2020/10/29 by Antonio Fazzi, Fazzi, Antonio, Nicola Guglielmi +3
Physics and Astronomy · Computer Science · #Quantum chaos and dynamical systems #Matrix Theory and Algorithms #Quantum many-body systems
paper · pdf · doi:10.48550/arxiv.2010.15954
We propose and study an algorithm for computing a nearest passive system to a\ngiven non-passive linear time-invariant system (with much freedom in the choice\nof the metric defining `nearest', which may be restricted to structured\nperturbations), and also a closely related algorithm for computing the\nstructured distance of a given passive system to non-passivity. Both problems\nare addressed by solving eigenvalue optimization problems for Hamiltonian\nmatrices that are constructed from perturbed system matrices. The proposed\nalgorithms are two-level methods that optimize the Hamiltonian eigenvalue of\nsmallest positive real part over perturbations of a fixed size in the inner\niteration, using a constrained gradient flow. They optimize over the\nperturbation size in the outer iteration, which is shown to converge\nquadratically in the typical case of a defective coalescence of simple\neigenvalues approaching the imaginary axis. For large systems, we propose a\nvariant of the algorithm that takes advantage of the inherent low-rank\nstructure of the problem. Numerical experiments illustrate the behavior of the\nproposed algorithms.\n