Computing resonant modes of accelerator cavities by solving nonlinear eigenvalue problems via rational approximation

Beeumen, Roel Van; Marques, Osni; Ng, Esmond G.; Yang, Chao; Bai, Zhaojun; Ge, Lixin; Kononenko, O.; Li, Zenghai; Ng, Cho-Kuen; Xiao, Liling

doi:10.1016/j.jcp.2018.08.017

Cited by 11 publications

(11 citation statements)

References 34 publications

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“…54-58; notably, the Compact Rational Krylov Method (CORK) 57 has been used to compute eigenmodes of a TESLA cavity cryomodule subject to external losses by solving the corresponding large-scale NLEVP. 59 In this work, we utilize Beyn's algorithm to solve the NLEVP (3) of the reduced SSM (2). The thereby found solutions are individually refined by a subsequent Newton iteration.…”

Section: Solving the Nonlinear Eigenvalue Problemmentioning

confidence: 99%

“…The initial pair (λ 0 , x 0 ) must be known a priori , however, the determination of a "good" initial guess remains an open question. 59 Commonly, the starting pairs are obtained by solving a linearized system or (additionally) sampling the domain of interest using a grid or Monte Carlo methods. 61 In this work, the solutions found by Beyn's algorithm serve as initial solutions.…”

Section: Computation Of Lossy Higher Order Modes In Complex Srf Cavitiesmentioning

confidence: 99%

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Computation of lossy higher order modes in complex SRF cavities using Beyn’s and Newton’s methods on reduced order models

Pommerenke

Heller

Zadeh

et al. 2019

Int. J. Mod. Phys. A

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Superconducting radio frequency cavities meet the demanding performance requirements of modern accelerators and high-brilliance light sources. Their design requires a precise knowledge of their electromagnetic resonances. A numerical solution of Maxwell's equations is required to compute the resonant eigenmodes, their frequencies and losses due to the complex cavity shape. The consideration of resonances damped by external losses leads to a nonlinear eigenvalue problem. Previous work showed that, using State-Space Concatenation to construct a reduced order model and Newton iteration to solve the arising eigenvalue problem, solutions can be obtained on workstation computers even for large-scale problems without extensive simplification of the structure itself. In this paper, we augment the solution workflow by Beyn's contour integral algorithm to increase the number of found eigenmodes. Numerical experiments are presented for one academic and two real-life superconducting cavities and partially compared to measurements.

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Section: Solving the Nonlinear Eigenvalue Problemmentioning

confidence: 99%

Section: Computation Of Lossy Higher Order Modes In Complex Srf Cavitiesmentioning

confidence: 99%

Computation of lossy higher order modes in complex SRF cavities using Beyn’s and Newton’s methods on reduced order models

Pommerenke

Heller

Zadeh

et al. 2019

Int. J. Mod. Phys. A

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“…Another example appears in the spectral analysis of delay differential equations [Michiels and Niculescu, 2014]. In some applications, the nonlinear character comes from using special boundary conditions for the solution of differential equations [Vandenberghe et al, 2014, van Beeumen et al, 2018.…”

Section: Introductionmentioning

confidence: 99%

Nep

Campos

Román

2021

ACM Trans. Math. Softw.

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SLEPc is a parallel library for the solution of various types of large-scale eigenvalue problems. Over the past few years, we have been developing a module within SLEPc, called NEP, that is intended for solving nonlinear eigenvalue problems. These problems can be defined by means of a matrix-valued function that depends nonlinearly on a single scalar parameter. We do not consider the particular case of polynomial eigenvalue problems (which are implemented in a different module in SLEPc) and focus here on rational eigenvalue problems and other general nonlinear eigenproblems involving square roots or any other nonlinear function. The article discusses how the NEP module has been designed to fit the needs of applications and provides a description of the available solvers, including some implementation details such as parallelization. Several test problems coming from real applications are used to evaluate the performance and reliability of the solvers.

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“…In this context, the elimination corresponds to an elimination of one of the domains. The elimination of an outer domain, in a way that directly leads to NEPs, by introduction of artificial boundary conditions is the origin of several standard NEPs in the literature, e.g., [44] and the electromagnetic cavity model in [48].…”

mentioning

confidence: 99%

Nonlinearizing Two-parameter Eigenvalue Problems

Ringh¹,

Jarlebring²

2021

SIAM J. Matrix Anal. Appl.

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We investigate a technique to transform a linear two-parameter eigenvalue problem into a nonlinear eigenvalue problem (NEP). The transformation stems from an elimination of one of the equations in the two-parameter eigenvalue problem, by considering it as a (standard) generalized eigenvalue problem. We characterize the equivalence between the original and the nonlinearized problem theoretically and show how to use the transformation computationally. Special cases of the transformation can be interpreted as a reversed companion linearization for polynomial eigenvalue problems, as well as a reversed (less known) linearization technique for certain algebraic eigenvalue problems with square-root terms. Moreover, by exploiting the structure of the NEP we present algorithm specializations for NEP methods, although the technique also allows general solution methods for NEPs to be directly applied. The nonlinearization is illustrated in examples and simulations, with focus on problems where the eliminated equation is of much smaller size than the other twoparameter eigenvalue equation. This situation arises naturally in domain decomposition techniques. A general error analysis is also carried out under the assumption that a backward stable eigensolver is used to solve the eliminated problem, leading to the conclusion that the error is benign in this situation.

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Computing resonant modes of accelerator cavities by solving nonlinear eigenvalue problems via rational approximation

Cited by 11 publications

References 34 publications

Computation of lossy higher order modes in complex SRF cavities using Beyn’s and Newton’s methods on reduced order models

Computation of lossy higher order modes in complex SRF cavities using Beyn’s and Newton’s methods on reduced order models

Nep

Nonlinearizing Two-parameter Eigenvalue Problems

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