Ian Sloan
University of New South Wales, Australia
Sergei and spherical approximation
Abstract:
This talk describes recent work on spherical approximation inspired in part by
Sergei Pereverzyev. In 2017 a paper by Cui, Pereverzyev, Sloan and Tkachenko
described a two-step approach to solving strongly ill-posed problems in
geomathematics, in which the first step was to obtain a smoothed version of
the data from satellite measurments, with the input data assumed to be noisy
and available only at specified points. In that work the smoothing was carried
out by a regularised least-squares approach, with the regularisation term being
the squared norm in some reproducing kernel Hilbert space, and moreover with
the approximate solution being forced to be a spherical polynomial. In recent
related work with Kerstin Hesse (Padeborn) and Rob Womersley (UNSW) the data
points are now free, the approximation is by radial basis functions, and
moreover (thanks to a discussion with Sergei) the data measurments can now be
incomplete, as is almost invariably the case with satellite data (because caps
centered at the two poles are missed by the satellite). If time permits, a
completely different spherical approximation problem, concerning the cosmic
microwave background, will be sketched.
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