Robust Successive Computation of Eigenpairs for Nonlinear Eigenvalue Problems

Effenberger, Cedric

doi:10.1137/120885644

Cited by 43 publications

(102 citation statements)

References 38 publications

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“…Instead, we include all converged eigenvectors in the space for the Rayleigh-Ritz projection and then select unconverged Ritz pairs as approximations for new eigenpairs. Such a strategy is called 'soft deflation', and it is also used to expand invariant pairs [4] in a robust manner for solving general (non-Hermitian) nonlinear eigenproblems of the form T (λ)v = 0 [13].…”

Section: Single-vector Methodsmentioning

confidence: 99%

Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. I. Extreme eigenvalues

Szyld

Xue

2016

Math. Comp.

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Efficient computation of extreme eigenvalues of large-scale linear Hermitian eigenproblems can be achieved by preconditioned conjugate gradient (PCG) methods. In this paper, we study PCG methods for computing extreme eigenvalues of nonlinear Hermitian eigenproblems of the form T ( λ ) v = 0 T(\lambda )v=0 that admit a nonlinear variational principle. We investigate some theoretical properties of a basic CG method, including its global and asymptotic convergence. We propose several variants of single-vector and block PCG methods with deflation for computing multiple eigenvalues, and compare them in arithmetic and memory cost. Variable indefinite preconditioning is shown to be effective to accelerate convergence when some desired eigenvalues are not close to the lowest or highest eigenvalue. The efficiency of variants of PCG is illustrated by numerical experiments. Overall, the locally optimal block preconditioned conjugate gradient (LOBPCG) is the most efficient method, as in the linear setting.

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Section: Single-vector Methodsmentioning

confidence: 99%

Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. I. Extreme eigenvalues

Szyld

Xue

2016

Math. Comp.

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“…In the recent literature we find several subspace based nonlinear eigensolvers [17][18][19][20]. This type of methods have the advantage that they are able to compute several eigenpairs at once.…”

Section: Rational Krylov Methods (Nleigs)mentioning

confidence: 99%

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

et al. 2014

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We present a nonlinear eigenvalue solver enabling the calculation of bound solutions of the Schrödinger 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 twodimensional 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 Pöschl-Teller potential, a two-dimensional potential describing a particle in a canyon and the multi-band Hamiltonian of a topological insulator.

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“…The NLEP (1.1) has been studied extensively in the literature and there exist specialized methods for different families of A(λ); see, e.g., [28,35]. Popular approaches can be roughly classified as Newton-type methods [29,6,22], methods based on contour integration [8,7,10], and methods based on approximations of A(λ) [28,11,35,21,36]. The method proposed in this paper belongs to the last class.…”

Section: 1)mentioning

confidence: 99%

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

Güttel

Beeumen²,

Meerbergen³

et al. 2014

SIAM J. Sci. Comput.

215

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A new rational Krylov method for the efficient solution of nonlinear eigenvalue problems, A(λ)x = 0, 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 function A(λ) 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 function A(λ) has singularities nearby the target set. Moreover, NLEIGS implements an automatic scaling procedure which makes it work robustly independently 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. From the numerical experiments we can recommend two variants of the algorithm for solving the nonlinear eigenvalue problem.

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Robust Successive Computation of Eigenpairs for Nonlinear Eigenvalue Problems

Cited by 43 publications

References 38 publications

Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. I. Extreme eigenvalues

Preconditioned eigensolvers for large-scale nonlinear Hermitian eigenproblems with variational characterizations. I. Extreme eigenvalues

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

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

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