scholarly journals On the Variational Eigenvalues Which Are Not of Ljusternik-Schnirelmann Type

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Pavel Drábek
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problem ◽  
Linear Case ◽  
Minimax Principle ◽  
Variational Characterization ◽  
Nonlinear Eigenvalue

We discuss nonlinear homogeneous eigenvalue problems and the variational characterization of their eigenvalues. We focus on the Ljusternik-Schnirelmann method, present one possible alternative to this method and compare it with the Courant-Fischer minimax principle in the linear case. At the end we present a special nonlinear eigenvalue problem possessing an eigenvalue which allows the variational characterization but is not of Ljusternik-Schnirelmann type.

scholarly journals On a New Characterization of Some Class Nonlinear Eigenvalue Problem

2019 ◽  
Vol 2019 ◽  
pp. 1-7 ◽  
Author(s):  
Lutfi Akin
Keyword(s):  
Asymptotic Behavior ◽  
Eigenvalue Problem ◽  
Nonlinear Eigenvalue Problem ◽  
Normal Mode Analysis ◽  
Mode Analysis ◽  
Electrical System ◽  
Eigenvalues And Eigenfunctions ◽  
Complete Study ◽  
Nonlinear Eigenvalue

A normal mode analysis of a vibrating mechanical or electrical system gives rise to an eigenvalue problem. Faber made a fairly complete study of the existence and asymptotic behavior of eigenvalues and eigenfunctions, Green’s function, and expansion properties. We will investigate a new characterization of some class nonlinear eigenvalue problem.

RESTARTING PROJECTION METHODS FOR RATIONAL EIGENPROBLEMS ARISING IN FLUID‐SOLID VIBRATIONS

2008 ◽  
Vol 13 (2) ◽  
pp. 171-182 ◽  
Author(s):  
Marta M. Betcke ◽  
Heinrich Voss
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Projection Methods ◽  
Nonlinear Eigenvalue Problems ◽  
Solid Structure ◽  
Minmax Characterization ◽  
Nonlinear Eigenvalue

For nonlinear eigenvalue problems T(λ)x = 0 satisfying a minmax characterization of its eigenvalues iterative projection methods combined with safeguarded iteration are suitable for computing all eigenvalues in a given interval. Such methods hit their limitations if a large number of eigenvalues is required. In this paper we discuss restart procedures which are able to cope with this problem, and we evaluate them for a rational eigenvalue problem governing vibrations of a fluid‐solid structure.

scholarly journals Inverse Nonlinear Eigenvalue Problem Framework for the Synthesis of Coupled-Resonator Filters With Non-Resonant Nodes and Arbitrary Frequency-Variant Couplings

2021 ◽  
Author(s):  
Michal Mrozowski ◽  
Adam Lamecki ◽  
Martyna Mul ◽  
Roberto Gómez-García
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problem ◽  
Scattering Parameters ◽  
Highly Nonlinear ◽  
Special Cases ◽  
Arbitrary Frequency ◽  
Nonlinear Eigenvalue ◽  
Coupled Resonator ◽  
Zeros And Poles

A novel, general circuit-level description of coupled-resonator microwave filters is introduced in this paper. Unlike well-established coupling-matrix models based on frequency-invariant couplings or linear frequency-variant couplings (LFVCs), a model with arbitrary frequency-variant coupling (AFVC) coefficients is proposed. The engineered formulation is more general than prior-art ones and can be treated as an extension of previous synthesis models, since constant or linear couplings are special cases of arbitrary frequency dependence. The suggested model is fully general, allows for AFVCs with highly nonlinear (even singular) characteristics, loaded or unloaded non-resonating nodes (NRNs), frequency-dependent source-load coupling, multiple frequency-variant cross-couplings, and{/}or multiple dispersive couplings for connecting the source and load to the filter network. The model is accompanied by a powerful synthesis technique that is based on the zeros and poles of the admittance or scattering parameters and the eigenvalues of properly defined eigenproblems. In the most general case, the zeros and poles of the admittance or scattering parameters are related to solutions of nonlinear eigenvalue problems. The synthesis is defined as an inverse nonlinear eigenvalue problem (INEVP) where the matrix is constructed from three sets of eigenvalues. This is accomplished by optimization using an iterative nonlinear least-squares solver with excellent convergence property. Finally, third- and fifth-order examples of bandpass filter topologies involving AFVCs are shown, and the experimental validation of the proposed theory is presented through the manufacturing and characterization of a microstrip filter prototype with transmission zeros (TZs)

An Adaptive Finite Element Method with Hybrid Basis for Singularly Perturbed Nonlinear Eigenvalue Problems

2016 ◽  
Vol 19 (2) ◽  
pp. 442-472
Author(s):  
Ye Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problem ◽  
Singularly Perturbed ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Eigenvalue

