Workshop Analysis on the Occasion of Jaap Korevaar's 95th Birthday
Jaap Korevaar, emeritus professor, is still active in mathematics research at Korteweg-de Vries Institute. In January he will celebrate his 95th birthday, and for that occasion there is a workshop Analysis, organised by KdV members Jan Wiegerinck and Han Peters. The speakers will be: Raymond Brummelhuis (Université de Reims-Champagne-Ardenne) and Charles K. Chui (Hong Kong Baptist University and Stanford University)
09:30 Welcome with coffee and tea
10:00 Raymond Brummelhuis (Université de Reims-Champagne-Ardenne)
Title: Interpolation by Radial Basis Functions and PDEs
Abstract: One can interpolate an arbitrary function on Euclidean space in an arbitrary finite set of points by a linear combination of translates of a given basis function if this basis function is positive definite or, more generally, conditionally positive definite. The basis function is often chosen to be radially symmetric (without this being strictly necessary), which is why the field studying this type of interpolation method has become known as that of Radial Basis Function (RBF) interpolation. The resulting interpolation scheme often has excellent approximation properties for sufficiently smooth functions: either one can show convergence as the set of interpolation points becomes dense, or one can achieve an arbitrarily small limiting error by appropriately scaling the basis function. The latter is known as approximate approximation. In this talk we will show how RBF interpolation can be used to construct numerical schemes for PDEs of evolution type. We will discuss the theoretical convergence and approximate approximation properties of such schemes, mainly through the example of the classical heat equation.
11.00: Charles K. Chui (Hong Kong Baptist University and Stanford University)
Title: Super-resolution approach to mathematics of big data
Abstract: Big data have been around since Big Bang and the beginning of life, but not yet explored till very recently. In addition, data are continually being generated in rapidly increasing volumes and complexity, everywhere, and by just about everything around us. The understanding of big data is indeed a most challenging endeavor to the communities of mathematicians and other scientists. On the other hand, the recent exciting advancement of green fluorescence protein (GFP) light microscopy, with the capability of viewing well below the hundredth nanometer scale, that enables the feasibility of viewing the molecular activities in human cells, is truly a profound breakthrough in super-resolution imaging. In this lecture, we will describe the background and development of the mathematics of superresolution and present both the frequency and spatial approaches to solving the inverse problem of superresolution, thereby preparing the way to the visualization and understanding of big data, from the “very small" to the “very big".
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