Inverse Obstacle Scattering with Non-Over-Determined Scattering Data

Erscheinungsjahr 2019
ISBN: 978-1-68173-590-0
Verlag: Morgan & Claypool Publishers

Buch, Englisch, 69 Seiten, Hardback, Format (B × H): 191 mm x 235 mm

Reihe: Synthesis Lectures on Mathematics and Statistics

The inverse obstacle scattering problem consists of finding the unknown surface of a body (obstacle) from the scattering A(B;a;k), where A(B;a;k) is the scattering amplitude, B;a E 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(B):= A(B;a0;k0). By sub-index 0 a fixed value of a variable is denoted.

It is proved in this book that the data A(B), known for all B 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.

Autoren/Hrsg.

Fachgebiete

• Mathematik | Informatik EDV | Informatik Business Application Mathematische & Statistische Software
• Mathematik | Informatik EDV | Informatik Informatik
• Mathematik | Informatik Mathematik Mathematische Analysis Moderne Anwendungen der Analysis

Inhaltsverzeichnis

- Preface
- Introduction
- The Direct Scattering Problem
- Inverse Obstacle Scattering
- Bibliography
- Author's Biography

Über Autor(innen)

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 a 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 wavescattering 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.