Solving Maxwell eigenvalue problems for accelerating cavities

Arbenz, Peter; Geus, Roman; Adam, Stefan

doi:10.1103/physrevstab.4.022001

Cited by 31 publications

(51 citation statements)

References 26 publications

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“…References [1,5,6] and references therein. One such application is the analysis of the electromagnetic cavity resonator.…”

Section: The Problemmentioning

confidence: 99%

“…We assume that a sparse basis C for the null space of A is available, as is the case in several applications; see for instance References [1][2][3] and their references. A relevant feature of the considered setting is that the dimension of the null space is very high, compared with the problem dimension, reaching up to, say, one-third of the entire problem dimension.…”

Section: The Problemmentioning

confidence: 99%

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Algebraic formulations for the solution of the nullspace‐free eigenvalue problem using the inexact Shift‐and‐Invert Lanczos method

Simoncini

2002

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SUMMARYGiven the generalized symmetric eigenvalue problem Ax = Mx, with A semideÿnite and M deÿnite, we analyse some algebraic formulations for the approximation of the smallest non-zero eigenpairs, assuming that a sparse basis for the null space is available. In particular, we consider the inexact version of the Shift-and-Invert Lanczos method, and we show that apparently di erent algebraic formulations provide the same approximation iterates, under some natural hypotheses. Our results suggest that alternative strategies need to be explored to really take advantage of the special problem setting, other than reformulating the algebraic problem. Experiments on a real application problem corroborate our theoretical ÿndings. Copyright ? 2002 John Wiley & Sons, Ltd.KEY WORDS: generalized eigenvalue problem; Shift-and-Invert Lanczos; iterative system solvers 1. THE PROBLEM We are interested in the analysis of some algebraic formulations for the computation of some of the smallest non-zero positive eigenvalues and associated eigenvectors of the large n × n generalized eigenvalue problemwith M symmetric positive deÿnite and A symmetric positive semideÿnite. We assume that a sparse basis C for the null space of A is available, as is the case in several applications; see for instance References [1-3] and their references. A relevant feature of the considered setting is that the dimension of the null space is very high, compared with the problem dimension, reaching up to, say, one-third of the entire problem dimension. The fact that C is a basis for the null space of A plays a crucial role in our analysis, therefore we emphasize the constraint in terms of null space orthogonality. When only the smallest non-zero eigenvalue is requested,

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“…References [1,5,6] and references therein. One such application is the analysis of the electromagnetic cavity resonator.…”

Section: The Problemmentioning

confidence: 99%

Section: The Problemmentioning

confidence: 99%

Algebraic formulations for the solution of the nullspace‐free eigenvalue problem using the inexact Shift‐and‐Invert Lanczos method

Simoncini

2002

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“…This algorithm is well-suited since it does not require the factorization of the matrices A or M . In [2,3,4] we found JDSYM to be the method of choice for this problem.…”

Section: The Eigensolvermentioning

confidence: 99%

On a parallel multilevel preconditioned Maxwell eigensolver

et al. 2006

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We report on a parallel implementation of the Jacobi-Davidson algorithm to compute a few eigenvalues and corresponding eigenvectors of a large real symmetric generalized matrix eigenvalue problemThe eigenvalue problem stems from the design of cavities of particle accelerators. It is obtained by the finite element discretization of the time-harmonic Maxwell equation in weak form by a combination of Nédélec (edge) and Lagrange (node) elements.We found the Jacobi-Davidson (JD) method to be a very effective solver provided that a good preconditioner is available for the correction equations that have to be solved in each step of the JD iteration. The preconditioner of our choice is a combination of a hierarchical basis preconditioner and the ML smoothed aggregation AMG preconditioner. It is close to optimal regarding iteration count.The parallel code makes extensive use of the Trilinos software framework. In our examples from accelerator physics we observe satisfactory speedups and efficiencies.

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“…This is very time consuming even if the conjugate gradient method is applied with a good preconditioner. In most of our experiments the JacobiDavidson algorithm was much more effective than the implicitly restarted Lanczos algorithm as implemented in ARPACK [5] for solving the cavity and other eigenvalue problems [2,3]. In this study we investigate three factorization-free algorithms for solving (13)…”

Section: Solving the Matrix Eigenvalue Problemmentioning

confidence: 99%

“…In earlier studies [2,3] we found that for large eigenvalue problems the JacobiDavidson algorithm [4] was superior to the Lanczos algorithm or the restarted Lanczos algorithm as implemented in ARPACK [5]. While the Jacobi-Davidson algorithm retains the high rate of convergence it only poses small accuracy requirements on the solution of the so-called correction equation, at least in the initial steps of iterations.…”

Section: Introductionmentioning

confidence: 99%