scholarly journals Compact Two-Sided Krylov Methods for Nonlinear Eigenvalue Problems

2018 ◽  
Vol 40 (5) ◽  
pp. A2801-A2829 ◽  
Author(s):  
Pieter Lietaert ◽  
Karl Meerbergen ◽  
Françoise Tisseur
Keyword(s):  
Eigenvalue Problems ◽  
Krylov Methods ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Eigenvalue

scholarly journals Automatic rational approximation and linearization of nonlinear eigenvalue problems

2021 ◽  
Author(s):  
Pieter Lietaert ◽  
Karl Meerbergen ◽  
Javier Pérez ◽  
Bart Vandereycken
Keyword(s):  
Rational Approximation ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Space Representation ◽  
Rational Approximations ◽  
Krylov Methods ◽  
Nonlinear Eigenvalue Problems ◽  
State Space Representation ◽  
Rational Polynomial ◽  
Nonlinear Eigenvalue

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.

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

2014 ◽  
Vol 36 (6) ◽  
pp. A2842-A2864 ◽  
Author(s):  
Stefan Güttel ◽  
Roel Van Beeumen ◽  
Karl Meerbergen ◽  
Wim Michiels
Keyword(s):  
Eigenvalue Problems ◽  
Krylov Methods ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Eigenvalue

MULTIPLE POSITIVE SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS ASSOCIATED TO A CLASS OF p-LAPLACIAN LIKE OPERATORS

2003 ◽  
Vol 05 (05) ◽  
pp. 737-759 ◽  
Author(s):  
NOBUYOSHI FUKAGAI ◽  
KIMIAKI NARUKAWA
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Eigenvalue Problems ◽  
Multiple Positive Solutions ◽  
Combined Effects ◽  
Nonlinear Eigenvalue Problems ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic ◽  
Nonlinear Eigenvalue

This paper deals with positive solutions of a class of nonlinear eigenvalue problems. For a quasilinear elliptic problem (#) - div (ϕ(|∇u|)∇u) = λf(x,u) in Ω, u = 0 on ∂Ω, we assume asymptotic conditions on ϕ and f such as ϕ(t) ~ tp0-2, f(x,t) ~ tq0-1as t → +0 and ϕ(t) ~ tp1-2, f(x,t) ~ tq1-1as t → ∞. The combined effects of sub-nonlinearity (p0> q0) and super-nonlinearity (p1< q1) with the subcritical term f(x,u) imply the existence of at least two positive solutions of (#) for 0 < λ < Λ.

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