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.
DAY ON DIFFRACTION '2003
June 24–27, 2003
Designing of an effective acceleration part of a supercollider generates some optimization problems , . 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) . 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.
Demmel Dzh. Vychislitel'naya lineynaya algebra. Teoriya i prilozheniya. — M.: Mir, 2001.