Method For Nonlinear Eigenvalue Problems Research Articles

A contour integral-based method for nonlinear eigenvalue problems for semi-infinite photonic crystals

In this study, we introduce an efficient method for determining isolated singular points of two-dimensional semi-infinite and bi-infinite photonic crystals, equipped with perfect electric conductor and quasi-periodic mixed boundary conditions. This specific problem can be modeled by a Helmholtz equation and is recast as a generalized eigenvalue problem involving an infinite-dimensional block quasi-Toeplitz matrix. Through an intelligent implementation of cyclic structure-preserving matrix transformations, the contour integral method is elegantly employed to calculate the isolated eigenvalue and to extract a component of the associated eigenvector. Moreover, a propagation formula for electromagnetic fields is derived. This formulation enables rapid computation of field distributions across the expansive semi-infinite and bi-infinite domains, thus highlighting the attributes of edge states. The preliminary MATLAB implementation is available at https://github.com/FAME-GPU/2D_Semi-infinite_PhC.

Derivative Interpolating Subspace Frameworks for Nonlinear Eigenvalue Problems

We first consider the problem of approximating a few eigenvalues of a proper rational matrix-valued function closest to a prescribed target. It is assumed that the proper rational matrix-valued function is expressed in the transfer function form $H(s) = C (sI - A)^{-1} B$, where the middle factor is large, whereas the number of rows of $C$ and the number of columns of $B$ are equal and small. We propose a subspace framework that performs two-sided projections on the state-space representation of $H(\cdot)$, commonly employed in model reduction and giving rise to a reduced transfer function. At every iteration, the projection subspaces are expanded to attain Hermite interpolation conditions at the eigenvalues of the reduced transfer function closest to the target, which in turn leads to a new reduced transfer function. We prove in theory that, when a sequence of eigenvalues of the reduced transfer functions converges to an eigenvalue of the full problem, it converges at least at a quadratic rate. In the second part, we extend the proposed framework to locate the eigenvalues of a general square large-scale nonlinear meromorphic matrix-valued function $T(\cdot)$, where we exploit a representation ${\mathcal R}(s) = C(s) A(s)^{-1} B(s) - D(s)$ defined in terms of the block components of $T(\cdot)$. The numerical experiments illustrate that the proposed framework is reliable in locating a few eigenvalues closest to the target point, and that, with respect to runtime, it is competitive to established methods for nonlinear eigenvalue problems.

The subspace iteration method for nonlinear eigenvalue problems occurring in the dynamics of structures with viscoelastic elements

The paper presents an extension of the subspace iteration method for application in systems with viscoelastic damping elements, which are described by both classical and fractional models. The presented method enables determination of only a certain number of eigenvalues and associated eigenvectors. This is very useful, especially when the considered problem has many degrees of freedom, in which case it would be very time-consuming or even impossible to determine all the eigenvalues. At the same time, from the engineering point of view, it is not necessary to know all the eigenvalues. In this paper, the subspace iteration method is used for the first time to solve the nonlinear eigenproblems which appear in the dynamic analysis of systems with viscoelastic damping elements. The proposed solution consists of assuming the number of eigenvalues to be determined, taking the initial point of iteration and then solving the nonlinear reduced eigenproblem with Hermitian matrices in each iterative loop. The solution to the reduced eigenproblem is obtained using the continuation method, and it is an additional novelty in the paper. The correctness and effectiveness of the proposed approach are illustrated with numerical examples.

A Riesz-projection-based method for nonlinear eigenvalue problems

We propose an algorithm for general nonlinear eigenvalue problems to compute physically relevant eigenvalues within a chosen contour. Eigenvalue information is explored by contour integration incorporating different weight functions. The gathered information is processed by solving a nonlinear system of equations of small dimension prioritizing eigenvalues with high physical impact. No auxiliary functions have to be introduced since linearization is not used. The numerical implementation is straightforward as the evaluation of the integrand can be regarded as a blackbox. We apply the method to a quantum mechanical problem and to two nanophotonic systems.

Compact Two-Sided Krylov Methods for Nonlinear Eigenvalue Problems

We describe a generalization of the compact rational Krylov (CORK) methods for polynomial and rational eigenvalue problems that usually, but not necessarily, come from polynomial or rational approximations of genuinely nonlinear eigenvalue problems. CORK is a family of one-sided methods that reformulates the polynomial or rational eigenproblem as a generalized eigenvalue problem. By exploiting the Kronecker structure of the associated pencil, it constructs a right Krylov subspace in compact form and thereby avoids the high memory and orthogonalization costs that are usually associated with linearizations of high degree matrix polynomials. CORK approximates eigenvalues and their corresponding right eigenvectors but is not suitable in its current form for the computation of left eigenvectors. Our generalization of the CORK method is based on a class of Kronecker structured pencils that include as special cases the CORK pencils, the transposes of CORK pencils, and the symmetrically structured linearizations by Robol, Vandebril, and Van Dooren [SIAM J. Matrix Anal. Appl., 38 (2017), pp. 188--216]. This class of structured pencils facilitates the development of a general framework for the computation of both right- and left-sided Krylov subspaces in compact form, and hence allows the development of two-sided compact rational Krylov methods for nonlinear eigenvalue problems. The latter are particularly efficient when the standard inner product is replaced by a cheaper to compute quasi-inner product. We show experimentally that convergence results similar to CORK can be obtained for a certain quasi-inner product.

