Speaker
Description
We consider the approximation of $B^T (A+sI)^{-1} B$ for large s.p.d. $A\in\mathbb R^{n\times n}$ with a dense spectrum and $B\in\mathbb R^{n\times p}$, $p\ll n$ using block-Lanczos recursion. We target the computations of MIMO transfer functions for large-scale discretizations of problems with continuous spectral measures, such as linear time-invariant (LTI) PDEs on unbounded domains.
While Krylov methods such as Lanczos and CG are near-optimal for problems with discrete, well-separated spectra, their spectral adaptation deteriorates in the dense-spectrum regime. We address this limitation using a square-root terminator framework (originated in quantum physics in 1970s), modifying the final Lanczos recursion coefficient in an s-dependent manner via a Krein–Nudelman damped string representation. The resulting approximants reproduce the same Stieltjes moments while yielding continuous spectral measures and significantly reduced approximation error through adaptive damping, that maximizes the relative outflow of energy.
Large-scale experiments for diffusion and wave PDEs demonstrate the competitiveness of the proposed approach for deterministic as well as randomized computations.