ISBN 978-3-8439-3169-4

978-3-8439-3169-4, Reihe Mathematik

Torge Schmidt
Approximation of Spectra and Pseudospectra on a Hilbert Space

192 Seiten, Dissertation Technische Universität Hamburg-Harburg (2017), Softcover, B5

Zusammenfassung / Abstract

In this work we study spectra and pseudospectra of bounded linear operators on the 2-norm sequence space over the integers. These operators are generally non-normal and their matrix representation has an off-diagonal decay.

Based on a result by Chandler-Wilde, Chonchaiya and Lindner for tridiagonal infinite matrices, we develop methods to approximate upper and lower bounds of pseudospectra of operators in the Wiener algebra over the 2-norm sequence space over the integers, which are called spectral inclusion sets. These sets are computed using finite projections of the operator and can be arbitrarily sharp as we present an upper bound of the approximation error that can be made as small as desired by increasing the size of the projections. The method is extended to band dominated operators although in this case the approximation error is difficult to estimate.

We discover that by reducing the problem to finite projections we need to solve problems that are equivalent to pseudospectra of rectangular matrices. In this context we analyze pseudospectra of rectangular matrices and prove that they can be represented as a lower semi-continuous set-valued map from the positive real numbers to the complex numbers.

In order to compute the spectral inclusion sets we have developed two approaches, denoted as approach A and approach B. Approach A is based on computing pseudospectra of multiple rectangular matrices and then merging them, whereas approach B is defined by computing level-sets of two functions, which involve the successive computation of the smallest singular value of a sequence of rectangular matrices with large overlaps in their entries.

To optimize approach A we use standard methods for computing pseudospectra of square matrices and enhance them to be applied to our case. We use PC-continuation methods and PL-continuation methods to approximate the boundary curves and enhance the methods used in the literature by applying higher order predictors and constructing methods to detect singular points along the curve.

To optimize approach B we develop and implement the QH-Shift method, which rigorously computes the desired singular values using only linear instead of quadratic number of flops w.r.t. the bandwidth for all but the first computation. This is especially efficient when applying it to operators with a high bandwidth.