Inverse Boundary Spectral Problems: 1st Edition (Hardback) book cover

Inverse Boundary Spectral Problems

1st Edition

By Alexander Kachalov, Yaroslav Kurylev, Matti Lassas

Chapman and Hall/CRC

312 pages

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Hardback: 9781584880059
pub: 2001-07-30
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Description

Inverse boundary problems are a rapidly developing area of applied mathematics with applications throughout physics and the engineering sciences. However, the mathematical theory of inverse problems remains incomplete and needs further development to aid in the solution of many important practical problems.

Inverse Boundary Spectral Problems develop a rigorous theory for solving several types of inverse problems exactly. In it, the authors consider the following:

"Can the unknown coefficients of an elliptic partial differential equation be determined from the eigenvalues and the boundary values of the eigenfunctions?"

Along with this problem, many inverse problems for heat and wave equations are solved.

The authors approach inverse problems in a coordinate invariant way, that is, by applying ideas drawn from differential geometry. To solve them, they apply methods of Riemannian geometry, modern control theory, and the theory of localized wave packets, also known as Gaussian beams. The treatment includes the relevant background of each of these areas.

Although the theory of inverse boundary spectral problems has been in development for at least 10 years, until now the literature has been scattered throughout various journals. This self-contained monograph summarizes the relevant concepts and the techniques useful for dealing with them.

Reviews

"[This book] contains a wealth of important methods and ideas, and the presentation is always very clear. … [A] very interesting and valuable contribution to the literature on inverse problems for partial differential equations."

- Zentralblatt MATH, Vol. 1037

Table of Contents

INTRODUCTION

ONE-DIMENSIONAL INVERSE PROBLEM

The Problem and the Main Result

Wave Equation

Controllability and Projectors

Gaussian Beams

BASIC GEOMETRICAL AND ANALYTICAL METHODS FOR INVERSE PROBLEMS

Basic Tools of Riemannian Geometry for Inverse Problems

Elliptic Operators on Manifolds and Gauge Transformation

Initial-Boundary Value Problem for Wave Equation

Gaussian Beams

Carleman Estimates and Unique Continuation

GEL'FAND INVERSE BOUNDARY SPECTRAL PROBLEM FOR MANIFOLDS

Formulation of the Problem and the Main Result

Fourier Coefficients of Waves

Domains of Influence

Global Unique Continuation from the Boundary

Gaussian Beams from the Boundary

Domains of Influence and Gaussian Beams

Boundary Distance Functions

Reconstruction of the Riemannian Manifold

Reconstruction of the Potential

INVERSE PROBLEMS FOR WAVE AND OTHER TYPES OF EQUATIONS

Inverse Problems with Different Types of Data

Dynamical Inverse Problem for the Wave Equation

Continuation of Data

Inverse problems with Data Given on a Part of the Boundary

Inverse Problems for Operators in Rm

BIBLIOGRAPHY

TABLE OF NOTATION

About the Series

Monographs and Surveys in Pure and Applied Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
SCI040000
SCIENCE / Mathematical Physics