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A reduced basis approach to large‐scale pseudospectra computation
Author(s) -
Sirković Petar
Publication year - 2019
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2222
Subject(s) - computation , mathematics , resolvent , eigenvalues and eigenvectors , singular value , basis function , norm (philosophy) , matrix (chemical analysis) , basis (linear algebra) , smoothness , spectrum (functional analysis) , algorithm , mathematical analysis , geometry , physics , materials science , quantum mechanics , political science , law , composite material
Summary For studying spectral properties of a nonnormal matrix A ∈ C n × 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, i.e. the level sets of the resolvent norm function g ( z ) = ‖ ( z I − A ) − 1‖ 2 . The computation of ε ‐pseudospectra requires determining the smallest singular values σ min ( z I − A ) for all z on a portion of the complex plane. In this work, we propose a reduced basis approach to pseudospectra computation, which provides highly accurate estimates of pseudospectra in the region of interest, in particular, for pseudospectra estimates in isolated parts of the spectrum containing few eigenvalues of A . It incorporates the sampled singular vectors of z I − 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. In addition, we propose a domain splitting technique for tackling numerically more challenging examples. We present a comparison of our algorithms to several existing approaches on a number of numerical examples, showing that our approach provides significant improvement in terms of computational time.