intrinsic heat which is stored in the accessible part of the Earth’s crust. Its importance among the renewable energy resources originates from the almost unlimited energy supply of the Earth and its independence from external influences such as seasonal or even daily climatic variability. Nevertheless, there are risks which have to be assessed. In particular, local depletion poses a significant risk during the industrial utilization of deep geothermal reservoirs. In order to reduce this risk, reliable techniques to predict the heat transport and the production temperature are required. To this end, a 3D-model to simulate the heat transport in hydrothermal systems is developed which is based on a transient advection-diffusion-equation for a 2-phase porous medium. The existence, uniqueness, and continuity of the weak solution of the resulting initial boundary value problem is verified. For the numerical realization, a linear Galerkin scheme is introduced on the basis of scalar kernels. Exemplary applications of this method are investigated for the biharmonic kernel as well as appropriate geometric representations of a hydrothermal reservoir. Moreover, numerical integration methods on geoscientifically relevant bounded regions in 3D are introduced and tested for the considered geometries. 156
DMV Tagung 2011 - Köln, 19. - 22. September Sergei Pereverzev Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz Multiparameter Regularization in Geodetic Data Processing We are going to discuss recent developments in multiparameter regularization. The need in this approach becomes apparent when several model uncertainties affect data processing. Focusing on the context of satellite geodesy we discuss theoretical and computational aspects of some multiparameter regularization schemes. Numerical illustrations with synthetic data will be also presented. Literatur Lu, S. and Pereverzev, S. (2010). Multiparameter Regularization in Downward Continuation of Satellite Data. Handbook of Geomathematics, Springer, 813 - 832, Chapter 27. Robert Plato Universität Siegen The regularizing properties of some quadrature methods for linear weakly singular Volterra integral equations of the first kind The subject of this talk is the stable quadrature of the following class of linear weakly singular Volterra integral equations of the first kind: � x 0 (x − y) −(1−α) k(x,y)u(y)dy = f (x) for 0 ≤ x ≤ 1, with some parameter 0 < α < 1 and a sufficiently smooth kernel function k : [0,1] × [0,1] → R. In addition, f : [0,1] → R denotes a given function, and u : [0,1] → R is the unknown function. Problems of this kind arise, e.g., in the inversion of seismic flat-earth travel times. The quadrature methods under consideration are the composite trapezoidal scheme and the composite midpoint rule. In the present talk we consider their regularizing properties, i. e., we discuss appropriate choices of the step size as a function of the noise level for the right-hand side of the considered equation. Different smoothness assumptions on the involved functions are taken into account. Finally some numerical results are presented. Literatur Eggermont, P.P.B. (1981). A new analysis of the trapezoidal-discretization method for the numerical solution of Abel-type integral equations. J. Integral Equations, 3, 317–332. Plato, R. (to appear). The regularizing properties of the composite trapezoidal method for weakly singular Volterra integral equations of the first kind. Adv. Comput. Math. 157
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Ulrich Derenthal Ludwig-Maximilians
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Eleutherius Symeonidis Katholische
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Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

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