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A Ritz method for solution of parametric generalized EVPs

Joanna Bisch, Antti Hannukainen

TL;DR

This work tackles the multiparametric generalized eigenvalue problem with affine parameter dependence by constructing a common Ritz subspace that captures both the low-energy modes of the average matrix and a correction operator capturing parameter-induced excursions. The correction operator is approximated via sparse polynomial (stochastic collocation) interpolation, enabling accurate Ritz approximations of eigenvalues in $(0,\Lambda)$ even in the presence of eigenvalue crossings. The authors prove error bounds linking Ritz eigenvalue accuracy to the interpolation quality and subspace closeness, and validate the approach with FEM-based numerical experiments across one and multiple parameter dimensions. The method provides a scalable, robust framework for rapid parametric eigenvalue computations with practical guidance on interpolation parameters and subspace construction.

Abstract

This work deals with approximate solution of generalized eigenvalue problem with coefficient matrix that is an affine function of d-parameters. The coefficient matrix is assumed to be symmetric positive definite and spectrally equivalent to an average matrix for all parameters in a given set. We develop a Ritz method for rapidly approximating the eigenvalues on the spectral interval of interest $(0,Λ)$ for given parameter value. The Ritz subspace is the same for all parameters and it is designed based on the observation that any eigenvector can be split into two components. The first component belongs to a subspace spanned by some eigenvectors of the average matrix. The second component is defined by a correction operator that is a d + 1 dimensional analytic function. We use this structure and build our Ritz subspace from eigenvectors of the average matrix and samples of the correction operator. The samples are evaluated at interpolation points related to a sparse polynomial interpolation method. We show that the resulting Ritz subspace can approximate eigenvectors of the original problem related to the spectral interval of interest with the same accuracy as the sparse polynomial interpolation approximates the correction operator. Bound for Ritz eigenvalue error follows from this and known results. Theoretical results are illustrated by numerical examples. The advantage of our approach is that the analysis treats multiple eigenvalues and eigenvalue crossings that typically have posed technical challenges in similar works.

A Ritz method for solution of parametric generalized EVPs

TL;DR

This work tackles the multiparametric generalized eigenvalue problem with affine parameter dependence by constructing a common Ritz subspace that captures both the low-energy modes of the average matrix and a correction operator capturing parameter-induced excursions. The correction operator is approximated via sparse polynomial (stochastic collocation) interpolation, enabling accurate Ritz approximations of eigenvalues in even in the presence of eigenvalue crossings. The authors prove error bounds linking Ritz eigenvalue accuracy to the interpolation quality and subspace closeness, and validate the approach with FEM-based numerical experiments across one and multiple parameter dimensions. The method provides a scalable, robust framework for rapid parametric eigenvalue computations with practical guidance on interpolation parameters and subspace construction.

Abstract

This work deals with approximate solution of generalized eigenvalue problem with coefficient matrix that is an affine function of d-parameters. The coefficient matrix is assumed to be symmetric positive definite and spectrally equivalent to an average matrix for all parameters in a given set. We develop a Ritz method for rapidly approximating the eigenvalues on the spectral interval of interest for given parameter value. The Ritz subspace is the same for all parameters and it is designed based on the observation that any eigenvector can be split into two components. The first component belongs to a subspace spanned by some eigenvectors of the average matrix. The second component is defined by a correction operator that is a d + 1 dimensional analytic function. We use this structure and build our Ritz subspace from eigenvectors of the average matrix and samples of the correction operator. The samples are evaluated at interpolation points related to a sparse polynomial interpolation method. We show that the resulting Ritz subspace can approximate eigenvectors of the original problem related to the spectral interval of interest with the same accuracy as the sparse polynomial interpolation approximates the correction operator. Bound for Ritz eigenvalue error follows from this and known results. Theoretical results are illustrated by numerical examples. The advantage of our approach is that the analysis treats multiple eigenvalues and eigenvalue crossings that typically have posed technical challenges in similar works.

Paper Structure

This paper contains 20 sections, 16 theorems, 98 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Review on Ritz method
  4. Application in FEM
  5. Generalized eigenvalue behaviour
  6. Average matrix
  7. Spectral separation
  8. Correction operator and the stochastic collocation method
  9. Correction formula
  10. Ritz subspace by polynomial approximation
  11. The model problem and hierarchic polynomial approximation
  12. The Stochastic collocation operator
  13. $L^2$ Convergence rate
  14. Error analysis
  15. Convergence rate
  16. ...and 5 more sections

Key Result

Lemma 2.1

Let $\sigma \in S$, $(\lambda_k(\sigma))_k$ be the eigenvalues of eq:evp, and $(\mu_m(\sigma))_m$ their Ritz approximation from $V$. In addition, let $E_{\leq \lambda_k(\sigma)}(\sigma)$ be the union of all eigenspaces related to $(0,\lambda_k(\sigma))$ and $P_{A(\sigma)}$ the $A(\sigma)$ orthogonal for any $\sigma \in S$. The multiplicative factor $C(\lambda_k(\sigma), V) = (1-e(\lambda_k(\sigma)

Figures (3)

  • Figure 1: Values of \ref{['Error_studied']} for the $10$ smallest exact eigenvalue solutions using RMCLI (black line), and reduced version (red line) with $\eta=0.5$, $\varepsilon=\eta/10$ and $q=1,\ldots,5$.
  • Figure 2: Value of \ref{['Error_studied']} for the $10$ smallest exact eigenvalues of \ref{['eq:evp']} for $10$ values of $\sigma$ using RMCLI with $\eta=0.5$, $\varepsilon = \eta/(10 k)$ and $q=2k$ (black line), Legendre points with $\eta=(0.5,0.25)^T$ and $\varepsilon = \eta_1/(10 k)$ (red dashed line) and Padua points (blue line) for $\rho\Lambda=320$, after $6$ refinements of mesh for $k=1,2,3$.
  • Figure 3: Value of \ref{['Error_studied']} in the $1,2,3$ and $4$d cases with $S=[-1;1]^d$ for the $10$ smallest eigenvalues with $\lambda_i(\sigma)$ for 10 value of $\sigma$ using RMCLI and its reduced version for $q=1,3$ and $5$.

Theorems & Definitions (43)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.1
  • proof : Proof of Lemma \ref{['Lemma_gen_res_bound_lamb_by_x']}
  • Corollary 2.1
  • Example 2.1
  • Example 2.2
  • Definition 3.1
  • Example 3.1
  • ...and 33 more