Abstract
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.
We consider the numerical computation of resonances for metallic grating structures with dispersive media and small slit holes. The underlying eigenvalue problem is nonlinear and the mathematical ...model is multiscale due to the existence of several length scales in problem geometry and material contrast. We discretize the partial differential equation model over the truncated domain using the finite element method and develop a multi-step contour integral eigensolver to compute the resonances. The eigensolver first locates eigenvalues using a spectral indicator and then computes eigenvalues by a subspace projection scheme. The proposed numerical method is robust and scalable, and does not require initial guess as the iteration methods. Numerical examples are presented to demonstrate its effectiveness.
Closed combustion devices like gas turbines and rockets are prone to thermoacoustic instabilities. Design engineers in the industry need tools to accurately identify and remove instabilities early in ...the design cycle. Many different approaches have been developed by the researchers over the years. In this work we focus on the Helmholtz wave equation based solver which is found to be relatively fast and accurate for most applications. This solver has been a subject of study in many previous works. The Helmholtz wave equation in frequency space reduces to a nonlinear eigenvalue problem which needs to be solved to compute the acoustic modes. Most previous implementations of this solver have relied on linearized solvers and iterative methods which as shown in this work are not very efficient and sometimes inaccurate. In this work we make use of specialized algorithms implemented in SLEPc that are accurate and efficient for computing eigenvalues of nonlinear eigenvalue problems. We make use of the n-tau model to compute the reacting source terms in the Helmholtz equation and describe the steps involved in deriving the Helmholtz eigenvalue equation and obtaining its solution using the SLEPc library.
•Software for the early identification of thermoacoustic instabilities in closed combustion devices with arbitrary geometry.•Helmholtz wave equation in frequency space leading to a nonlinear eigenvalue formulation.•SLEPc's nonlinear eigensolver is fast and accurate, and can be run in parallel for large-scale problems.
In this article, we present an overview of different a posteriori error analysis and post-processing methods proposed in the context of nonlinear eigenvalue problems, e.g. arising in electronic ...structure calculations for the calculation of the ground state and compare them. We provide two equivalent error reconstructions based either on a second-order Taylor expansion of the minimized energy, or a first-order expansion of the nonlinear eigenvalue equation. We then show how several a posteriori error estimations as well as post-processing methods can be formulated as specific applications of the derived reconstructed errors, and we compare their range of applicability as well as numerical cost and precision.
•Nonlinear eigenvalue problems are ubiquitous in different fields.•A posteriori estimators are important to validate the numerical approximations.•Linear post-treatments may improve the precision of numerical approximations.
Linearizable Eigenvector Nonlinearities Claes, Rob; Jarlebring, Elias; Meerbergen, Karl ...
SIAM journal on matrix analysis and applications,
01/2022, Letnik:
43, Številka:
2
Journal Article
Recenzirano
Odprti dostop
We present a method to linearize, without approximation, a specific class of eigenvalue problems with eigenvector nonlinearities (NEPv), where the nonlinearities are expressed by scalar functions ...that are defined by a quotient of linear functions of the eigenvector. The exact linearization relies on an equivalent multiparameter eigenvalue problem (MEP) that contains the exact solutions of the NEPv. Due to the characterization of MEPs in terms of a generalized eigenvalue problem this provides a direct way to compute all NEPv solutions for small problems, and it opens up the possibility to develop locally convergent iterative methods for larger problems. Moreover, the linear formulation allows us to easily determine the number of solutions of the NEPv. We propose two numerical schemes that exploit the structure of the linearization: inverse iteration and residual inverse iteration. We show how symmetry in the MEP can be used to improve reliability and reduce computational cost of both methods. Two numerical examples verify the theoretical results, and a third example shows the potential of a hybrid scheme that is based on a combination of the two proposed methods.
On the basis of the generalized argument principle, here we develop a numerical scheme for locating zeros and poles of a meromorphic function. A subdivision-transformation-calculation scheme is ...proposed to ensure the algorithm stability. A novel feature of this algorithm is the ability to estimate the error level automatically. Numerical examples are also presented, with an emphasis on potential applications to plasma physics.
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