Krylov type methods for linear systems exploiting properties of the quadratic numerical range. ETNA - Electronic Transactions on Numerical Analysis
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30 novembre 2020
- handle: 10670/1.hkp7t0
- issn: 1068-9613
- doi: 10.1553/etna_vol53s541
info:eu-repo/semantics/openAccess
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Values, Field of Range, Numerical Field of valuesCiter ce document
Andreas Frommer et al., « Krylov type methods for linear systems exploiting properties of the quadratic numerical range. ETNA - Electronic Transactions on Numerical Analysis », Elektronisches Publikationsportal der Österreichischen Akademie der Wissenschafte, ID : 10.1553/etna_vol53s541
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Résumé
The quadratic numerical range $W^2(A)$ is a subset of thestandard numerical range of a linear operator, which still contains itsspectrum. It arises naturally in operators that have a $2 \times 2$ blockstructure, and it consists of at most two connected components, none of whichnecessarily convex. The quadratic numerical range can thus reveal spectralgaps, and it can in particular indicate that the spectrum of an operator isbounded away from $0$.We exploit this property in the finite-dimensional setting to derive Krylovsubspace-type methods to solve the system $Ax = b$, in which the iteratesarise as solutions of low-dimensional models of the operator whosequadratic numerical range is contained in $W^2(A)$. This implies that theiterates are always well-defined and that, as opposed to standard FOM, largevariations in the approximation quality of consecutive iterates are avoided,although $0$ lies within the convex hull of the spectrum. We also considerGMRES variants that are obtained in asimilar spirit. We derive theoretical results on basic properties ofthese methods, review methods on how to compute the required bases in a stablemanner, and present results of several numerical experiments illustratingimprovements over standard FOM and GMRES.