Scalable Pseudospectral Analysis via Low-Rank Approximations of Dynamical Systems

Abstract: Pseudospectral analysis is fundamental for quantifying the sensitivity and transient behavior of nonnormal matrices, yet its computational cost scales cubically with dimension, rendering it prohibitive for large-scale systems. While existing research on scalable pseudospectral computation has focused on exploiting sparsity structures, common in discretizations of differential operators, these approaches are ill-suited for machine learning and data-driven dynamical systems, where operators are typically dense but approximately low-rank. In this paper, we develop a comprehensive low-rank framework that dramatically reduces this computational burden. Our core theoretical contribution is an exact characterization of the pseudospectrum of arbitrary low-rank matrices, reducing the evaluation of resolvent norms to eigenvalue problems of dimension proportional to the rank. Building on this foundation, we derive rigorous inclusion sets for the pseudospectra of general matrices via truncated and randomized low-rank approximations, with explicit perturbation bounds. These results enable efficient estimators for key stability quantities, including distance to instability and Kreiss constants, at a cost that scales with the effective rank rather than the ambient dimension. We further demonstrate how our framework naturally extends to data-driven settings, providing pseudospectral analysis of transfer operators learned from nonlinear and stochastic dynamical systems. Numerical experiments confirm orders-of-magnitude speedups while preserving accuracy, opening pseudospectral analysis to previously intractable high-dimensional problems in computational PDEs, control theory, and data-driven dynamics.