Nonlinear Eigenvalue Problem Research Articles

Asymptotic expansion of oscillatory bifurcation curves of ODEs with nonlinear diffusion

We consider the nonlinear eigenvalue problem $$ [D(u)u']' + \lambda f(u) = 0, \ \ u(t) > 0, \ \ t \in I := (0,1), \ \ u(0) = u(1) = 0, $$ where $D(u) = u^p$, $f(u) = u^{q} + \sin (u^n)$ and $\lambda > 0$ is a bifurcation parameter. Here, $p \ge 0$, $n > 0$ and $q > 0$ are given constants and $k:=(p + q + 1)/2 \in \mathbb{N}$. This equation is motivated by the mathematical model of animal dispersal and invasion and $\lambda$ is parameterized by the maximum norm $\alpha = \Vert u_\lambda\Vert_\infty$ of the solution $u_\lambda$ associated with $\lambda$ and is written as $\lambda = \lambda(\alpha)$. We establish new precise asymptotic expansion formulas for $\lambda(\alpha)$ as $\alpha \to \infty$. In particular, we obtain the precise asymptotic behavior of $\lambda(\alpha)$ which tends to $0$ with oscillation as $\alpha \to \infty$.

Computing several eigenvalues of nonlinear eigenvalue problems by selection

Computing more than one eigenvalue for (large sparse) one-parameter polynomial and general nonlinear eigenproblems, as well as for multiparameter linear and nonlinear eigenproblems, is a much harder task than for standard eigenvalue problems. We present simple but efficient selection methods based on divided differences to do this. Selection means that the approximate eigenpair is picked from candidate pairs that satisfy a certain suitable criterion. The goal of this procedure is to steer the process away from already detected pairs. In contrast to locking techniques, it is not necessary to keep converged eigenvectors in the search space, so that the entire search space may be devoted to new information. The selection techniques are applicable to many types of matrix eigenvalue problems; standard deflation is feasible only for linear one-parameter problems. The methods are easy to understand and implement. Although the use of divided differences is well known in the context of nonlinear eigenproblems, the proposed selection techniques are new for one-parameter problems. For multiparameter problems, we improve on and generalize our previous work. We also show how to use divided differences in the framework of homogeneous coordinates, which may be appropriate for generalized eigenvalue problems with infinite eigenvalues. While the approaches are valuable alternatives for one-parameter nonlinear eigenproblems, they seem the only option for multiparameter problems.

Saturation phenomena for some classes of nonlinear nonlocal eigenvalue problems

Let us consider the following minimum problem \lambda_\alpha(p,r)=\min_{\substack{u\in W_{0}^{1,p}(-1,1)\\u\not\equiv0}}\frac{\int_{-1}^{1}|u'|^{p}dx+\alpha\left|\int_{-1}^{1}|u|^{r-1}u\, dx\right|^{\frac pr}}{\int_{-1}^{1}|u|^{p}dx}, where \alpha\in\mathbb R , p\ge 2 and \frac p2 \le r \le p . We show that there exists a critical value \alpha_C=\alpha_C (p,r) such that the minimizers have constant sign up to \alpha=\alpha_{C} and then they are odd when \alpha > \alpha_{C} .

A framework for second-order eigenvector centralities and clustering coefficients.

We propose and analyse a general tensor-based framework for incorporating second-order features into network measures. This approach allows us to combine traditional pairwise links with information that records whether triples of nodes are involved in wedges or triangles. Our treatment covers classical spectral methods and recently proposed cases from the literature, but we also identify many interesting extensions. In particular, we define a mutually reinforcing (spectral) version of the classical clustering coefficient. The underlying object of study is a constrained nonlinear eigenvalue problem associated with a cubic tensor. Using recent results from nonlinear Perron-Frobenius theory, we establish existence and uniqueness under appropriate conditions, and show that the new spectral measures can be computed efficiently with a nonlinear power method. To illustrate the added value of the new formulation, we analyse the measures on a class of synthetic networks. We also give computational results on centrality and link prediction for real-world networks.

