Solutions Of Nonlinear Eigenvalue Problems Research Articles

Inverse Nonlinear Eigenvalue Problem Framework for the Synthesis of Coupled-Resonator Filters With Nonresonant Nodes and Arbitrary Frequency-Variant Reactive Couplings

A novel, general circuit-level description of coupled-resonator microwave filters is introduced in this article. Unlike well-established coupling-matrix models based on frequency-invariant couplings or linear frequency-variant couplings (LFVCs), a model with arbitrary reactive frequency-variant coupling (AFVC) networks is proposed. The engineered formulation is more general than prior-art ones—with the only restriction that the coupling network is a reactive-type two-port circuit—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, which allows for AFVCs with highly nonlinear (even singular) characteristics, loaded or unloaded nonresonating 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, the 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).

Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems

This paper presents a definition for local linearizations of rational matrices and studies their properties. This definition allows to introduce matrix pencils associated to a rational matrix that preserve its structure of zeros and poles in subsets of any algebraically closed field and also at infinity. This new theory of local linearizations captures and explains rigorously the properties of all the different pencils that have been used from the 1970's until 2020 for computing zeros, poles and eigenvalues of rational matrices. Particular attention is paid to those pencils that have appeared recently in the numerical solution of nonlinear eigenvalue problems through rational approximation.

On one variant of strongly nonlinear Gagliardo–Nirenberg inequality involving Laplace operator with application to nonlinear elliptic problems

We obtain the inequality \int_{\Omega}|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_{\Omega} \left( \sqrt{ |\Delta u(x)||{\mathcal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx, where \Omega\subset \mathbb R^n is a bounded Lipschitz domain, u\in W^{2,1}_{\mathrm {loc}}(\Omega) is positive and obeys some additional assumptions, \Delta u is the Laplace operator, {\mathcal T}_{h,C}(\cdot) is certain transformation of the continuous function h(\cdot) . We also explain how to apply such inequality to deduce regularity for solutions of nonlinear eigenvalue problems of elliptic type for degenerated PDEs, with the illustration within the model of electrostatic micromechanical systems (MEMS).

Global Bifurcation of Positive Solutions of Semi-Linear Elliptic Partial Differential Equations with Indefinite Weight

In this paper we consider global bifurcation of solutions of nonlinear eigenvalue problems for second order uniformly elliptic equations with an indefinite weight function and the Dirichlet boundary condition. We show the existence of an unbounded continua contained in the classes of positive and negative functions and bifurcating from the points of the line of trivial solutions corresponding to the principal eigenvalues of the linearized problem.

Conversions between barycentric, RKFUN, and Newton representations of rational interpolants

We derive explicit formulas for converting between rational interpolants in barycentric, rational Krylov (RKFUN), and Newton form. We show applications of these conversions when working with rational approximants produced by the AAA algorithm Y. Nakatsukasa et al. (2018) [6] within the Rational Krylov Toolbox and for the solution of nonlinear eigenvalue problems.

Verified solutions of delay eigenvalue problems

A numerical algorithm for enclosing solutions of nonlinear eigenvalue problems arising in delay differential equations is proposed. This algorithm verifies the uniqueness of the enclosed eigenvalue when it is simple, and is applicable even for multiple eigenvalues. Numerical results show properties of the algorithm.

Computation of pseudospectral abscissa for large-scale nonlinear eigenvalue problems

