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

Vandenberghe, William G.; Fischetti, Massimo V.; Beeumen, Roel Van; Meerbergen, Karl; Michiels, Wim; Effenberger, Cedric

doi:10.1007/s10825-014-0597-5

Cited by 4 publications

(7 citation statements)

References 23 publications

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“…where H, S j ∈ R 16281×16281 ; see Vandenberghe et al (2014);Van Beeumen (2015). Matrix H is symmetric and matrices S j have low rank.…”

Section: Bound States In Semiconductor Devicesmentioning

confidence: 99%

“…For approximating the nonlinear functions with Leja-Bagby points, in Vandenberghe et al (2014), a transformation is used that removes the branch cut between two predetermined, subsequent branch points, i.e., for λ ∈ [α i−1 , α i ]. The interpolant based on these Leja-Bagby points is only valid for λ -values within this interval.…”

Section: Bound States In Semiconductor Devicesmentioning

confidence: 99%

“…The interpolant based on these Leja-Bagby points is only valid for λ -values within this interval. For interval [α 0 , α 1 ], a rational approximation with 50 poles was used in Vandenberghe et al (2014). This corresponds to the green triangular marker in Figure 2.…”

Section: Bound States In Semiconductor Devicesmentioning

confidence: 99%

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Automatic rational approximation and linearization of nonlinear eigenvalue problems

Lietaert

Meerbergen

Pérez

et al. 2021

IMA Journal of Numerical Analysis

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We present a method for solving nonlinear eigenvalue problems (NEPs) using rational approximation. The method uses the Antoulas–Anderson algorithm (AAA) of Nakatsukasa, Sète and Trefethen to approximate the NEP via a rational eigenvalue problem. A set-valued variant of the AAA algorithm is also presented for building low-degree rational approximations of NEPs with a large number of nonlinear functions. The rational approximation is embedded in the state-space representation of a rational polynomial by Su and Bai. This procedure perfectly fits the framework of the compact rational Krylov methods (CORK and TS-CORK), allowing solve large-scale NEPs to be efficiently solved. One advantage of our method, compared to related techniques such as NLEIGS and infinite Arnoldi, is that it automatically selects the poles and zeros of the rational approximations. Numerical examples show that the presented framework is competitive with NLEIGS and usually produces smaller linearizations with the same accuracy but with less effort for the user.

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“…where H, S j ∈ R 16281×16281 ; see Vandenberghe et al (2014);Van Beeumen (2015). Matrix H is symmetric and matrices S j have low rank.…”

Section: Bound States In Semiconductor Devicesmentioning

confidence: 99%

Section: Bound States In Semiconductor Devicesmentioning

confidence: 99%

See 1 more Smart Citation

Automatic rational approximation and linearization of nonlinear eigenvalue problems

Lietaert

Meerbergen

Pérez

et al. 2021

IMA Journal of Numerical Analysis

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“…The Schrödinger equation is discretized on a 4 × 10-nm 2 grid. For more information about the physics we refer to [37].…”

Section: Particle In a Canyon Problemmentioning

confidence: 99%

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

Güttel

Beeumen²,

Meerbergen³

et al. 2014

SIAM J. Sci. Comput.

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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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“…Another example appears in the spectral analysis of delay differential equations [Michiels and Niculescu, 2014]. In some applications, the nonlinear character comes from using special boundary conditions for the solution of differential equations [Vandenberghe et al, 2014, van Beeumen et al, 2018.…”

Section: Introductionmentioning

confidence: 99%

Nep

Campos

Román

2021

ACM Trans. Math. Softw.

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SLEPc is a parallel library for the solution of various types of large-scale eigenvalue problems. Over the past few years, we have been developing a module within SLEPc, called NEP, that is intended for solving nonlinear eigenvalue problems. These problems can be defined by means of a matrix-valued function that depends nonlinearly on a single scalar parameter. We do not consider the particular case of polynomial eigenvalue problems (which are implemented in a different module in SLEPc) and focus here on rational eigenvalue problems and other general nonlinear eigenproblems involving square roots or any other nonlinear function. The article discusses how the NEP module has been designed to fit the needs of applications and provides a description of the available solvers, including some implementation details such as parallelization. Several test problems coming from real applications are used to evaluate the performance and reliability of the solvers.

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

Cited by 4 publications

References 23 publications

Automatic rational approximation and linearization of nonlinear eigenvalue problems

Automatic rational approximation and linearization of nonlinear eigenvalue problems

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

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