MATHICSE Technical Report : A reduced basis approach to large-scale pseudospectra computation

For studying spectral properties of a non-normal matrix A ∈ Cn×n, information about its spectrum σ(A) alone is usually not enough. Effects of perturbations on σ(A) can be studied by computing ε-pseudospectra, that is the level-sets of the resolvent norm function g(z) = ‖(zI − A)−1‖2. The computation of ε-pseudospectra requires determining the smallest singular values σmin(zI − A) on for all z on a portion of the complex plane. In this work, we propose a reduced basis approach to pseudospectra computation that provides highly accurate estimates of pseudospectra in the region of interest. It incorporates the sampled singular vectors of zI − A for different values of z and implicitly exploits their smoothness properties. It provides rigorous upper and lower bounds for the pseudospectra in the region of interest. We also present a comparison of our approach to several existing approaches on a number of numerical examples, showing that our approach provides significant improvement in terms of computational time.