A. Iserles

A. Iserles, G. R. W. Quispel

Since its emergence, GNI has become the new paradigm in numerical solution of ODEs, while making significant inroads into numerical PDEs. As often, yesterday's revolutionaries became the new establishment. This is an excellent moment to pause and take stock. Have all the major challenges been achieved, all peaks scaled, leaving just a tidying-up operation? Is there still any point to GNI as a separate activity or should it be considered as a victim of its own success and its practitioners depart to fields anew - including new areas of activity that have been fostered or enabled by GNI?

A. S. Fokas, A. Iserles, V. Marinakis

The modern imaging techniques of Positron Emission Tomography and of Single Photon Emission Computed Tomography are not only two of the most important tools for studying the functional characteristics of the brain, but they now also play a vital role in several areas of clinical medicine, including neurology, oncology and cardiology. The basic mathematical problems associated with these techniques are the construction of the inverse of the Radon transform and of the inverse of the so called attenuated Radon transform respectively. We first show that, by employing mathematical techniques developed in the theory of nonlinear integrable equations, it is possible to obtain analytic formulas for these two inverse transforms. We then present algorithms for the numerical implementation of these analytic formulas, based on approximating the given data in terms of cubic splines. Several numerical tests are presented which suggest that our algorithms are capable of producing accurate reconstruction for realistic phantoms such as the well known Shepp--Logan phantom.