Author(s): Meerbergen, K; Michiels, W; Van Beeumen, R; Mengi, E | Abstract: We present an algorithm to compute the pseudospectral abscissa for a nonlinear eigenvalue problem. The algorithm relies on global under-estimator and over-estimator functions for the eigenvalue and singular value functions involved. These global models follow from eigenvalue perturbation theory. The algorithm has three particular features. First, it converges to the globally rightmost point of the pseudospectrum, and it is immune to nonsmoothness. The global convergence assertion is under the assumption that a global lower bound is available for the second derivative of a singular value function depending on one parameter. It may not be easy to deduce such a lower bound analytically, but assigning large negative values works robustly in practice. Second, it is applicable to large-scale problems since the dominant cost per iteration stems from computing the smallest singular value and associated singular vectors, for which efficient iterative solvers can be used. Furthermore, a significant increase in computational efficiency can be obtained by subspace acceleration, that is, by restricting the domains of the linear maps associated with the matrices involved to small but suitable subspaces, and solving the resulting reduced problems. Occasional restarts of these subspaces further enhance the efficiency for large-scale problems. Finally, in contrast to existing iterative approaches based on constructing low-rank perturbations and rightmost eigenvalue computations, the algorithm relies on computing only singular values of complex matrices. Hence, the algorithm does not require solutions of nonlinear eigenvalue problems, thereby further increasing efficiency and reliability. This work is accompanied by a robust implementation of the algorithm that is publicly available.

Solving large‐scale nonlinear eigenvalue problems by rational interpolation and resolvent sampling based Rayleigh–Ritz method

SummaryNumerical solution of nonlinear eigenvalue problems (NEPs) is frequently encountered in computational science and engineering. The applicability of most existing methods is limited by the matrix structures, properties of the eigen‐solutions, sizes of the problems, etc. This paper aims to remove those limitations and develop robust and universal NEP solvers for large‐scale engineering applications. The novelty lies in two aspects. First, a rational interpolation approach (RIA) is proposed based on the Keldysh theorem for holomorphic matrix functions. Comparing with the existing contour integral approach, the RIA provides the possibility to select sampling points in more general regions and has advantages in improving the accuracy and reducing the computational cost. Second, a resolvent sampling scheme using the RIA is proposed to construct reliable search spaces for the Rayleigh–Ritz procedure, based on which a robust eigen‐solver, called resolvent sampling based Rayleigh–Ritz method (RSRR), is developed for solving general NEPs. The RSRR can be easily implemented and parallelized. The advantages of the RIA and the performance of the RSRR are demonstrated by a variety of benchmark and application examples. Copyright © 2016 John Wiley & Sons, Ltd.

A Multigrid Method for Ground State Solution of Bose-Einstein Condensates

AbstractA multigrid method is proposed to compute the ground state solution of Bose-Einstein condensations by the finite element method based on the multilevel correction for eigenvalue problems and the multigrid method for linear boundary value problems. In this scheme, obtaining the optimal approximation for the ground state solution of Bose-Einstein condensates includes a sequence of solutions of the linear boundary value problems by the multigrid method on the multilevel meshes and some solutions of nonlinear eigenvalue problems some very low dimensional finite element space. The total computational work of this scheme can reach almost the same optimal order as solving the corresponding linear boundary value problem. Therefore, this type of multigrid scheme can improve the overall efficiency for the simulation of Bose-Einstein condensations. Some numerical experiments are provided to validate the efficiency of the proposed method.

NLEIGS: A Class of Fully Rational Krylov Methods for Nonlinear Eigenvalue Problems

A new rational Krylov method for the efficient solution of nonlinear eigenvalue problems is proposed. This iterative method, called fully rational Krylov method for nonlinear eigenvalue problems (abbreviated as NLEIGS), is based on linear rational interpolation and generalizes the Newton rational Krylov method proposed in [R. Van Beeumen, K. Meerbergen, and W. Michiels, SIAM J. Sci. Comput., 35 (2013), pp. A327-A350]. NLEIGS utilizes a dynamically constructed rational interpolant of the nonlinear operator and a new companion-type linearization for obtaining a generalized eigenvalue problem with special structure. This structure is particularly suited for the rational Krylov method. A new approach for the computation of rational divided differences using matrix functions is presented. It is shown that NLEIGS has a computational cost comparable to the Newton rational Krylov method but converges more reliably, in particular, if the nonlinear operator has singularities nearby the target set. Moreover, NLEIGS implements an automatic scaling procedure which makes it work robustly independent of the location and shape of the target set, and it also features low-rank approximation techniques for increased computational efficiency. Small- and large-scale numerical examples are included.

One Approach to Construction of Bilateral Approximations Methods for Solution of Nonlinear Eigenvalue Problems

In this paper a new approach to construction of iterative methods of bilateral approximations of eigenvalue is proposed and investigated. The conditions on initial approximation, which ensure the convergence of iterative processes, are obtained.

