Stable Approximate Evaluation of Unbounded Operators

ISBN-13: 9783540399421 / Angielski / Miękka / 2006 / 144 str.

Charles W. Groetsch

Opis

This book teams up the spectral theory of bounded linear operators with von Neumann's theory of unbounded operators to provide a framework for the study of stable methods for the evaluation of unbounded operators. The text presents numerous illustrations of unbounded linear operators that arise in various inverse problems of mathematical physics. It also offers an extensive exposition of background material from the theory of operators on Hilbert space.

Recenzje

From the reviews:

"This interesting monograph is devoted to the stable evaluation of the action of unbounded operators defined on Hilbert spaces. This problem is considered as an abstract mathematical problem within the scope of operator approximation theory. To motivate the discussion, the mathematical theory of inverse problems is briefly introduced. ... The monograph is reasonably self-contained and elegantly written. It gradually invites the reader to learn more about the difficulties of solving ill-posed problems." (Antonio C. G. Leitão, Mathematical Reviews, Issue 2008 a)

"The author does an excellent job of covering the subject of the title in 120 pages and five chapters ... . the monograph would make a nice graduate course or seminar in applied mathematics." (John R. Cannon, SIAM Review, Vol. 52 (2), 2010)

Spis treści

Some Problems Leading to Unbounded Operators.- Hilbert Space Background.- A General Approach to Stabilization.- The Tikhonov-Morozov Method.- Finite-Dimensional Approximations.

Blurb

Spectral theory of bounded linear operators teams up with von Neumann’s theory of unbounded operators in this monograph to provide a general framework for the study of stable methods for the evaluation of unbounded operators. An introductory chapter provides numerous illustrations of unbounded linear operators that arise in various inverse problems of mathematical physics. Before the general theory of stabilization methods is developed, an extensive exposition of the necessary background material from the theory of operators on Hilbert space is provided. Several specific stabilization methods are studied in detail, with particular attention to the Tikhonov-Morozov method and its iterated version.