Disguised and new quasi-Newton methods for nonlinear eigenvalue problems

In this paper, we take a quasi-Newton approach to nonlinear eigenvalue problems (NEPs) of the type M(λ)v = 0, where M:mathbb {C}rightarrow mathbb {C}^{ntimes n} is a holomorphic function. We investigate which types of approximations of the Jacobian matrix lead to competitive algorithms, and provide convergence theory. The convergence analysis is based on theory for quasi-Newton methods and Keldysh’s theorem for NEPs. We derive new algorithms and also show that several well-established methods for NEPs can be interpreted as quasi-Newton methods, and thereby, we provide insight to their convergence behavior. In particular, we establish quasi-Newton interpretations of Neumaier’s residual inverse iteration and Ruhe’s method of successive linear problems.

Restarting iterative projection methods for Hermitian nonlinear eigenvalue problems with minmax property

In this work we present a new restart technique for iterative projection methods for nonlinear eigenvalue problems admitting minmax characterization of their eigenvalues. Our technique makes use of the minmax induced local enumeration of the eigenvalues in the inner iteration. In contrast to global numbering which requires including all the previously computed eigenvectors in the search subspace, the proposed local numbering only requires a presence of one eigenvector in the search subspace. This effectively eliminates the search subspace growth and therewith the super-linear increase of the computational costs if a large number of eigenvalues or eigenvalues in the interior of the spectrum are to be computed. The new restart technique is integrated into nonlinear iterative projection methods like the Nonlinear Arnoldi and Jacobi-Davidson methods. The efficiency of our new restart framework is demonstrated on a range of nonlinear eigenvalue problems: quadratic, rational and exponential including an industrial real-life conservative gyroscopic eigenvalue problem modeling free vibrations of a rolling tire. We also present an extension of the method to problems without minmax property but with eigenvalues which have a dominant either real or imaginary part and test it on two quadratic eigenvalue problems.

Matrix iteration method for nonlinear eigenvalue problems with applications

A simple and intuitive matrix iteration method for solving nonlinear eigenvalue problems is described and demonstrated in detail by two problems: (i) the boundary value problem associated with large deflection of a flexible rod, and (ii) the initial value problem associated with normal mode motion of a double pendulum. The two problems are solved by two approaches, the finite difference approach and a continuous realization approach which is similar in spirit to the Rayleigh–Ritz method.

Numerical methods for nonlinear two-parameter eigenvalue problems

We introduce nonlinear two-parameter eigenvalue problems and generalize several numerical methods for nonlinear eigenvalue problems to nonlinear two-parameter eigenvalue problems. As a motivation we consider the computation of critical delays of delay-differential equations with multiple delays.

A successive quadratic approximations method for nonlinear eigenvalue problems

Numerical methods for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter are discussed. We propose a successive quadratic approximations method, which reduces the nonlinear eigenvalue problem into a sequence of quadratic problems. The convergence for the new method is investigated. Numerical experiments illustrate the effectiveness of the method.

Determining bound states in a semiconductor device with contacts using a nonlinear eigenvalue solver

We present a nonlinear eigenvalue solver enabling the calculation of bound solutions of the Schrodinger equation in a system with contacts. We discuss how the imposition of contacts leads to a nonlinear eigenvalue problem and discuss the numerics for a one- and two-dimensional potential. We reformulate the problem so that the eigenvalue problem can be efficiently solved by the recently proposal rational Krylov method for nonlinear eigenvalue problems, known as NLEIGS. In order to improve the convergence of the method, we propose a holomorphic extension such that we can easily deal with the branch points introduced by a square root. We use our method to determine the bound states of the one-dimensional Poschl---Teller potential, a two-dimensional potential describing a particle in a canyon and the multi-band Hamiltonian of a topological insulator.

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.

An adaptive multidomain Chebyshev method for nonlinear eigenvalue problems: Application to self-similar solutions of gas dynamics equations with nonlinear heat conduction

An adaptive multidomain Chebyshev collocation method for handling nonlinear eigenvalue problems, such as those arising in the computation of self-similar solutions, is presented. The algorithm is made of two stages, the second one using an iterative method. The method is applied to the time-dependent one-dimensional self-similar solutions of the Euler equations with nonlinear heat conduction in which a thermal front follows a shock wave. It leads to (highly) accurate solutions, for solutions stiffness ranging from 1 to 10−7. The method is quite general and may be applied to a large class of problems.

