# Phase Optimization Problems

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März 2010

## Beschreibung

### Beschreibung

This is the only book available in English language to consider inverse and optimization problems in which phase field distributions are used as optimizing functions. The mathematical technique used relates to nonlinear integral equations, with numerical methods developed and applied to concrete problems. Written by a team of outstanding and renowned experts in the field, this monograph will appeal to all those dealing with the investigation, design, and optimization of electromagnetic and acoustic radiating and transmitting devices and systems, while also being of interest to mathematicians working on the theory of nonlinear integral equations.### Inhaltsverzeichnis

Preface1. Introduction

2. Formulation of physical problems

2.1. Forming field of given structure

2.1.1. Optimization of beam power transmission lines

2.1.2. Focusing field in the middle zone

2.1.3. Two-element phase field transformers

2.1.4. Multi-element phase field transformers

2.2. Antenna synthesis problems

2.2.1. Aperture antenna synthesis by amplitude radiation pattern

2.2.2. Phase array synthesis

2.2.3. Multi-element beam transformers

2.2.3. Volume antenna synthesis

2.3. Various field optimization problems

2.3.1 Two-criterion optimization problems

2.3.2 Antenna synthesis with minimization of field in certain areas

2.3.3 Creating zeroth in radiation pattern

2.3.4 Mirror shape optimization in quasioptical resonator

3. Mathematical formulation of the problems

3.1. Variational problems without amplitude-phase restrictions on sought functions

3.1.1. Case of isometric operator. Nonlinear equations of Hammerstein type as Lagrange-Euler equations

3.1.2. Solution branching

3.1.3. Case of compact operator

3.2. Variational problems with amplitude-phase restrictions on sought function

3.2.1. Case of given argument (phase) of sought function

3.2.2. Case of given modulus (amplitude) of sought function

3.2.3. Problems with composed operators

4. Analytical solutions

4.1. Analytical solutions of a general class of nonlinear integral equations with free phase

4.1.1. Finite-parametric representation of solutions

4.1.2. Solution branching

4.2. Special case of equation related to one-dimensional Fourier transformation

4.2.1. Finite-parametric representation of solutions

4.2.2. Additional theoretical results

4.2.3. Numerical investigation of solution branching

4.3. Other special cases of nonlinear integral equations

4.3.1. Finite-parametric representation of solutions in the case of discrete Fourier transformations

4.3.2. Numerical investigation of solution branching

4.3.3. Finite-parametric representation of solutions in the case of Hankel transformation connected with two-dimensional Fourier transformation

5. Numerical methods, algorithms, and results

5.1. Direct minimization of functionals

5.1.1. Methods of steepest descent with controlled step

5.1.2. Combined gradient methods

5.1.3. Opposite directions method

5.1.4. Numerical results

5.2. Simplest successive approximation methods for nonlinear equations of Hammerstein type with free phase

5.2.1. Iterative methods for equations with free phase

5.2.2. Convergence of iterative methods for the isometric operator case

5.2.3. Convergence of iterative methods for the compact operator case

5.2.4. Numerical results

5.3. Methods of the Newton type

5.3.1. Specified scheme of the Newton type methods

5.3.2. Peculiarities of the methods in the case of continual bifurcations

5.3.3. Convergence of the Newton type methods

5.3.4. Numerical results

5.4. Method of generalized separation of variables

5.4.1. General scheme of the method in the linear case

5.4.2. Extension of the methods to the multidimensional problems with free phase

5.4.3. Numerical results

6. Conclusions

References

### Portrait

Olena O. Bulatsyk is a scientific researcher at the Department of Numerical Methods in Mathematical Physics at the Institute of Applied Problems in Mechanics and Mathematics (IAPMM) in Lviv, Ukraine. She received her M.S. degree in mathematics from Lviv State University in 1997 and her Ph.D. degree in mathematical modelling from IAPMM in 2004. Dr. Bulatsyk has authored over 10 scientific publications and is a member of the IEEE and AMS.Boris Z. Katsenelenbaum studied at Moscow State University. He received his Ph.D. and D.Sc. degrees from the Institute of Radio Engineering and Electronics (IRE)/ USSR Academy of Sciences (Moscow) in 1948 and 1960, respectively, where since 1954 he has held numerous positions at the latest being that of Chief Scientist. From 1962 to 1985, he served also as a Professor at the Moscow Institute of Physics and Technology, from which he received his professor diploma in 1965. Prof. Katsenelenbaum has written ten books and about 160 articles on various problems of high frequency electrodynamics and diffraction theory, and received the Ukrainian State Award for Science and Technology in 1989. Since 1998 he has been living in Nahariya, Israel.Yury P. Topolyuk is a senior scientific researcher at the Department of Numerical Methods in Mathematical Physics at IAPMM in Lviv, Ukraine. He received his M.S. degree in 1984 and his Ph.D. degree in applied and numerical mathematics in 1994, both from Lviv State University. Dr. Topolyuk has authored over 40 scientific publications. He is a member of the IEEE and AMS.Nikolai N. Voitovich is the head of the Department of Numerical Methods in Mathematical Physics at IAPMM in Lviv, Ukraine. He received his Ph.D. degree in 1968 and his D.Sc. degree in 1982, both in radiophysics, from IRE (Moscow) and Kharkiv State University, respectively. From 1961 to 1971 he worked at IRE, then took up a position at the IAPMM. Prof. Voitovich has authored six books and over 150 scientific papers, and received the Ukrainian State Award for Science and Technology in 1989. He is a member of the IEEE, AMS, and of editorial boards of several scientific journals.EAN: 9783527407996

ISBN: 3527407995

Untertitel: Applications in Wave Field Theory.
1. Auflage.
70 schwarz-weiße Abbildungen.
Sprache: Englisch.

Verlag: Wiley VCH Verlag GmbH

Erscheinungsdatum: März 2010

Seitenanzahl: X

Format: gebunden

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