Adapting the Lanczos algorithm to matrices with almost continuous spectra
We consider approximating $B^T(A+sI)^{-1}B$, where $A\in\mathbb{R}^{n\times n}$ is large, symmetric positive definite, and $B\in\mathbb{R}^{n\times p}$ with $p\ll n$. We focus on the case where $A$ has an almost continuous (dense) spectrum: its eigenvalues fill one or more intervals so densely that Krylov methods cannot resolve them individually. Our target is computing multiple-input multiple-output transfer functions arising from large-scale discretizations of problems with continuous spectral measures, such as linear time-invariant PDEs on unbounded domains. Traditional Krylov methods, such as Lanczos or conjugate gradients, resolve individual eigenvalues of a dense discretization while ignoring the underlying continuous spectral measure these points approximate. We argue it is more efficient to model the operator's inherent branch cut than to exhaustively resolve the artificial point spectrum induced by discretization. We adapt the framework of Kreĭn-Nudelman semi-infinite strings to the block Lanczos algorithm, with parameters chosen adaptively by maximizing the energy absorbed at the string termination relative to the energy stored in the string. This yields a low-rank modification to the block Lanczos matrix, dependent on $\sqrt{s}$, at an additional $O(n)$ cost. We show significant error reductions for large-scale self-adjoint PDE discretizations in unbounded domains, including two- and three-dimensional Maxwell's equations in diffusive regimes. The method is especially advantageous for computing state-space solutions for wave propagation, specifically for 2D wave and 3D Maxwell operators. Replacing the conventional Lanczos spectral decomposition with the continuous Kreĭn-Nudelman spectrum yields a qualitative improvement in finite-difference approximations, transforming standing-wave artifacts into outgoing propagating waves.