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An inexact Matrix-Newton method for solving NEPv

Tom Werner

Abstract

In this paper, an inexact Newton method for solving real-valued nonlinear eigenvalue problems with eigenvector dependency (NEPv) is introduced that is able to solve the problem on a matrix level. Our main contribution is to derive a variant of Newton's method that uses global Krylov methods such as global GMRES to solve the linear operator equation necessary to compute the Newton correction in a matrix-free way. The advantages that this second order method has over the well-established SCF algorithm are explained and visualized by a variety of numerical experiments.

An inexact Matrix-Newton method for solving NEPv

Abstract

In this paper, an inexact Newton method for solving real-valued nonlinear eigenvalue problems with eigenvector dependency (NEPv) is introduced that is able to solve the problem on a matrix level. Our main contribution is to derive a variant of Newton's method that uses global Krylov methods such as global GMRES to solve the linear operator equation necessary to compute the Newton correction in a matrix-free way. The advantages that this second order method has over the well-established SCF algorithm are explained and visualized by a variety of numerical experiments.
Paper Structure (19 sections, 4 theorems, 73 equations, 7 figures, 5 algorithms)

This paper contains 19 sections, 4 theorems, 73 equations, 7 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Newton's Method for nonlinear matrix equations
  3. Inexact Newton methods
  4. The inexact Matrix-Newton method for NEPv
  5. Pre and Postprocessing by SCF
  6. Local convergence analysis
  7. Implementation details
  8. GL-GMRES for NEPv
  9. Forcing terms and damping
  10. Numerical Experiments
  11. The Kohn-Sham NEPv
  12. A simple 1D model
  13. A 3D model
  14. The robust LDA GNEPv
  15. The sum of trace ratios
  16. ...and 4 more sections

Key Result

Lemma 1

Let $F:\mathbb{R}^{n,k}\rightarrow\mathbb{R}^{n,k}$ be Fréchet differentiable at $X\in\mathbb{R}^{n,k}$ with Fréchet derivative $L_F(X):\mathbb{R}^{n,k}\rightarrow\mathbb{R}^{n,k}$ and Kronecker form $K_F(X)\in\mathbb{R}^{n\cdot k,n\cdot k}$. Then, for any $B\in\mathbb{R}^{n,k}$, if neither iteratio after $\ell$ steps of the global GMRES method and the approximate solution $y_\ell\in\mathbb{R}^{n\

Figures (7)

  • Figure 4.1: Convergence results for simple KS-Problem by JFNK(top left), JFNGK(top right) and SCF(bottom left) as well as relative timing(bottom right). The convergence plots show results for $\gamma=0.5$($\diamond$), $\gamma=0.6$(x), $\gamma=0.7$($\square$), $\gamma=0.75$($\ast$), $\gamma=0.8$(+), $\gamma=0.85$($\otimes$) and $\gamma=0.9$($\triangle$)
  • Figure 4.2: Convergence results for KS-problem, $n=32^3$, $\gamma=1$, $k=2$(left) and $k=8$(right)
  • Figure 4.3: Consequences of switching to Newton too early(left) or too late(right)
  • Figure 4.4: Convergence results for different choices of $\alpha$, Ionosphere-Data-Set
  • Figure 4.5: Convergence results for different choices of $\alpha$, Sonar-Data-Set
  • ...and 2 more figures

Theorems & Definitions (7)

  • Definition 1: Fréchet derivative and Kronecker form
  • Lemma 1
  • Remark 1
  • Definition 2: Dimension of a zero and semi-regularity (nonisolated)
  • Lemma 2
  • Lemma 3: Consistency of the update equation
  • Theorem 1: Convergence of Newton's method for NEPv