scholarly journals Sobolev Gradient Flow for the Gross--Pitaevskii Eigenvalue Problem: Global Convergence and Computational Efficiency

2020 ◽  
Vol 58 (3) ◽  
pp. 1744-1772 ◽  
Author(s):  
Patrick Henning ◽  
Daniel Peterseim
Keyword(s):  
Global Convergence ◽  
Eigenvalue Problem ◽  
Computational Efficiency ◽  
Gradient Flow ◽  
Sobolev Gradient

scholarly journals A New Jacobian-Like Method for the Polyhedral Cone-Constrained Eigenvalue Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
Guo Sun
Keyword(s):  
Global Convergence ◽  
Eigenvalue Problem ◽  
Numerical Experiments ◽  
Quadratic Convergence ◽  
Polyhedral Cone ◽  
System Of Equations ◽  
Local Quadratic Convergence ◽  
Ncp Function

The eigenvalue problem over a polyhedral cone is studied in this paper. Based on the F-B NCP function, we reformulate this problem as a system of equations and propose a Jacobian-like method. The global convergence and local quadratic convergence of the proposed method are established under suitable assumptions. Preliminary numerical experiments for a special polyhedral cone are reported in this paper to show the validity of the proposed method.

scholarly journals A speed preserving Hilbert gradient flow for generalized integral Menger curvature

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Jan Knappmann ◽  
Henrik Schumacher ◽  
Daniel Steenebrügge ◽  
Heiko von der Mosel
Keyword(s):  
Gradient Descent ◽  
Gradient Flow ◽  
Optimization Methods ◽  
The Self ◽  
Initial Curve ◽  
Long Time Existence ◽  
Sobolev Gradient ◽  
Long Time ◽  
Menger Curvature ◽  
Time Existence

Abstract We establish long-time existence for a projected Sobolev gradient flow of generalized integral Menger curvature in the Hilbert case and provide C 1 , 1 C^{1,1} -bounds in time for the solution that only depend on the initial curve. The self-avoidance property of integral Menger curvature guarantees that the knot class of the initial curve is preserved under the flow, and the projection ensures that each curve along the flow is parametrized with the same speed as the initial configuration. Finally, we describe how to simulate this flow numerically with substantially higher efficiency than in the corresponding numerical L 2 L^{2} gradient descent or other optimization methods.

Solution of a Nonlinear Eigenvalue Problem Using Signed Singular Values

2017 ◽  
Vol 7 (4) ◽  
pp. 799-809
Author(s):  
Kouhei Ooi ◽  
Yoshinori Mizuno ◽  
Tomohiro Sogabe ◽  
Yusaku Yamamoto ◽  
Shao-Liang Zhang
Keyword(s):  
Eigenvalue Problem ◽  
Computational Efficiency ◽  
Numerical Experiments ◽  
Nonlinear Eigenvalue Problem ◽  
Singular Values ◽  
Singular Value ◽  
Scaling Exponent ◽  
Strong Nonlinearity ◽  
Constant Matrix ◽  
Nonlinear Eigenvalue

AbstractWe propose a robust numerical algorithm for solving the nonlinear eigenvalue problem A(ƛ)x = 0. Our algorithm is based on the idea of finding the value of ƛ for which A(ƛ) is singular by computing the smallest eigenvalue or singular value of A(ƛ) viewed as a constant matrix. To further enhance computational efficiency, we introduce and use the concept of signed singular value. Our method is applicable when A(ƛ) is large and nonsymmetric and has strong nonlinearity. Numerical experiments on a nonlinear eigenvalue problem arising in the computation of scaling exponent in turbulent flow show robustness and effectiveness of our method.

scholarly journals Analysis of the convergence properties of Rubenstein's method for the determination of the lower modes of vibration of a multi-degree of freedom system

1964 ◽  
Vol 54 (6A) ◽  
pp. 1757-1766
Author(s):  
M. E. J. O'Kelly
Keyword(s):  
Global Convergence ◽  
Eigenvalue Problem ◽  
Degree Of Freedom ◽  
Convergence Properties ◽  
Modes Of Vibration

abstract This paper analyzes the convergence properties of a method, recently proposed by Rubenstein, for the determination of the lower modes of vibration of a multi-degree of freedom system from a reduced eigenvalue problem. It is shown that under certain conditions the method converges to the exact eigenvalues. It does not have global convergence and hence some care must be exercised when using it.

Eigenvalue Problem for the Second Order Differential Equation with Nonlocal

2006 ◽  
Vol 11 (1) ◽  
pp. 13-32 ◽  
Author(s):  
B. Bandyrskii ◽  
I. Lazurchak ◽  
V. Makarov ◽  
M. Sapagovas
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Variable Coefficient ◽  
Order Differential Equation ◽  
Integral Condition ◽  
Second Order ◽  
Computational Results ◽  
Discrete Method ◽  
Complex Eigenvalues ◽  
Nonlocal Integral Condition

The paper deals with numerical methods for eigenvalue problem for the second order ordinary differential operator with variable coefficient subject to nonlocal integral condition. FD-method (functional-discrete method) is derived and analyzed for calculating of eigenvalues, particulary complex eigenvalues. The convergence of FD-method is proved. Finally numerical procedures are suggested and computational results are schown.

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