Positive Solutions of Nonlinear Eigenvalue Problems for a Nonlocal Fractional Differential Equation

By using the fixed point theorem, positive solutions of nonlinear eigenvalue problems for a nonlocal fractional differential equation are considered, where 1 < α ≤ 2 is a real number, λ is a positive parameter, is the standard Riemann‐Liouville differentiation, and ξi ∈ (0,1), αi ∈ [0, ∞) with , a(t) ∈ C([0,1], [0, ∞)), f(t, u) ∈ C([0, ∞), [0, ∞)).

Estimates on Pull-In Distances in Microelectromechanical Systems Models and Other Nonlinear Eigenvalue Problems

Motivated by certain mathematical models for microelectromechanical systems (MEMS), we give upper and lower $L^\infty$ estimates for the minimal solutions of nonlinear eigenvalue problems of the form $-\Delta u=\lambda f(x)F(u)$ on a smooth bounded domain $\Omega$ in $\mathbb{R}^N$. We are mainly interested in the pull-in distance, that is, the $L^\infty$-norm of the extremal solution $u^*$ and how it depends on the geometry of the domain, the dimension of the space, and the so-called permittivity profile f. In particular, our results provide mathematical proofs for various observed phenomena as well as rigorous derivations for several estimates obtained numerically by Pelesko [SIAM J. Appl. Math., 62 (2002), pp. 888–908], Guo, Pan, and Ward [SIAM J. Appl. Math., 66 (2005), pp. 309–338], and others in the case of the MEMS nonlinearity $F(u)=\frac{1}{(1-u)^2}$ and for power-law permittivity profiles $f(x)=|x|^\alpha$.

Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method

A general analytic approach for nonlinear eigenvalue problems is described. Two physical problems are used as examples to show the validity of this approach for eigenvalue problems with either periodic or non-periodic eigenfunctions. Unlike perturbation techniques, this approach is independent of any small physical parameters. Besides, different from all other analytic techniques, it provides a simple way to ensure the convergence of series of eigenvalues and eigenfunctions so that one can always get accurate enough approximations. Finally, unlike all other analytic techniques, this approach provides great freedom to choose an auxiliary linear operator so as to approximate the eigenfunction more effectively by means of better base functions. This approach provides us a new way to investigate eigenvalue problems with strong nonlinearity.

Variational methods in relativistic quantum mechanics

This review is devoted to the study of stationary solutions of lin- ear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy func- tional. Contrary to the Laplacian appearing in the equations of nonrelativistic quantum mechanics, the Dirac operator has a negative continuous spectrum which is not bounded from below. This has two main consequences. First, the energy functional is strongly indefinite. Second, the Euler-Lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap (between the negative and positive continuous spectra). Moreover, since we work in the space domain R 3 , the Palais-Smale condition is not satisfied. For these reasons, the problems discussed in this review pose a challenge in the Calculus of Variations. The existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems. In the first part, we consider the fixed eigenvalue problem for models of a free self-interacting relativistic particle. They allow to describe the localized state of a spin-1/2 particle (a fermion) which propagates without changing its shape. This includes the Soler models, and the Maxwell-Dirac or Klein- Gordon-Dirac equations. The second part is devoted to the presentation of min-max principles al- lowing to characterize and compute the eigenvalues of linear Dirac operators with an external potential, in the gap of their essential spectrum. Many con- sequences of these min-max characterizations are presented, among them a new kind of Hardy-like inequalities and a stable algorithm to compute the eigenvalues. In the third part we look for normalized solutions of nonlinear eigenvalue problems. The eigenvalues are Lagrange multipliers, lying in a spectral gap. We review the results that have been obtained on the Dirac-Fock model which is a nonlinear theory describing the behavior of N interacting electrons in an external electrostatic field. In particular we focus on the problematic definition of the ground state and its nonrelativistic limit. In the last part, we present a more involved relativistic model from Quan- tum Electrodynamics in which the behavior of the vacuum is taken into ac- count, it being coupled to the real particles. The main interesting feature of this model is that the energy functional is now bounded from below, providing us with a good definition of a ground state.

