Nonlinear Rayleigh functionals

Schwetlick, Hubert; Schreiber, Kathrin

doi:10.1016/j.laa.2010.06.048

Cited by 35 publications

(24 citation statements)

References 23 publications

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“…1 shows the graph of the function ( ) γ µ of the parametric eigenvalue problem (11) and the minimal simple eigenvalue 1.3248 λ = of the nonlinear eigenvalue problem (10). We see that the experimental results are consistent with the theoretical results in Theorems 1 and 2.…”

Section: Variational Statement Of the Problemsupporting

confidence: 73%

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Finite difference approximation of electron balance problem in the stationary high-frequency induction discharges

́ev

Chebakova

2017

MATEC Web Conf.

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Abstract. The problem of finding the minimal eigenvalue corresponding to a positive eigenfunction of the nonlinear eigenvalue problem for the ordinary differential equation with coefficients depending on a spectral parameter is investigated. This problem arises in modeling the plasma of radio-frequency discharge at reduced pressures. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. A sufficient condition for the existence of a minimal eigenvalue corresponding to a positive eigenfunction of the finite difference nonlinear eigenvalue problem is established. Error estimates for the approximate eigenvalue and the corresponding approximate positive eigenfunction are proved. Investigations of this paper generalize well known results for eigenvalue problems with linear dependence on the spectral parameter.

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Section: Variational Statement Of the Problemsupporting

confidence: 73%

“…Numerical methods for solving matrix eigenvalue problems with nonlinear dependence on the parameter were constructed and investigated in the papers [5][6][7][8][9][10][11][12][13]. Mesh methods for solving differential nonlinear eigenvalue problems were studied in [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Finite difference approximation of electron balance problem in the stationary high-frequency induction discharges

́ev

Chebakova

2017

MATEC Web Conf.

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“…Several nonlinear eigensolvers that were applied to the problem were compared in [5], which in turn partly motivated the research in [6][7][8][9][10]. Each of these solvers required the solution of many symmetric indefinite tridiagonal systems.…”

Section:   R Nmentioning

confidence: 99%

Designing Optical Fibers: Fitting the Derivatives of a Nonlinear Pde-Eigenvalue Problem

Kaufman¹,

Bang²,

Heacook³

et al. 2012

AJCM

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When trying to fit data to functions of the eigensystem of a PDE-eigenvalue problem, such as Maxwell's equation, numerical differentiation is ineffective and analytic gradients must be supplied. In our motivating example of trying to determine the chemical composition of the layers of specialty optical fibers, the function involved fitting the higher order derivatives with respect to frequency of the positive eigenvalues. The computation of the gradient was the most time consuming part of the minimization problem. It was realized that if one interchanged the order of differentiation, and differentiated first with respect to the design parameters, fewer derivatives of the eigenvectors would be required and one could take full advantage that each grid point was affected by only a few variables. As the model was expanded to cover a fiber wrapped around a spool, the bandwidth of the linearized symmetric eigenvalue problem increased. At the heart of each of the iterative methods used to find the few positive eigenvalues was a symmetric, banded, indefinite matrix. Here we present an algorithm for this problem which reduces a symmetric banded matrix to a block diagonal matrix of 1 × 1 and 2 × 2 blocks. Fillin outside the band because of pivoting for stability is prevented by a sequence of planar transformations. Computationally the algorithm is compared to the block unsymmetric banded solver and the block positive definite symmetric band solver in LAPACK.

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“…Thus, either a shift-and-invert technique that transforms the problem and maps the desired eigenvalue to the periphery is necessary or other techniques like Newton's method [37,38] or the Jacobi-Davidson method [39,40] have to be used. All these methods require the solution of linear systems of the form (λ 2 M +λD + K)x = b with a given shift-pointλ.…”

Section: Complex Symmetric Quadratic Eigenvalue Problemsmentioning

confidence: 99%

Nonlinear eigenvalue and frequency response problems in industrial practice

Mehrmann

Schröder

2011

Mathematics in Industry

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Background: We discuss the numerical solution of large scale nonlinear eigenvalue problems and frequency response problems that arise in the analysis, simulation and optimization of acoustic fields. We report about the cooperation with the company SFE in Berlin. We present the challenges in the current industrial problems and the state-of-the-art of current methods. Results: The difficulties that arise with current off-the-shelf methods are discussed and several industrial examples are presented. Conclusions: It is documented that industrial cooperation is by no means a one-way street of transfer from academia to industry but the challenges arising in industrial practice also lead to new mathematical questions which actually change the mathematical theory and methods.Keywords Nonlinear eigenvalue problem · frequency response problem · complex symmetric linear system · block-Arnoldi method · acoustic field computation · automotive industry AMS Subject Classification 65F18 · 15A18 Background IntroductionTraffic noise emissions by transport vehicles, such as cars, trains or airplanes are one of the key factors restricting the quality of life in urban areas. Acoustic waves in V Mehrmann ( ) · C Schröder

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Nonlinear Rayleigh functionals

Cited by 35 publications

References 23 publications

Finite difference approximation of electron balance problem in the stationary high-frequency induction discharges

Finite difference approximation of electron balance problem in the stationary high-frequency induction discharges

Designing Optical Fibers: Fitting the Derivatives of a Nonlinear Pde-Eigenvalue Problem

Nonlinear eigenvalue and frequency response problems in industrial practice

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