Inverse Problems for Partial Differential Equations

ISBN-13: 9780387253640 / Angielski / Twarda / 2005 / 346 str.

Victor Isakov

Opis

In 8 years after publication of the ?rst version of this book, the rapidly progre- ing ?eld of inverse problems witnessed changes and new developments. Parts of the book were used at several universities, and many colleagues and students as well as myself observed several misprints and imprecisions. Some of the research problems from the ?rst edition have been solved. This edition serves the purposes of re?ecting these changes and making appropiate corrections. I hope that these additions and corrections resulted in not too many new errors and misprints. Chapters 1 and 2 contain only 2-3 pages of new material like in sections 1.5, 2.5. Chapter 3 is considerably expanded. In particular we give more convenient de?nition of pseudo-convexity for second order equations and included bou- ary terms in Carleman estimates (Theorem 3.2.1 ) and Counterexample 3.2.6. We give a new, shorter proof of Theorem 3.3.1 and new Theorems 3.3.7, 3.3.12, and Counterexample 3.3.9. We revised section 3.4, where a new short proof of exact observability inequality in given: proof of Theorem 3.4.1 and Theorems 3.4.3, 3.4.4, 3.4.8, 3.4.9 are new. Section 3.5 is new and it exposes recent progress on Carleman estimates, uniqueness and stability of the continuation for systems. In Chapter 4 we added to sections 4.5, 4.6 some new material on size evaluation of inclusionsandonsmallinclusions.Chapter5containsnewresultsonidenti?cation

Recenzje

From the reviews of the second edition:

"The first edition of this excellent book appeared in 1998 and became a standard reference for everyone interested in analysis and numerics of inverse problems in partial differential equations. ... The second edition is considerably expanded and reflects important recent developments in the field ... . Some of the research problems from the first edition have been solved ... ." (Johannes Elschner, Zentralblatt MATH, Vol. 1092 (18), 2006)

Spis treści

Inverse Problems.- Ill-Posed Problems and Regularization.- Uniqueness and Stability in the Cauchy Problem.- Elliptic Equations: Single Boundary Measurements.- Elliptic Equations: Many Boundary Measurements.- Scattering Problems.- Integral Geometry and Tomography.- Hyperbolic Problems.- Inverse parabolic problems.- Some Numerical Methods.

O autorze

Since the first publication of this book, the rapidly progressing field of inverse problems witnessed changes and new developments. This book reflects these changes and describes the contemporary state of the theory and some numerical aspects of inverse problems in partial differential equations.
Currently, there are hundreds of publications containing new and interesting results on the topic of inverse problems. This book successfully collects and presents many of them in a readable and informative form. This second edition is considerably expanded and some concepts (like pseudo-convexity) or proofs are simplified. New material is added to reflect recent progress in theory of inverse problems.
This useful and stimulating material is intended for a reader with a moderate knowledge of partial differential equations, of the Fourier transform, and of basic functional analysis. It formulates basic inverse problems, discusses regularization, gives a short review of uniqueness in the Cauchy problem, and includes several exercises. Applications include recovery of inclusions from anomalies of their gravity fields, reconstruction of the interior of the human body from exterior electrical, ultrasonic, and magnetic measurement. Parts of the book in a preliminary form have been presented as graduate courses at a number of universities.

Blurb

The topic of the inverse problems is of substantial and rapidly growing interest for many scientists and engineers. The second edition covers most important recent developments in the field of inverse problems, describing theoretical and computational methods, and emphasizing new ideas and techniques. It also reflects new changes since the first edition, including some corrections. This edition is considerably expanded, with some concepts such as pseudo-convexity, and proofs simplified. New material is added to reflect recent progress in theory of inverse problems.

This book is intended for mathematicians working with partial differential equations and their applications, and physicists, geophysicists and engineers involved with experiments in nondestructive evaluation, seismic exploration, remote sensing and tomography.