Class Of Nonlinear Eigenvalue Problems Research Articles

Analyticity and hp discontinuous Galerkin approximation of nonlinear Schrödinger eigenproblems

We study a class of nonlinear eigenvalue problems of Schrödinger type, where the potential is singular on a set of points. Such problems are widely present in physics and chemistry, and their analysis is of both theoretical and practical interest. In particular, we study the regularity of the eigenfunctions of the operators considered, and we propose and analyze the approximation of the solution via an isotropically refined [Formula: see text] discontinuous Galerkin (dG) method. We show that, for weighted analytic potentials and for up-to-quartic polynomial nonlinearities, the eigenfunctions belong to analytic-type non-homogeneous weighted Sobolev spaces. We also prove quasi optimal a priori estimates on the error of the dG finite element method; when using an isotropically refined [Formula: see text] space, the numerical solution is shown to converge with exponential rate towards the exact eigenfunction. We conclude with a series of numerical tests to validate the theoretical results.

Fractional eigenvalues in Orlicz spaces with no Δ2 condition

We study the eigenvalue problem for the g-Laplacian operator in fractional order Orlicz-Sobolev spaces, where g=G′ and neither G nor its conjugated function satisfy the Δ2 condition. Our main result is the existence of a nontrivial solution to such a problem; this is achieved by first showing that the corresponding minimization problem has a solution and then applying a generalized Lagrange multiplier theorem to get the existence of an eigenvalue. Further, we prove closedness of the spectrum and some properties of the eigenvalues and, as an application, we show existence for a class of nonlinear eigenvalue problems.

Sharp Estimation of Convergence Rate for Self-Consistent Field Iteration to Solve Eigenvector-Dependent Nonlinear Eigenvalue Problems

We present a comprehensive convergence analysis for the self-consistent field (SCF) iteration to solve a class of nonlinear eigenvalue problems with eigenvector dependency (NEPvs). Using the tangent-angle matrix as an intermediate measure for approximation error, we establish new formulas for two fundamental quantities that characterize the local convergence behavior of the plain SCF: the local contraction factor and the local asymptotic average contraction factor. In comparison with previously established results, new convergence rate estimates provide much sharper bounds on the convergence speed. As an application, we extend the convergence analysis to a popular SCF variant---the level-shifted SCF. The effectiveness of the convergence rate estimates is demonstrated numerically for NEPvs arising from solving the Kohn--Sham equation in electronic structure calculation and the Gross--Pitaevskii equation for modeling of the Bose--Einstein condensation.

Global Bifurcation of Fourth-Order Nonlinear Eigenvalue Problems’ Solution

In this paper, we study the global bifurcation of infinity of a class of nonlinear eigenvalue problems for fourth-order ordinary differential equations with nondifferentiable nonlinearity. We prove the existence of two families of unbounded continuance of solutions bifurcating at infinity and corresponding to the usual nodal properties near bifurcation intervals.

Eigenfunction behavior and adaptive finite element approximations of nonlinear eigenvalue problems in quantum physics

In this paper, we investigate a class of nonlinear eigenvalue problems resulting from quantum physics. We first prove that for any open set G, there exists an eigenfunction that cannot be a polynomial on G, which may be reviewed as a refinement of the classic unique continuation property. Then we apply the non-polynomial behavior of the eigenfunction to show that the adaptive finite element approximations are convergent even if the initial mesh is not fine enough. We finally remark that similar arguments can be applied to a class of linear eigenvalue problems that improve the relevant existing results.

Nonlinear eigenvalue problems for nonhomogeneous Leray\u2013Lions operators

This paper deals with the mathematical analysis of a class of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator. We are concerned both with the coercive and the noncoercive (and nonresonant) cases, which are in relationship with two associated Rayleigh quotients. The proof combines critical point theory arguments and the dual variational principle. The arguments developed in this paper can be extended to other classes of nonlinear eigenvalue problems with nonstandard growth.

Nonlinear and nonlocal eigenvalue problems: variational existence, decay properties, approximation, and universal scaling limits

We study a class of nonlinear eigenvalue problems which involves a convolution operator as well as a superlinear nonlinearity. Our variational existence proof is based on constrained optimization and provides a one-parameter family of solutions with positive eigenvalues and unimodal eigenfunctions. We also discuss the decay properties and the numerical computations of those eigenfunctions, and conclude with two asymptotic results concerning small and large eigenvalues.