Multiple positive solutions for nonlinear eigenvalue problems with the p-Laplacian

In this paper we consider a nonlinear eigenvalue problem driven by the p -Laplacian differential operator and with a nonsmooth potential. Using degree theoretic arguments based on the degree map for certain operators of monotone type, we show that the problem has at least two nontrivial positive solutions as the parameter λ > 0 varies in a half-line.

Constant-sign and sign-changing solutions for nonlinear eigenvalue problems

We prove the existence of multiple constant-sign and sign-changing solutions for a nonlinear elliptic eigenvalue problem under Dirichlet boundary condition involving the p -Laplacian. More precisely, we establish the existence of a positive solution, of a negative solution, and of a nontrivial sign-changing solution when the eigenvalue parameter λ is greater than the second eigenvalue λ 2 of the negative p -Laplacian, extending results by Ambrosetti–Lupo, Ambrosetti–Mancini, and Struwe. Our approach relies on a combined use of variational and topological tools (such as, e.g., critical points, Mountain-Pass theorem, second deformation lemma, variational characterization of the first and second eigenvalue of the p -Laplacian) and comparison arguments for nonlinear differential inequalities. In particular, the existence of extremal nontrivial constant-sign solutions plays an important role in the proof of sign-changing solutions.

Existence results for time scale boundary value problem

In this paper, we establish some existence results for positive solutions to a class of singular boundary value problem on time scale by using the Krasnosel'skii fixed point theorem. Two examples are presented as applications. The conditions we used in this paper are different from those in [D.R. Anderson, Eigenvalue intervals for a two-point boundary value problem on a measure chain, J. Comput. Appl. Math. 141 (2002) 57–64; C.J. Chyan, J. Henderson, Eigenvalues problems for nonlinear differential equations on a measure chain, J. Math. Anal. Appl. 245 (2000) 547–559; L.H. Erbe, A. Peterson Positive solutions for nonlinear differential equation on a measure chain, Math. Comput. Modelling 32 (2000) 571–585; L.H. Erbe, H.Y. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994) 743–748; J. Henderson, H.Y. Wang, Positive solutions for nonlinear eigenvalue problems, J. Math. Anal. Appl. 208 (1997) 252–259; C.H. Hong, C.C. Yeh, Positive solutions for eigenvalue problems on a measure chain, Nonlinear Anal. 51 (2002) 499–507; W.C. Lian; W.F. Wong; C.C. Yeh, On the existence of positive solutions of nonlinear differential equations, Proc. Amer. Math. Soc. 124 (1996) 1117–1126; J. Liang, T.J. Xiao, Z.C. Hao, Positive solutions of singular differential equations on measure chain, Comput. Math. Appl. 49 (2005) 651–663].

The dynamic element method for analysis of frame and cable type structures

With the dynamic element method (DEM), results more accurate than with the conventional finite element method (FEM) are obtained with the same number of degrees of freedom. This is due to the introduction of shape functions of polynomial type, introducing frequency dependence into the mass matrix expressions. It is demonstrated how this affects free vibration analysis, including the solution of nonlinear eigenvalue problems. Various numerical techniques for solving these polynomial problems are discussed. The polynomial matrix formulations for stiffness and mass matrices are given for a beam, a bar and a cable element. Numerical examples demonstrate how the DEM can be implemented for modelling of frame type structures and its efficiency is compared to that of the conventional FEM.

Solution Algorithm of Large Scale Eigenvalue Analysis with Damping.

This paper presents a solution algorithm of nonlinear eigenvalue problem which occurs in an eigenvalue analysis of a structure with damping. The solution of nonlinear eigenvalue problems usually require operations of a complex matrix and complex vectors to generate Krylov subspace. The proposed algorithm is especially effective in terms of large scale analysis, because it does not use complex matrix nor complex vectors. Therefore, the number of operations and the size of memory used in the analysis is reduced. Furthermore, the accuracy of the solutions is confirmed to be satisfactory.