INVERSE PROBLEMS NEWSLETTER

Abstract

The Newsletter is a key element in further enhancing the value of the journal to the inverse problems community. So why not be a part of this exciting forum by sending to our Bristol office material suitable for inclusion under any of the categories mentioned above. Your contributions will be very welcome. Book reviewSome Newton Type Methods for the Regularization of Nonlinear Ill-Posed Problems Schriften der Johannes-Kepler-Universität Linz, Reihe C: Technik und Naturwissenschaften, Band 15 B Blaschke 1996 Linz: Universitätsverlag Rudolf Trauner 145 pp ISBN 3-85320-816-9 öS248.00, DM34.00, sFr31.50 The book under review is the PhD thesis of Barbara Kaltenbacher-Blaschke prepared at the Institut für Mathematik of the Johannes-Kepler-Universität Linz, Austria. This book summarizes the author's papers [3,4,6]. In this book Newton type methods for the solution of nonlinear ill-posed problems are analysed. Bakushinskii [1] was most probably the first to analyse a Newton type method for the solution of nonlinear ill-posed problems. If is a Fréchet-differentiable operator, Bakushinskii established local convergence for the iteratively regularized Gauss - Newton technique if a solution of the operator equation (relative to the initial guess) satisfies a source-wise representation In (2) denotes a given perturbation of the exact data, which satisfies Assuming a source-wise representation (3) of the solution relative to the initial guess is in many applications inappropriate. Therefore the author studies convergence of Newton type methods without assuming source-wise representations of the solution. Assuming in (2) and convergence of the iteratively regularized Gauss - Newton technique for one finds that the limit satisfies i.e., is a critical point. The aim of the author is to prove convergence of Newton type methods to a solution of (1) (and not to a critical point), and therefore assumptions on the operator F have to be posed, which exclude (at least locally) that a critical point is not a solution of (2). In this book two conditions on the operator F are studied and The first condition depends on the actual value of in (3). If is Lipschitz continuous then locally (5) is more restrictive the smaller is. For , R = I, and , (4) is equivalent to Lipschitz continuity of . The second condition (6) is a Newton - Mysovskii condition as studied e.g. in the books by Kantorowitsch and Akilov [5] and Deuflhard and Hohmann [7] (see also the references quoted therein). The author studies in her book a class of Newton type methods defined by where for small is an approximation of and is an approximation of . As special cases of (7) the iteratively regularized Gauss - Newton technique and a Newton - Landweber method (introduced in this book) can be considered. The Newton - Landweber iteration is a method where the linear equations occurring in each Newton step are solved approximately with a Landweber iteration combined with an appropriate stopping criterion. A similar approach has been suggested recently by Hanke [2], who uses a conjugate gradient technique instead of the Landweber method for the inner iteration. The author studies extensively approximation properties and convergence results of the iterates of (7). In the case of measurement errors, stopping criteria are developed which stabilize the output of (7). The theory developed in this book applies to the particular inverse problem of reconstructing the diffusion parameter in a quasi-linear elliptic differential equation from transient measurements. [1]Bakushinskii A B 1992 The problem of the iteratively regularized Gauss - Newton method Comput. Math. Math. Phys. 32 1353 - 9[2]Hanke M 1997 Regularizing properties of a truncated Newton - CG algorithm for nonlinear inverse problems Preprint No 280 Universität Kaiserslautern[3]Kaltenbacher B 1998 A posteriori parameter choice strategies for some Newton type methods for the regularization of nonlinear ill-posed problems (submitted)[4]Kaltenbacher B 1997 Some Newton type methods for the solution of nonlinear ill-posed problems Inverse Problems 13 729 - 53[5]Kantorowitsch L W and Akilov G P 1964 Funktionalanalysis in Normierten Räumen (Berlin: Akademie)[6]Blaschke B, Neubauer A and Scherzer O 1997 On convergence rates for the iteratively regularized Gauss - Newton method IMA J. Numer. Anal. to appear[7]Deuflhard P and Hohmann A 1995 Numerical Analysis. A First Course in Scientific Computation (Berlin, New York: de Gruyter) O Scherzer Universität Linz