Eigenvalue problems with unbalanced growth: Nonlinear patterns and standing wave solutions

We consider two classes of nonlinear eigenvalue problems with double-phase energy and lack of compactness. We establish existence and non-existence results and related properties of solutions. Our analysis combines variational methods with the generalized Pucci–Serrin maximum principle.

Computing resonant modes of accelerator cavities by solving nonlinear eigenvalue problems via rational approximation

We present an efficient and reliable algorithm for solving a class of nonlinear eigenvalue problems arising from the modeling of particle accelerator cavities. The eigenvalue nonlinearity in these problems results from the use of waveguides to couple external power sources or to allow certain excited electromagnetic modes to exit the cavity. We use a rational approximation to reduce the nonlinear eigenvalue problem first to a rational eigenvalue problem. We then apply a special linearization procedure to turn the rational eigenvalue problem into a larger linear eigenvalue problem with the same eigenvalues, which can be solved by existing iterative methods. By using a compact scheme to represent both the linearized operator and the eigenvectors to be computed, we obtain a numerical method that only involves solving linear systems of equations of the same dimension as the original nonlinear eigenvalue problem. We refer to this method as a compact rational Krylov (CORK) method. We implemented the CORK method in the Omega3P module of the Advanced Computational Electromagnetic 3D Parallel (ACE3P) simulation suite and validated it by comparing the computed cavity resonant frequencies and damping Q factors of a small model problem to those obtained from a fitting procedure that uses frequency responses computed by another ACE3P module called S3P. We also used the CORK method to compute trapped modes damped in an ideal eight 9-cell SRF cavity cryomodule. This was the first time it was possible to compute these modes directly. The damping Q factors of the computed modes match well with those measured in experiments and the difference in resonant frequencies is within the range introduced by cavity imperfection. Therefore, the CORK method is an extremely valuable tool for computational cavity design.

A Riemannian Newton Algorithm for Nonlinear Eigenvalue Problems

We give the formulation of a Riemannian Newton algorithm for solving a class of nonlinear eigenvalue problems by minimizing a total energy function subject to the orthogonality constraint. Under some mild assumptions, we establish the global and quadratic convergence of the proposed method. Moreover, the positive definiteness condition of the Riemannian Hessian of the total energy function at a solution is derived. Some numerical tests are reported to illustrate the efficiency of the proposed method for solving large-scale problems.

Stress-strain state in the vicinity of a crack tip under mixed loading

A method is proposed to calculate the eigenvalues of the class of nonlinear eigenvalue problems resulting from the problem of determining the stress-strain state in the vicinity of a crack tip in power-law materials over the entire range of mixed modes of deformation, from the opening mode to pure shear. The proposed approach was used to found eigenvalues of the problem that differ from the well-known eigenvalue corresponding to the Hutchinson-Rice-Rosengren solution. The resulting asymptotic form of the stress field is a self-similar intermediate asymptotic solution of the problem of a crack in a damaged medium under mixed loading. Using the new asymptotic form of the stress field and introducing a self-similar variable, we obtained an asymptotic solution of the problem of a crack in a damaged medium and constructed the regions of dispersed material near the crack.

Fast algorithms for computing the distance to instability of nonlinear eigenvalue problems, with application to time-delay systems

A continuous dynamical system is stable if all eigenvalues lie strictly in the left half of the complex plane. However, this is not a robust measure because stability is no longer guaranteed when the system parameters are slightly perturbed. Therefore, the pseudospectrum of a matrix and its pseudospectral abscissa are studied. Mostly, one is often interested in computing the distance to instability, because it is a robust measure for stability against perturbations. As a first contribution, this paper presents two iterative methods for computing the distance to instability, considering complex perturbations. The first one is based on locating a zero of the pseudospectral abscissa function. This method is particularly suitable for large sparse matrices as it is based on repeated eigenvalue computations, where the original matrix is perturbed with a rank one matrix. The second method is based on a recently proposed global optimization technique. The advantages of both methods can be combined in a hybrid algorithm. As a second contribution we show that the methods apply to a broad class of nonlinear eigenvalue problems, in particular eigenvalue problems inferred from linear delay-differential equations, and, therefore, they are useful for a wide range of problems. In the numerical examples the standard eigenvalue problem, the quadratic eigenvalue problem and the delay eigenvalue problem are addressed.