Positive solutions to a nonlinear eigenvalue problem

Positive solutions to a nonlinear eigenvalue problem

SOLUTION BRANCHES OF NONLINEAR EIGENVALUE PROBLEMS ON RESTRICTED DOMAINS

We extend bifurcation results of nonlinear eigenvalue problems from real Banach spaces to any neighbourhood of a given point. For points of odd multiplicity on these restricted domains, we establish that the component of solutions through the bifurcation point either is unbounded, admits an accumulation point on the boundary, or contains an even number of odd-multiplicity points. In the simple-multiplicity case, we show that branches of solutions in the directions of corresponding eigenvectors satisfy similar conditions on such restricted domains.

Offshore-to-Nearshore Transformation of Wave Conditions and Directional Extremes with Application to Port Resonances in the Bay of Sitia-Crete

For coastal engineering studies and the efficient design of ports and harbors, reliable information concerning wave conditions in nearshore and coastal sites is needed. In the absence of long-term wave data at the site of interest, this becomes possible by using offshore data, which are usually available in the nearby geographical area, in addition to bathymetric and coastline information concerning the nearshore area and the local site. The latter are used in conjunction with a suitable wave model, which calculates the offshore-to-nearshore transformation of wave conditions and incorporates the relevant shallow-water phenomena. In the present work, the above methodology is applied to calculate the nearshore wave conditions in the Bay of Sitia, Crete, by applying the Simulating WAves Nearshore (SWAN) model. The interesting feature of the Bay of Sitia is its vulnerability due to strong erosion, which downgrades the touristic value of the beach. Furthermore, nearshore wave data offer valuable information concerning further coastal and port engineering studies. In this context, results from directional extreme value analysis of the nearshore wave conditions in the Sitia Bay are derived and used to investigate resonances in the enclosed marina of the Sitia port, by taking into account the depth variations inside the basin. To this end, a novel method was developed based on the modified mild-slope equation, in conjunction with the Finite Element Model, for the solution of the nonlinear eigenvalue problem.

Investigation of radiation damping in sandwich structures using finite and boundary element methods and a nonlinear eigensolver.

The fully coupled vibroacoustic interaction of sandwich panels is studied using the finite and the boundary element methods. The extent of radiation damping is quantified for various configurations based on both harmonic response analyses and modal analyses. The underlying nonlinear eigenvalue problem is solved using a projection method based on contour integration yielding the shifted (wet) eigenfrequencies, modal radiation loss factors, and air-loaded structural modes. The numerical results clearly illustrate the relevance of air-loading when studying the vibration of sandwich structures. Further, the numerically obtained estimates for radiation damping are compared to both theoretical expressions and experimental results found in the literature. Although good agreement is observed in general, the comparison indicates the limited applicability of commonly used theoretical expressions when coincidence occurs in a frequency range where the modes are still well separated. Moreover, possible sources of error when experimentally determining radiation damping are discussed in detail. The results presented in this paper provide deep insights into the phenomenon of acoustic radiation damping and help to estimate its relevance in future research.

Solution of Thermoacoustic Eigenvalue Problems With a Noniterative Method

Abstract Gas turbine combustors are prone to undesirable combustion dynamics in the form of thermoacoustic oscillations. Analysis of the stability of thermoacoustic systems in the frequency domain leads to nonlinear eigenvalue problems (NLEVP); here, “nonlinear” refers to the fact that the eigenvalue, the complex oscillation frequency, appears in a nonlinear fashion. In this paper, we employ a noniterative strategy based on contour integration in the complex eigenvalue plane, which returns all eigenvalues inside the contour. An introduction to the technique is given, and is complemented with guidelines for the specific application to thermoacoustic problems. Two prototypical nonlinear eigenvalue problems are considered: a network model of the classical Rijke tube with an analytic flame response model and a finite element discretization of an annular model combustor with an experimental flame transfer function (FTF). Computation of all eigenvalues in a domain of interest is vital to assess stability of these systems. We demonstrate that this is generally challenging for iterative strategies. An eigenvalue solver based on contour integration, in contrast, provides a reliable, noniterative method to achieve this goal.

