Speaker
Description
Linear operators with a continuous spectrum often lurk behind complex physical phenomena in nature, from wave attractors in geometrically confined fluids to topological bifurcations in dynamical systems. However, they are notoriously tricky to learn from data. For example, finite-dimensional approximations of the operator must “discretize” the continuous spectrum into finitely many points and cannot converge uniformly.
In this talk, we explore two principled approaches to data-driven approximation of unitary operators with continuous spectrum. The first approach synthesizes recent advances in computational spectral theory with stable algorithms for Gauss-Szego quadrature rules on the unit circle. The second approach leverages new algorithms for quadrature rules that adaptively deform and discretize the contour inside the unit circle, resembling classical resonance expansions for wave operators. We discuss strong convergence rates for the approximate operators, data-driven aspects of the approximation, and highlight the unifying power of quadrature rules derived from rational approximations.