Analytic perturbation theory and its applications (Q2871892)

This book is about the inverse of analytically perturbed matrices and linear operators of the form NEWLINE\[NEWLINE A(z) = A_0 +z A_1 + z^2 A_2 \dots, NEWLINE\]NEWLINE with emphasis on the case that \(A_0\) is not invertible but \(A(z)\) has an inverse for \(z \neq 0\) sufficiently small. In the case that \(A_0\) is invertible, it is well-known that \(A^{-1}(z)\) is also analytic near the origin. In the present case \(A^{-1}(z)\) can only be expanded as a Laurent series.NEWLINENEWLINEThe book is divided into three parts following an introductory and motivating first section: Part I: Finite Dimensional Perturbations (3 chapters, 102 pp.), Part II: Applications to Optimization and Markov Processes (3 chapters, 136 pp.), Part III: Infinite Dimensional Perturbations (2 chapters, 113 pp.).NEWLINENEWLINEPart I: In Chapter 2, the analytic perturbation of linear systems is considered. Three methods are presented for computing the coefficients of the Laurent series and the order of the pole. Various special cases, such as linear or polynomial perturbations, are treated separately. It is a nice feature of the book that a large number of examples and problems to solve are given at the end of each chapter as well as detailed bibliographic notes containing remarks on the historical development and other existing research concerning the topic of the chapter.NEWLINENEWLINEChapter 3 is devoted to the perturbation of null spaces, eigenvectors, and generalized inverses, assuming that the eigenvalue of \(A(0)\) remains to be an eigenvalue of \(A(z)\) for \(z \neq 0\) with the same or lower multiplicity (the case of regular \(A(z)\) is considered in Chapter 2). A special case are regular perturbations, where the multiplicity remains to be constant in a neighborhood of \(z = 0\). For singular perturbations, a reduction process is developed. Also, the perturbation of generalized inverses is treated.NEWLINENEWLINEThe following Chapter 4 is dedicated to polynomial perturbations of algebraic nonlinear systems. The main technique relies on Gröbner bases and Buchberger's algorithm for reducing the given system to a decoupled system of bivariate polynomials. As an efficient method for computing series coefficients of a solution of a perturbed polynomial equation, the Newton polygon method is presented.NEWLINENEWLINEPart II: The aim of Chapter 5 is to describe the asymptotic behaviour of solutions to a generic perturbed mathematical program. There are three sections corresponding to the following three levels of generality: A. Asymptotic linear programming; B. Asymptotic polynomial programming; C. Asymptotic analytic programming. The corresponding sections are entitled ``Asymptotic simplex method'', ``Asymptotic gradient projection method'' and ``Asymptotic analysis for general nonlinear programming'', respectively. The next chapter is concerned with applications to Markov chains. The form of the stationary distribution matrix of an analytically perturbed probability transition matrix of the Markov chain is studied. An important role play irreducible probability transition matrices and nearly completely decomposible Markov chains. In a final subsection, the Google PageRank in its web search machine is subsumed under perturbed Markov chains.NEWLINENEWLINEChapter 7 is called ``Applications to Markov decision processes''. The authors describe the content of this section as follows:NEWLINENEWLINEWhereas Markov chains form a good description of some discrete event stochastic processes, they are not automatically equipped with a capability to model situations where there may be a ``decision-maker'' or a ``controller'' who can influence the trajectory of the process. Hence, in this chapter, the authors consider discrete Markov decision processes with finite state and action spaces and study the dependence of optimal policies/controls of these decision processes on certain important parameters.NEWLINENEWLINE An application to the Hamiltonian cycle problem to find a simple cycle that contains all vertices of a directed graph is given.NEWLINENEWLINEPart III: The main Chapter 8 deals with analytic perturbations of linear operators in infinite dimensional Hilbert or Banach spaces. This part has to some extent the character of a brief introduction to basic concepts of normed spaces, and a surprisingly detailed definition of the Hilbert space \(L^2(-\pi,\pi)\) and Fourier analysis in \(L^2(-\pi,\pi)\), culminating in the definition of the Sobolev space \(H_0^1(\mathcal{R})\), is given.NEWLINENEWLINEIn a motivating section in Chapter 8 the input retrieval in linear control systems and singularly perturbed Markov processes are presented as possible applications for the obtained later results. Linearly perturbed operators in a Hilbert space \(H\) are considered in the three situations that \(A_0\) is not one-to-one, \(A_0(H)\) is closed but not surjective and \(A_0(H)\) is not closed, respectively. Next, linearly perturbed operators in a Banach space are considered. In separate subsections, the existence of an inverse with first, second or higher order poles is studied.NEWLINENEWLINEThe bibliography consists of 164 items, the most recent ones from 2013. The book is clearly written. The aim of each section is explained at the beginning which helps to get through the, sometimes involved, material, as do the already mentioned numerous examples. Although the topic of the book probably will not fit into the mainstream of mathematics courses at many universities, the authors give a couple of recommendations how to use parts of the book as a textbook.