Abstract
Inverse problems often consist in finding a physical quantity (density, diffusivity, conductivity) defined in a certain domain (‘support’). The determination of a solution always requires a mathematical modelling of the corresponding (source or parameter) functions and domains involved. This process can be understood as a certain manner of probing or sounding. In this meaning ‘sounding’ can be directed to the approximation of one or several function values, e.g. by means of the known Backus-Gilbert method, but it is also done by the decomposition procedures to be considered here.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams, R.A. (1975): Sobolev Spaces. Academic Press, New York
Anger, G. (1990): Inverse Problems in Differential Equations. Akademie Verlag, Berlin, and Plenum Press, London
Auscher, P. (1992): Wavelets with Boundary Conditions on the Interval, in: Wavelets - A Tutorial in Theory and Applications, Chui, C. K.(ed.), Academic Press, Boston, pp. 217–236
Ballani, L., Stromeyer, D. and Barthelmes, F. (1993a): Decomposition Principles for Linear Source Problems, in: Inverse Problems: Principles and Applications in Geophysics, Technology, and Medicine, Anger, G. et al (eds.), (Mathematical Research, Vol. 74 ), Akademie Verlag, Berlin, pp. 45–59
Ballani, L., Engels, J. and Grafarend, E. (1993b): Global base functions for the mass density in the interior of a massive body (Earth). Manuscr. Geod., Vol. 18, pp. 99–114
Barthelmes, F., Ballani, L. and Klees, R. (1994): On the Application of Wavelets in Geodesy, (these proceedings )
Batha, L., Benciolini, B. and Zatelli, P. (1994): Geodetic applications of wavelets: proposals and simple numerical experiments, (these proceedings)
Beylkin, G., Coifman, R. R. and Rokhlin, V. (1991): Fast Wavelet Transforms and Numerical Algorithms I. Comm. on Pure and Applied Math., Vol. 44, pp. 141–183
Beylkin, G. (1992): On the Representation of Operators in Bases of Compactly Supported Wavelets. SIAM J. Numer. Anal., Vol. 6, pp. 1716–1740
Chui, C.K.(ed.) (1992): Wavelets - A Tutorial in Theory and Applications. (Wavelet Analysis and Its Applications Series, Vol. 2 ), Academic Press, Boston
Cohen, A. and Daubechies, I. (1993): Non-separable bidimensional wavelet bases. Revista Mat. Iberoamericana, Vol. 9, pp. 51–137
Daubechies, I. (1992): Ten Lectures on Wavelets. (CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61 ), SIAM Press, Philadelphia
Dicken, V. (1994): Die Behandlung Inverser Probleme durch Wavelet-Zerlegungen. Fachbereich Mathematik, Philipps-Universität Marburg, Diploma Theses
Donoho, D. L. (1992): Nonlinear Solution of Linear Inverse Problems by Wavelet -Vaguelette Decomposition. Tech. Rep. 403, Stat. Dep., Stanford Univ., 46 p.
Freeden, W. and Schreiner, M. (1993): Nonorthogonal Expansions on the Sphere. Univ. of Kaiserslautern, (preprint, to appear in Math. Meth. in the Appl. Sci. )
Freeden, W. and Schreiner, M. (1994): New Wavelet Methods for Approximating Harmonic Functions, (these proceedings)
Freeden, W. and Windheuser, U. (1994): Earth’s Gravitational Potential and It’s MRA Approximation by Harmonic Singular Integrals, (to appear in Zeitschr. Angew. Math. Mech. (ZAMM))
Jaffard, S. and Meyer, Y. (1989): Bases d’Ondelettes dans des Ouverts de IRn. Journ. de Math. Pures et Appl. Vol. 68, pp. 95–108
Jaffard, S. and Laurengot, P. (1992): Orthonormal Wavelets, Analysis of Operators, and Applications to Numerical Analysis, in: Wavelets - A Tutorial in Theory and Applications. Chui, C.K. (ed.), Academic Press, Boston, pp. 543–601
Jawerth, B. and Sweldens, W. (1993): An Overview of Wavelet Based Multiresolution Analyses, (submitted to SIAM Review)
Louis, A. K. (1993): A Wavelet Approach to Identification Problems. University of Saarbrücken (preprint)
Maaß, P. (1992): Families of orthogonal 2D-wavelets with compact support. University of Saarbrücken (preprint)
Maaß, P. (1994): Wavelet-projection methods for inverse problems. University of Potsdam (preprint)
Meyer, Y. (1992): Wavelets and Operators. (Cambridge Studies in Advanced Mathematics, Vol. 37 ) Cambridge University Press, Cambridge
Sickel, W. (1991): A remark on orthonormal bases of compactly supported wavelets in Triebel-Lizorkin spaces. The case 0 < p, q <∞., Arch. Math., Vol. 57, pp. 281–289
Strichartz, R. S. (1993): How to make wavelets. Amer. Math. Monthly, Vol. 100, pp. 539–556
Weck, N. (1972): Zwei inverse Probleme in der Potentialtheorie. Mitt. Inst. Theor. Geod. Univ. Bonn, Nr. 4, pp. 27–36
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ballani, L. (1995). Solving the Inverse Gravimetric Problem: On the Benefit of Wavelets. In: Sansò, F. (eds) Geodetic Theory Today. International Association of Geodesy Symposia, vol 114. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-79824-5_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-79824-5_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-59421-5
Online ISBN: 978-3-642-79824-5
eBook Packages: Springer Book Archive