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A Newton method for solving locally definite multiparameter eigenvalue problems by multiindex

Henrik Eisenmann

TL;DR

This work addresses solving locally definite multiparameter eigenvalue problems (MEP) by a semismooth Newton method applied to functions $F_{\mathbf i}$ that encode the eigenvalue multiindex. By exploiting a generalized Jacobian and the concept of signed multiindices, the method achieves local quadratic convergence and, for extreme eigenvalues, global linear convergence, with complexity per eigenpair on the order of $\mathcal{O}\left(\sum_{k=1}^m n_k^3 + m^3\right)$. The framework extends to non-Hermitian discretizations via a Hermitian transform and is applicable to multiparameter Sturm–Liouville problems, ellipsoidal domains, and large-scale problems with many parameters. Numerical experiments on an ellipsoidal wave equation and randomly generated examples illustrate accurate recovery of targeted eigenvalues and favorable scalability, often outperforming standard subspace methods for computing many eigenpairs.

Abstract

We present a new approach to compute eigenvalues and eigenvectors of locally definite multiparameter eigenvalue problems by its signed multiindex. The method has the interpretation of a semismooth Newton method applied to certain functions that have a unique zero. We can therefore show local quadratic convergence, and for certain extreme eigenvalues even global linear convergence of the method. Local definiteness is a weaker condition than right and left definiteness, which is often considered for multiparameter eigenvalue problems. These conditions are naturally satisfied for multiparameter Sturm-Liouville problems that arise when separation of variables can be applied to multidimensional boundary eigenvalue problems.

A Newton method for solving locally definite multiparameter eigenvalue problems by multiindex

TL;DR

This work addresses solving locally definite multiparameter eigenvalue problems (MEP) by a semismooth Newton method applied to functions that encode the eigenvalue multiindex. By exploiting a generalized Jacobian and the concept of signed multiindices, the method achieves local quadratic convergence and, for extreme eigenvalues, global linear convergence, with complexity per eigenpair on the order of . The framework extends to non-Hermitian discretizations via a Hermitian transform and is applicable to multiparameter Sturm–Liouville problems, ellipsoidal domains, and large-scale problems with many parameters. Numerical experiments on an ellipsoidal wave equation and randomly generated examples illustrate accurate recovery of targeted eigenvalues and favorable scalability, often outperforming standard subspace methods for computing many eigenpairs.

Abstract

We present a new approach to compute eigenvalues and eigenvectors of locally definite multiparameter eigenvalue problems by its signed multiindex. The method has the interpretation of a semismooth Newton method applied to certain functions that have a unique zero. We can therefore show local quadratic convergence, and for certain extreme eigenvalues even global linear convergence of the method. Local definiteness is a weaker condition than right and left definiteness, which is often considered for multiparameter eigenvalue problems. These conditions are naturally satisfied for multiparameter Sturm-Liouville problems that arise when separation of variables can be applied to multidimensional boundary eigenvalue problems.
Paper Structure (13 sections, 10 theorems, 82 equations, 5 figures, 1 table, 4 algorithms)

This paper contains 13 sections, 10 theorems, 82 equations, 5 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Definite multiparameter eigenvalue problems
  3. Multiparameter Sturm–Liouville problems
  4. A Newton-type method
  5. Extreme eigenvalues
  6. Handling non-Hermitian matrices
  7. Numerical experiments
  8. Ellipsoidal wave equation
  9. Randomly generated examples
  10. Randomly generated three-parameter eigenvalue problems
  11. Randomly generated multiparameter eigenvalue problems
  12. Locally definite problem
  13. Conclusion and outlook

Key Result

Theorem 3

Volkmer88 Let the MEP eq:HomMEP be locally definite. Then for every multiindex $\mathbf i\in \{1,\ldots,n_1\}\times \ldots\times \{1,\ldots,n_m\}$ and every sign $\sigma\in\{+,-\}$ there is a unique eigenvalue $\lambda\in \mathcal{P}^\sigma$ of multiindex $\mathbf i$. We say $\lambda$ is of signed i

Figures (5)

  • Figure 1: Experiment for the ellipsoidal wave equation as described in \ref{['sec: numerical experiment Ellipsoidal wave equation']}. This figure shows the order of computation.
  • Figure 2: Experiment for the ellipsoidal wave equation as described in \ref{['sec: numerical experiment Ellipsoidal wave equation']}. The left figure compares the computational complexity with threepareigs_jd from multipareig and the right figure the acquired accuracy of eigenvalues.
  • Figure 3: Comparison of the performance of \ref{['alg:Newton']} or \ref{['alg:NewtonGlobalized']} respectively and threepareig from multipareig. The multiparameter eigenvalue problems are generated randomly as described in \ref{['sec: randomly generated examples']} with $n\times n$ matrices and varying size $n$. Here, all eigenvalues are computed. The first two graphs showcase the case of increasingly bad conditioned problems generated with Laguerre polynomials \ref{['eq: Laguerre random matrices']}, the second two graphs show the uniformly well conditioned problems \ref{['eq: uniformly well conditioned matrices']}.
  • Figure 4: Showcase of the performance of \ref{['alg:NewtonGlobalized']} for $n\times n$ matrices and varying number of parameters $m$. The multiparameter eigenvalue problems were generated randomly as described in \ref{['sec: randomly generated examples']}. All eigenvalues were computed. The first two graphs show the case of increasingly bad conditioned problems generated with Laguerre polynomials \ref{['eq: Laguerre random matrices']}, the second two graphs showcase the uniformly well conditioned problems \ref{['eq: uniformly well conditioned matrices']} As a comparison, we show the observed costs $(mn^2+m^3)n^m$.
  • Figure 5: Showcase of the performance of \ref{['alg:NewtonGlobalized']} for $n\times n$ matrices and varying number of parameters $m$. The multiparameter eigenvalue problems were generated randomly as described in \ref{['sec: randomly generated examples']}. The first two graphs show the case of increasingly bad conditioned problems generated with Laguerre polynomials \ref{['eq: Laguerre random matrices']}, the second two graphs show the uniformly well conditioned problems \ref{['eq: uniformly well conditioned matrices']} As a comparison, we show the observed rates for the costs $mn^2+m^3$ and $mn^{2.5}+m^3$, respectively.

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Definition 4
  • Lemma 5
  • Lemma 1
  • Proof 1
  • Proposition 2
  • Proof 2
  • Proposition 3
  • ...and 11 more