Prof. Mariarosa Mazza

Fast Solvers for Fractional Operators via Rational Approximations

Fractional derivatives are non-local operators increasingly used in engineering and applied sciences, as they capture multiscale dynamics and long-memory effects in materials. Their numerical treatment, however, is far more demanding than for classical integer-order

problems, motivating the development of fast and efficient solvers.

In this talk, we introduce preconditioners for the Riesz operator, a fractional operator generalizing the classical Laplacian and modeling anomalous diffusion. Such preconditioners are constructed by approximating the Riesz operator as a fractional power of the Laplacian,

with this fractional power then approximated through suitable quadrature formulas, such as Gauss–Jacobi and sinc rules, yielding rational approximations of the underlying operator.  Their properties are analyzed through approximation error estimates and a spectral study of

the associated discretization matrices. This combined analysis allows us to select the key parameters and rank the proposed preconditioners according to their convergence behavior. Numerical experiments confirm the theoretical predictions.

Prof. Michael Multerer

Samplets: Multiresolution methods for scattered data

We introduce samplets, a multiresolution analysis of discrete signed measures with prescribed vanishing moments. Samplets enable efficient data compression, feature detection and adaptive representations. The decay of samplet coefficients can be employed to quantify pointwise regularity within the framework of microlocal spaces introduced by Jaffard. In reproducing kernel Hilbert spaces, samplets induce bases of subspaces spanned by kernel translates via the Riesz isometry. The corresponding Gramian, that is, the kernel matrix expressed in samplet coordinates, becomes quasi-sparse for quasi-uniform data sites. As an application, we present multiscale interpolation schemes for scattered data using globally supported radial basis functions.