Title Inverse Obstacle Scattering with Non-Over-Determined Scattering Data
Subtitle (Synthesis Lectures on Mathematics and Statistics)
Author Alexander G. Ramm
ISBN 9781681735887
List price USD 34.95
Price outside India Available on Request
Original price
Binding Paperback
No of pages 70
Book size 191 X 235 mm
Publishing year 2019
Original publisher Morgan & Claypool Publishers (Eurospan Group)
Published in India by .
Exclusive distributors Viva Books Private Limited
Sales territory India, Sri Lanka, Bangladesh, Pakistan, Nepal, .
Status New Arrival
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Description:

The inverse obstacle scattering problem consists of finding the unknown surface of a body (obstacle) from the scattering A(ß;a;k), where A(ß;a;k) is the scattering amplitude, ß;a ? S2 is the direction of the scattered, incident wave, respectively, S2 is the unit sphere in the R3 and k > 0 is the modulus of the wave vector. The scattering data is called non-over-determined if its dimensionality is the same as the one of the unknown object. By the dimensionality one understands the minimal number of variables of a function describing the data or an object. In an inverse obstacle scattering problem this number is 2, and an example of non-over-determined data is A(ß) := A(ß;a0;,k0). By sub-index 0 a fixed value of a variable is denoted.

It is proved in this book that the data A(ß), known for all ß in an open subset of S2, determines uniquely the surface S and the boundary condition on S. This condition can be the Dirichlet, or the Neumann, or the impedance type.

The above uniqueness theorem is of principal importance because the non-over-determined data are the minimal data determining uniquely the unknown S. There were no such results in the literature, therefore the need for this book arose. This book contains a self-contained proof of the existence and uniqueness of the scattering solution for rough surfaces.


Contents:

Preface

Chapter1. Introduction

Chapter 2. The Direct Scattering Problem • Statement of the Problem • Uniqueness of the Scattering Solution • Existence of the Scattering Solution • Properties of the Scattering Amplitude

Chapter 3. Inverse Obstacle Scattering • Statement of the Problem • Uniqueness of the Solution to Obstacle Inverse Scattering Problem with the Data A(ß ) • Uniqueness of the Solution to the Inverse Obstacle Scattering Problem with Fixed-Energy Data • Uniqueness of the Solution to Inverse Obstacle Scattering Problem with Non-Over-Determined Data • Numerical Solution of the Inverse Obstacle Scattering Problem with Non-Over-Determined Data

A. Existence and Uniqueness of the Scattering Solutions in the Exterior of Rough Domains • Introduction • Notations and Assumptions • Function Spaces • Statement of the Problem • The Weak Formulation of the Scattering Problem • Uniqueness Theorem • Existence of the Scattering Solutions for Compactly Supported Potentials • Existence for the Equation with the Absorption • The Limiting Absorption Principle • Existence of the Scattering Solution for Decaying Potentials • Existence for the Equation with Absorption • The Limiting Absorption Principle

Bibliography

Author’s Biography


About the Author:

Alexander G. Ramm, Ph.D., was born in Russia, immigrated to the U.S. in 1979, and is a U.S. citizen. He is Professor of Mathematics with broad interests in analysis, scattering theory, inverse problems, theoretical physics, engineering, signal estimation, tomography, theoretical numerical analysis, and applied mathematics. He is an author of 690 research papers, 16 monographs, and an editor of 3 books. He has lectured in many universities throughout the world, presented approximately 150 invited and plenary talks at various conferences, and has supervised 11 Ph.D. students. He was Fulbright Research Professor in Israel and in Ukraine, distinguished visiting professor in Mexico and Egypt, Mercator professor, invited plenary speaker at the 7th PACOM, won the Khwarizmi international award, and received other honors. Recently he solved inverse scattering problems with non-over-determined data and the many-body wave-scattering problem when the scatterers are small particles of an arbitrary shape; Dr. Ramm used this theory to give a recipe for creating materials with a desired refraction coefficient. He gave a solution to the refined Pompeiu problem, and proved the refined Schiffer’s conjecture.


Target Audience:

This book contains a self-contained proof of the existence and uniqueness of the scattering solution for rough surfaces. This book is useful for people interested in numerical analysis, statistics and computer science.

 

 
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