Analyzing the convergence factor of residual inverse iteration

We will establish here a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well-known inverse iteration. The formula for the convergence factor is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows for some freedom in the choice of a vector w k and we can use the formula for the convergence factor to analyze how it depends on the choice of w k . We also use the formula to illustrate the convergence when the shift is close to the eigenvalue. Finally, we explain the slow convergence for double eigenvalues by showing that under generic conditions, the convergence factor is one, unless the eigenvalue is semisimple. If the eigenvalue is semisimple, it turns out that we can expect convergence similar to the simple case.

Convergence factors of Newton methods for nonlinear eigenvalue problems

Consider a complex sequence {λk}k=0∞ convergent to λ∗∈C with order p∈N. The convergence factor is typically defined as the fraction ck:=(λk+1-λ∗)/(λk-λ∗)p in the limit k→∞. In this paper, we prove formulas characterizing ck in the limit k→∞ for two different Newton-type methods for nonlinear eigenvalue problems. The formulas are expressed in terms of the left and right eigenvectors.The two treated methods are called the method of successive linear problems (MSLP) and augmented Newton and are widely used in the literature. We prove several explicit formulas for ck for both methods. Formulas for both methods are found for simple as well as double eigenvalues. In some cases, we observe in examples that the limit ck as k→∞ does not exist. For cases where this limit does not appear to exist, we prove other limiting expressions such that a characterization of ck in the limit is still possible.

A Jacobi–Davidson like method for nonlinear eigenvalue problems based on singularity theory

AbstractWe present a Jacobi–Davidson like correction formula for left and right eigenvector approximations for non‐Hermitian nonlinear eigenvalue problems. It exploits techniques from singularity theory for characterizing singular points of nonlinear equations. Unlike standard nonlinear Jacobi‐Davidson, the correction formula does not contain derivative information and works with orthogonal projectors only. Moreover, the basic method is modified in that the new eigenvalue approximation is taken as a nonlinear Rayleigh functional obtained as root of a certain scalar nonlinear equation the existence of which – as well as a first order perturbation expansion – is shown. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Photonic band structure calculations using nonlinear eigenvalue techniques

This paper considers the numerical computation of the photonic band structure of periodic materials such as photonic crystals. This calculation involves the solution of a Hermitian nonlinear eigenvalue problem. Numerical methods for nonlinear eigenvalue problems are usually based on Newton’s method or are extensions of techniques for the standard eigenvalue problem. We present a new variation on existing methods which has its derivation in methods for bifurcation problems, where bordered matrices are used to compute critical points in singular systems. This new approach has several advantages over the current methods. First, in our numerical calculations the new variation is more robust than existing techniques, having a larger domain of convergence. Second, the linear systems remain Hermitian and are nonsingular as the method converges. Third, the approach provides an elegant and efficient way of both thinking about the problem and organising the computer solution so that only one linear system needs to be factorised at each stage in the solution process. Finally, first- and higher-order derivatives are calculated as a natural extension of the basic method, and this has advantages in the electromagnetic problem discussed here, where the band structure is plotted as a set of paths in the ( ω, k) plane.

A Jacobi–Davidson-type projection method for nonlinear eigenvalue problems

This paper discusses a projection method for nonlinear eigenvalue problems. The subspace of approximants is constructed by a Jacobi–Davidson-type approach, and the arising eigenproblems of small dimension are solved by safeguarded iteration. The method is applied to a rational eigenvalue problem governing the vibrations of tube bundle immersed in an inviscid compressible fluid.

A numerical method for nonlinear eigenvalue/boundary-value problems by HIDM.

A powerful numerical method for nonlinear eigenvalue problems and for boundary value problems is developed based on a shooting scheme using the new numerical method HIDM (Higher order implicit difference method). Numerical stability of the shooting scheme is achieved by “tournament multi-divided shooting” method.The method is applied to several problems and proved its good performance.

Finite element methods for nonlinear eigenvalue problems and the postbuckling behavior of elastic plates

In the following paper new variational methods for the solution of problems of postbuckling of elastic plates are formulated, and approximate solutions for various cases are obtained using finite element methods. Initially, the postbuckling problem is phrased as a nonlinear eigenvalue problem. Variational principles for the nonlinear eigenvalue problem of the postbuckling of structures are developed. These variational principles have a form which can be characterized as a nonlinear Rayleigh principle, finite-element models are introduced and are used to implement the postbuckling variational principle. Incremental amplitude solution schemes are developed to solve the corresponding finite-element equations. Sample results are presented for the postbuckling behavior of circular plates.