Pseudospektrer og normvurderinger

Abstract

For en begrænset operator A har udviklingen af ||p_n(A)||, hvor p_n er et polynomium af grad n, stor interesse både for et givet n og asymptotisk. I denne forbindelse bevises bl.a. Kreiss' matrixs ætning og nogle forskellige generaliseringer heraf. Denne del af rapportens resultater tager udgangspunkt i kompleks funktionsteori, og Dunfordkalkulen bliver en vigtig metode til at forbinde kompleks funktionsteori med Hilbertrum. Udvidelsen af Kreiss' matrixsætning kræver kendskab til Faberpolynomier, som er tæt forbundet med Riemanns afbildningssætning. I den forbindelse bevises en række resultater fra kompleks funktionsteori. Det har stor interesse at bestemme pseudospektrer for matricer, da der er sammenhæng mellem pseudospektrer og operatorers og dynamiske systemers opførsel. Et eksempel på en operator er differentialoperatoren, der kan diskretiseres ved hjælp af spektrale differentiationsmetoder. Som et eksempel herpå betragtes Chebyshevdifferentiationsmatricer. Dele af teorien bag beskrives, og forskellige egenskaber vises. Endvidere beskrives implementering heraf i Maple og matlab, og pseudospektrerne betragtes. Desuden gennemgås teori om og eksempler på numerisk bestemmelse af den pseudospektrale abscisse ved hjælp af de såkaldte criss-crossalgoritmer. Disse algoritmer er baseret på egenskaber ved singulære værdier og hamiltoniske matricer.

Let A be a bounded operator in a Hilbert space. It is of interest to give estimates for ||f(A)||, in particular when f is a polynomial. The objective of this report is to give such bounds for ||f(A)|| based on pseudospectra and the Kreiss matrix theorem, of which the latter is generalized to bounded operators with spectrum \sigma(A) in arbitrary compact sets \sigma(A) \subseteq \Omega \subset C, and to address the question of computation of quantities related to norm bounds, particularly related to pseudospectra. To associate an operator with a continuous function, the Dunford calculus is used, and complex analysis and in particular the Cauchy integral formula will play an important role. In the first chapter we will present some classical results from complex analysis, e.g. the maximum modulus principle, the open mapping principle and the Riemann mapping theorem. In the second chapter we associate pseudospectra and the Kreiss constant, and through the conformal mapping from the Riemann mapping theorem another definition of the Kreiss constant is presented, and it is shown that the two definitions are equivalent in a sense. The Faber polynomials, too, are related to the Kreiss constant in the sense that they, too, are defined through the conformal mapping from the Riemann mapping theorem. These are defined in the third chapter, and generalizations of the Kreiss matrix theorem to Faber polynomials and polynomials in general are proven. Pseudospectra are particularly useful for non-normal operators. In chapter four a discretization of the differential operator by Chebyshev differential methods is considered, and implementation and some properties of the non-normal Chebyshev differentiation matrix are presented. The last chapter considers computation of the pseudospectral abscissa, which e.g. yields a lower bound for the transient behavior of ||exp(tA)||. The criss-cross algorithm is presented, and some properties concerning convergence are proven.

A master thesis from Aalborg University

Pseudospektrer og normvurderinger

[Pseudospectra and norm bounds]