Multiscale Tree Sampling Regularization of Inverse Spherical Pseudodifferential Equations

Willi Freeden¹, M. Zuhair Nashed², Michael Schreiner³

This chapter is concerned with multiscale tree-algorithmic sampling regularization of ill-posed inverse spherical pseudodifferential equations by (a certain class of scale discrete) spherical wavelets. The goal of the work is to preferably convince members from mathematics, but also from geosciences, that spherically oriented wavelet sampling regularization methodologically provides a rich mathematical cornucopia that has much to offer to a large palette of applications. Geomathematically, spherical tree-algorithmic sampling regularization reflects all scientific problems dealing with the approximate shape of the Earth’s surface and/or the typical spherical satellite geometry of a low Earth orbiter (LEO). The essential interest is in bandlimited as well as non-bandlimited wavelet sampling regularization corresponding to observables naturally occurring in practice and leading canonically to invariant spherical pseudodifferential operators of polynomially or exponentially growing spectra. All in all, the paper represents a synopsis of ideas and concepts developed in various publications of the authors over the last three decades such as Freeden (Multiscale modelling of spaceborne geodata, 1999. B.G. Teubner, Leipzig); Freeden et al. (GEM Int J Geomath, 9:199–264, 2018); Freeden and Schreiner (Constr Approx 14:493–515, 1997); Freeden and Schreiner (Spherical functions of mathematical geosciences—A scalar, vecterial, and tensorial setup, 2009. Springer, Heidelberg); Freeden and Schreiner (Spherical functions of mathematical geosciences—A scalar, vecterial, and tensorial setup, 2nd edn. Geosystems Mathematics, 2022. Birkhäuser, Springer Nature, Switzerland); Freeden et al. (Constructive approximation on the sphere (with applications to Geomathematics), 1998. Oxford Science Publications, Oxford); Freeden et al. (Spherical sampling. Geosystems Mathematics, 2018. Springer International Publishing, Basel).

¹University of Kaiserslautern-Landau, Kaiserslautern, Germany
²University of Central Florida, Orlando, FL, USA
³RhySearch, Werdenbergstr. 4, 9471 Buchs, Switzerland

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