Accelerating the Sinkhorn–Knopp iteration by Arnoldi-type methods

It is shown that the problem of balancing a nonnegative matrix by positive diagonal matrices can be recast as a constrained nonlinear multiparameter eigenvalue problem. Based on this equivalent formulation some adaptations of the power method and Arnoldi process are proposed for computing the dominant eigenvector which defines the structure of the diagonal transformations. Numerical results illustrate that our novel methods accelerate significantly the convergence of the customary Sinkhorn-Knopp iteration for matrix balancing in the case of clustered dominant eigenvalues.

Cascadic adaptive finite element method for nonlinear eigenvalue problem based on complementary approach

In this paper, a type of cascadic adaptive finite element method for nonlinear eigenvalue problem is proposed based on the complementary approach. In this new scheme, we only need to do some smoothing steps for the linearized boundary value problems on a series of adaptive finite element spaces and solve some nonlinear eigenvalue problems on a low dimensional space. Hence the efficiency can be improved since we do not need to solve the nonlinear eigenvalue problems on each adaptive space directly. Besides, the complementary error estimate for nonlinear eigenvalue problem will be introduced in our paper. This estimate can not only provide an accurate error estimate for the nonlinear eigenvalue problem but also provide the way to refine mesh and control the number of smoothing steps for the cascadic adaptive algorithm. Some numerical examples are presented to validate the efficiency of the proposed algorithm in this paper.

Local and Parallel Multigrid Method for Nonlinear Eigenvalue Problems

In this paper, a new type of local and parallel algorithm is proposed to solve nonlinear eigenvalue problem based on multigrid discretization. Instead of solving the nonlinear eigenvalue problem directly in each level mesh, our method converts the nonlinear eigenvalue problem in the finest mesh to a linear boundary value problem on each level mesh and some nonlinear eigenvalue problems on the coarsest mesh. Further, the involved linear boundary value problems are solved using the local and parallel strategy. As no nonlinear eigenvalue problem is being solved directly on the fine spaces, which is time-consuming, this new type of local and parallel multigrid method evidently improves the efficiency of nonlinear eigenvalue problem solving. We provide a rigorous theoretical analysis for our algorithm and present details on numerical simulations to support our theory.

Spectrum-based Stability Analysis and Stabilization of Time-periodic Time-delay Systems

We develop an eigenvalue-based approach for the stability assessment and stabilization of linear systems with multiple delays and periodic coefficient matrices. Delays and period are assumed commensurate numbers, such that the Floquet multipliers can be characterized as eigenvalues of the monodromy operator and by the solutions of a finite-dimensional non-linear eigenvalue problem, where the evaluation of the characteristic matrix involves solving an initial value problem. We demonstrate that such a dual interpretation can be exploited in a two-stage approach for computing dominant Floquet multipliers, where global approximation is combined with local corrections. Correspondingly, we also propose two novel characterizations of left eigenvectors. Finally, from the nonlinear eigenvalue problem formulation, we derive computationally tractable expressions for derivatives of Floquet multipliers with respect to parameters, which are beneficial in the context of stability optimization. Several numerical examples show the efficacy and applicability of the presented results.

A periodic-parabolic Droop model for two species competition in an unstirred chemostat

We study a periodic-parabolic Droop model of two species competing for a single-limited nutrient in an unstirred chemostat, where the nutrient is added to the culture vessel by way of periodic forcing function in time. For the single species model, we establish a threshold type result on the extinction/persistence of the species in terms of the sign of a principal eigenvalue associated with a nonlinear periodic eigenvalue problem. In particular, when diffusion rate is sufficiently small or large, the sign can be determined. We then show that for the competition model, when diffusion rates for both species are small, there exists a coexistence periodic solution.

