Saint Petersburg 2003

Saint Petersburg'2003

Здесь находятся тезисы доклада, сделанного Богомоловым Яковом Леонидовичем на международном семинаре «Дни дифракции — 2003», проходившем 24–27 июня 2003 года в Санкт-Петербурге. Соавторами доклада являются Юнаковский Алексей Дмитриевич и Семёнов Евгений Сергеевич.

INTERNATIONAL SEMINAR

DAY ON DIFFRACTION '2003

Saint Petersburg
June 24–27, 2003
ABSTRACTS

UNIVERSITAS
PETROPOLITANA
MDCCXXIV

p. 19

Singular value decomposition as a tool for solving of spectral problems arisen in supercollider simulation

Ya. L. Bogomolov, E. S. Semenov and A. D. Yunakovsky

Institute of Applied Physics RAS, N. Novgorod, Russia,
E-mail: bogomol(a)appl.sci-nnov.ru

Designing of an effective acceleration part of a supercollider generates some optimization problems [1], [2]. One of them is to transform an entrance quasicylindrical electric field into amplified (as much as possible) one of structure desired in near axes domain of a supercollider. For this purpose an electrodynamical system includes irregular waveguide channels that provide an amplification property. Parameters of these channels are required to be optimized. In other words, we deal with a spectral problem: to find profiles of waveguide channels providing the “zero” eigenvalue for Helmholtz type operator under corresponding boundary conditions.

The problem being of resonance type (our aim is to construct a resonator), it requires a suitable numerical technique. To be more precise, we need a numerical procedure, which can obtain a resonance eigenfunction corresponding to zero eigenvalue. The situation is essentially worse in the case of a chain of resonators. A multiple zero eigenvalue can occur.

For this “zero” problem there exists a very powerful technique known as singular value decomposition (SVD) [3]. This numerical tool allows to solve effectively a set of homogeneous simultaneous linear equations in the case that a matrix (a discrete approximation of the operator considered) is singular. Moreover, a solution is obtained by means of SVD immediately.

Several types of boundary profiles for amplifying waveguide channels of a supercollider are considered. Optimal parameters for these profiles are obtained.

This work was partially supported by the RFBR grant 01-01-00577.

References
[1] Ya. L. Bogomolov and A. D. Yunakovsky. Simulation of Electrodynamical Systems of Energy Accumulation. “Modern methods in theory of boundary problems”, Proceedings of Voronezh spring mathematical school, P. 1. — Voronezh: Voronezhskiy gosuniversitet, 2000. — pp. 37–47.
[2] Ya. L. Bogomolov and A. D. Yunakovsky. Scattering of Electromagnetic Waves in a Channel with a Step-like Boundary. Proceedings of International Seminar “Day on Diffraction — 2001”. — St. Petersburg, Universitas Petropolitana, 2001. — pp. 26–37.
[3] Demmel J. W. Applied Numerical Linear Algebra. — Philadelphia: SIAM, 1997.
Demmel Dzh. Vychislitel'naya lineynaya algebra. Teoriya i prilozheniya. — M.: Mir, 2001.

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