AbstractIn this paper, we propose an uniformly convergent adaptive finite element method with hybrid basis (AFEM-HB) for the discretization of singularly perturbed nonlinear eigenvalue problems under constraints with applications in Bose-Einstein condensation (BEC) and quantum chemistry. We begin with the time-independent Gross-Pitaevskii equation and show how to reformulate it into a singularly perturbed nonlinear eigenvalue problem under a constraint. Matched asymptotic approximations for the problem are reviewed to confirm the asymptotic behaviors of the solutions in the boundary/interior layer regions. By using the normalized gradient flow, we propose an adaptive finite element with hybrid basis to solve the singularly perturbed nonlinear eigenvalue problem. Our basis functions and the mesh are chosen adaptively to the small parameter ε. Extensive numerical results are reported to show the uniform convergence property of our method. We also apply the AFEM-HB to compute the ground and excited states of BEC with box/harmonic/optical lattice potential in the semiclassical regime (0 <ε≪C 1). In addition, we give a detailed error analysis of our AFEM-HB to a simpler singularly perturbed two point boundary value problem, show that our method has a minimum uniform convergence order

scholarly journals Inverse Nonlinear Eigenvalue Problem Framework for the Synthesis of Coupled-Resonator Filters With Non-Resonant Nodes and Arbitrary Frequency-Variant Couplings

2021 ◽  
Author(s):  
Michal Mrozowski ◽  
Adam Lamecki ◽  
Martyna Mul ◽  
Roberto Gómez-García
Keyword(s):  
Eigenvalue Problem ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problem ◽  
Scattering Parameters ◽  
Highly Nonlinear ◽  
Special Cases ◽  
Arbitrary Frequency ◽  
Nonlinear Eigenvalue ◽  
Coupled Resonator ◽  
Zeros And Poles

A novel, general circuit-level description of coupled-resonator microwave filters is introduced in this paper. Unlike well-established coupling-matrix models based on frequency-invariant couplings or linear frequency-variant couplings (LFVCs), a model with arbitrary frequency-variant coupling (AFVC) coefficients is proposed. The engineered formulation is more general than prior-art ones and can be treated as an extension of previous synthesis models, since constant or linear couplings are special cases of arbitrary frequency dependence. The suggested model is fully general, allows for AFVCs with highly nonlinear (even singular) characteristics, loaded or unloaded non-resonating nodes (NRNs), frequency-dependent source-load coupling, multiple frequency-variant cross-couplings, and{/}or multiple dispersive couplings for connecting the source and load to the filter network. The model is accompanied by a powerful synthesis technique that is based on the zeros and poles of the admittance or scattering parameters and the eigenvalues of properly defined eigenproblems. In the most general case, the zeros and poles of the admittance or scattering parameters are related to solutions of nonlinear eigenvalue problems. The synthesis is defined as an inverse nonlinear eigenvalue problem (INEVP) where the matrix is constructed from three sets of eigenvalues. This is accomplished by optimization using an iterative nonlinear least-squares solver with excellent convergence property. Finally, third- and fifth-order examples of bandpass filter topologies involving AFVCs are shown, and the experimental validation of the proposed theory is presented through the manufacturing and characterization of a microstrip filter prototype with transmission zeros (TZs)

A remark on multiple solutions for a nonlinear eigenvalue problem

2008 ◽  
pp. 497-507
Author(s):  
Gen-Qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Eigenvalue Problem ◽  
Multiple Solutions ◽  
Nonlinear Eigenvalue Problem ◽  
Nonlinear Eigenvalue

Variational characterization of real eigenvalues in linear viscoelastic oscillators

2017 ◽  
Vol 23 (10) ◽  
pp. 1377-1388 ◽  
Author(s):  
Seyyed Abbas Mohammadi ◽  
Heinrich Voss
Keyword(s):  
Degrees Of Freedom ◽  
Nonlinear Eigenvalue Problem ◽  
Eigenvalues And Eigenvectors ◽  
Variational Characterization ◽  
Viscoelastic System ◽  
New Approach ◽  
Motion Equations ◽  
Multiple Degrees Of Freedom ◽  
Linear Viscoelastic

This paper proposes a new approach for computing the real eigenvalues of a multiple-degrees-of-freedom viscoelastic system in which we assume an exponentially decaying damping. The free-motion equations lead to a nonlinear eigenvalue problem. If the system matrices are symmetric, the eigenvalues allow for a variational characterization of maxmin type, and the eigenvalues and eigenvectors can be determined very efficiently by the safeguarded iteration, which converges quadratically and, for extreme eigenvalues, monotonically. Numerical methods demonstrate the performance and the reliability of the approach. The method succeeds where some current approaches, with restrictive physical assumptions, fail.

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