Qualitative Properties of Eigenvectors Related to Multivalued Operators and some Existence Results

In this work, we study a class of nonlinear eigenvalue problems related to fully discontinuous operators. In particular, we prove the existence of a critical point for two distinct problems. Connected with this problem, we also study a minimization problem with constraint, and we investigate the existence of solutions for a resonant case near zero. Moreover, we give some estimates and qualitative properties of solutions by using the relative rearrangement theory.

An Iterative Method for Computing the Pseudospectral Abscissa for a Class of Nonlinear Eigenvalue Problems

We consider the following class of nonlinear eigenvalue problems: $\left(\sum_{i=1}^m A_i p_i(\lambda)\right) v= 0,$ where $A_1, \ldots,A_m$ are given $n \times n$ matrices and the functions $p_1, \ldots, p_m$ are assumed to be entire. This does not only include polynomial eigenvalue problems but also eigenvalue problems arising from systems of delay differential equations. Our aim is to compute the $\epsilon$-pseudospectral abscissa, i.e., the supremum of the real parts of the points in the $\epsilon$-pseudospectrum, which is the complex set obtained by joining all solutions of the eigenvalue problem under perturbations $\{ \delta A_i \}_{i=1}^{m}$, of norm at most $\epsilon$, of the matrices $\{ A_i \}_{i=1}^{m}$. Under mild assumptions, guaranteeing the existence of a globally rightmost point of the $\epsilon$-pseudospectrum, we prove that it is sufficient to restrict the analysis to rank-one perturbations of the form $\delta A_i = \beta_i u v^*$, where $u \in \mathbb{C}^n$ and $v \in \mathbb{C}^n$ wit...

Adaptive Finite Element Approximations for a Class of Nonlinear Eigenvalue Problems in Quantum Physics

AbstractIn this paper, we study an adaptive finite element method for a class of nonlinear eigenvalue problems resulting from quantum physics that may have a nonconvex energy functional. We prove the convergence of adaptive finite element approximations and present several numerical examples of micro-structure of matter calculations that support our theory.

Finite element approximations of nonlinear eigenvalue problems in quantum physics

In this paper, we study finite element approximations of a class of nonlinear eigenvalue problems arising from quantum physics. We derive both a priori and a posteriori finite element error estimates and obtain optimal convergence rates for both linear and quadratic finite element approximations. In particular, we analyze the convergence and complexity of an adaptive finite element method. In our analysis, we utilize certain relationship between the finite element eigenvalue problem and the associated finite element boundary value approximations. We also present several numerical examples in quantum physics that support our theory.

Solving Rational Eigenvalue Problems via Linearization

The rational eigenvalue problem is an emerging class of nonlinear eigenvalue problems arising from a variety of physical applications. In this paper, we propose a linearization-based method to solve the rational eigenvalue problem. The proposed method converts the rational eigenvalue problem into a well-studied linear eigenvalue problem, and meanwhile, exploits and preserves the structure and properties of the original rational eigenvalue problem. For example, the low-rank property leads to a trimmed linearization. We show that solving a class of rational eigenvalue problems is just as convenient and efficient as solving linear eigenvalue problems.

On Ricceri’s conjecture for a class of nonlinear eigenvalue problems

In this work we give a negative answer to a conjecture proposed by B. Ricceri in the reference [B. Ricceri, A remark on a class of nonlinear eigenvalue problems, Nonlinear Anal. 69 (2008) 2964–2967] for a class of nonlinear elliptic eigenvalue problems.

Eigenvalue problems in anisotropic Orlicz–Sobolev spaces

We establish sufficient conditions for the existence of solutions to a class of nonlinear eigenvalue problems involving nonhomogeneous differential operators in Orlicz–Sobolev spaces. To cite this article: M. Mihăilescu et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

KSSOLV—a MATLAB toolbox for solving the Kohn-Sham equations

We describe the design and implementation of KSSOLV, a MATLAB toolbox for solving a class of nonlinear eigenvalue problems known as the Kohn-Sham equations . These types of problems arise in electronic structure calculations, which are nowadays essential for studying the microscopic quantum mechanical properties of molecules, solids, and other nanoscale materials. KSSOLV is well suited for developing new algorithms for solving the Kohn-Sham equations and is designed to enable researchers in computational and applied mathematics to investigate the convergence properties of the existing algorithms. The toolbox makes use of the object-oriented programming features available in MATLAB so that the process of setting up a physical system is straightforward and the amount of coding effort required to prototype, test, and compare new algorithms is significantly reduced. All of these features should also make this package attractive to other computational scientists and students who wish to study small- to medium-size systems.