Quasi-Interpolation
Quasi-interpolation is one of the most useful and often applied methods for the approximation of functions and data in mathematics and applications. Its advantages are manifold: quasi-interpolants are able to approximate in any number of dimensions, they are efficient and relatively easy to formulate for scattered and meshed nodes and for any number of data. In our numerical analysis group, research into quasi-interpolation with multivariate approximation algorithms provides a field for work of graduate students and contributions from researchers, using all the mathematical background and methods of implementation. The mathematical analysis of quasi-interpolation is studied in three directions, namely on the basis (spline spaces, radial basis functions) from which the approximation is taken, on the form and computation of the quasi-interpolants (point evaluations, averages, least squares), and on the mathematical properties (existence, locality, convergence questions, precision). We study which type of quasi-interpolation to use in different contexts and how to optimise its features to suit applications in medicine, physics and engineering.