On global continua of solutions bifurcating from zero and infinity of some nonlinear fourth order eigenvalue problems

On global continua of solutions bifurcating from zero and infinity of some nonlinear fourth order eigenvalue problems

On the numerical solution of nonlinear eigenvalue problems for the Monge-Ampère operator

In this article, we report the results we obtained when investigating the numerical solution of some nonlinear eigenvalue problems for the Monge-Ampère operator v → det D2v. The methodology we employ relies on the following ingredients: (i) a divergence formulation of the eigenvalue problems under consideration. (ii) The time discretization by operator-splitting of an initial value problem (a kind of gradient flow) associated with each eigenvalue problem. (iii) A finite element approximation relying on spaces of continuous piecewise affine functions. To validate the above methodology, we applied it to the solution of problems with known exact solutions: The results we obtained suggest convergence to the exact solution when the space discretization step h → 0. We considered also test problems with no known exact solutions.

Mathematical and Numerical Analysis of the Generalized Complex-Frequency Eigenvalue Problem for Two-Dimensional Optical Microcavities

The current paper proposes a parametric eigenvalue problem for the Helmholtz equation on the plane based on two well-known physical models of emission from two-dimensional (2-D) microcavity lasers. The first model has been developed for passive, either lossy or lossless, open cavities and is usually referred to as the Complex-Frequency Eigenvalue Problem. The second model, named the Lasing Eigenvalue Problem (LEP), has been tailored to characterize emission from open cavities, filled in with gain material, on the threshold of nonattenuating in time radiation. Our generalized model contains both of them as special cases. We reduce the original problem to a nonlinear eigenvalue problem for a set of boundary integral equations with weakly singular kernels and formulate it as a parametric eigenvalue problem for a holomorphic Fredholm operator-valued function, which is convenient for mathematical and numerical analysis. Using this formulation and fundamental results of the theory of holomorphic operator-valued functions, we study the properties of the spectrum. For numerical solution of the problem, we propose a Nyström method, prove its convergence, and derive error estimates in the eigenvalue approximation. Combining this discretization with the residual inverse iteration technique, we compute approximate solutions of LEP and compare them with the exact ones and with the results obtained using other numerical methods.

Sobolev Gradient Flow for the Gross--Pitaevskii Eigenvalue Problem: Global Convergence and Computational Efficiency

We propose a new normalized Sobolev gradient flow for the Gross--Pitaevskii eigenvalue problem based on an energy inner product that depends on time through the density of the flow itself. The gradient flow is well-defined and converges to an eigenfunction. For ground states we can quantify the convergence speed as exponentially fast where the rate depends on spectral gaps of a linearized operator. The forward Euler time discretization of the flow yields a numerical method which generalizes the inverse iteration for the nonlinear eigenvalue problem. For sufficiently small time steps, the method reduces the energy in every step and converges globally in $H^1$ to an eigenfunction. In particular, for any nonnegative starting value, the ground state is obtained. A series of numerical experiments demonstrates the computational efficiency of the method and its competitiveness with established discretizations arising from other gradient flows for this problem.

A Cascadic Adaptive Finite Element Method for Nonlinear Eigenvalue Problems in Quantum Physics

This paper introduces a cascadic adaptive finite element method for nonlinear eigenvalue equations arising from quantum physics following the multilevel correction strategy. Instead of the classical scheme, which requires solving nonlinear eigenvalue equations on a series of adaptive spaces directly, the new scheme consists of several smoothing processes on an adaptive space sequence and nonlinear eigenvalue equations being solved in a very low dimensional space. The main feature of the proposed scheme is that large-scale nonlinear eigenvalue problem solving is avoided, and the associated smoothing process can be executed efficiently based on the appropriate number of smoothing steps. Thus, efficiency can be enhanced by the proposed cascadic adaptive method. The good performance of the new finite element strategy is examined by various numerical experiments.

A remark on the first <i>p</i>-buckling eigenvalue with an adhesive constraint

We consider a fourth order nonlinear eigenvalue problem with an adhesive constraint. The problem is regarded as a generalization of the buckling eigenvalue problem with the clamped boundary condition. We prove the existence of the first eigenvalue of the problem and show that the corresponding eigenfunction does not have flat core of adhesion type.