The sphere is an important setting for applied mathematics, yet the underlying approximation theory and numerical analysis needed for serious applications (such as, for example, global weather models) is much less developed than, for example, for the cube.
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This lecture will apply recent developments in approximation theory on the sphere to two different problems in scientific computing.
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First, in geodesy there is often the need to evaluate integrals using data selected from the vast amount collected by orbiting satellites. Sometimes the need is for quadrature rules that integrate exactly all spherical polynomials up to a specified degree $n$ (or equivalently, that integrate exactly all spherical harmonies $Y_{\ell ,k}(\theta ,\phi)$ with $\ell \le n).$ We shall demonstrate (using results of M. Reimer, I. Sloan and R. Womersley in collaboration with
W. Freeden) that excellent quadrature rules of this kind can be obtained from recent results on polynomial interpolation on the sphere, if the interpolation points (and thus the quadrature points) are chosen to be points of a so-called extremal fundamental system.
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The second application is to the scattering of sound by smooth three-dimensional objects, and to the inverse problem of finding the shape of a scattering object by observing the pattern of the scattered sound waves. For these problems a methodology has been developed, in joint work with I.G. Graham, M. Ganesh and R. Womersley, by applying recent results on constructive polynomial approximation on the sphere. (The scattering object is treated as a deformed sphere.)