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Stability of Quadratic Projection Methods

Lyonell Boulton, Michael Strauss

TL;DR

This paper addresses spectral pollution that arises when approximating the discrete spectrum of a self-adjoint operator via Galerkin methods. It introduces the pollution-free quadratic projection method, formulating a matrix polynomial $M(z)=A_0-2zA_1+z^2A_2$ whose zeros yield reliable spectral information, with a key stability bound $\mathrm{dist}[\mathrm{Re}\,\zeta, \mathrm{Spec}(A)] \le |\mathrm{Im}\,\zeta|$. The authors establish stability results under coefficient perturbations and show convergence of the quadratic method under reasonable subspace approximations (even when $\mathcal{L}_n\nsubseteq {\rm Dom}(A^2)$ by appropriate refinements). A finite-rank perturbation case study demonstrates nonpolluting retrieval of isolated eigenvalues and highlights the practical robustness of structured versus unstructured perturbations, supported by numerical experiments. Overall, the work provides a general, reliable framework for pollution-free spectral approximation applicable to broad self-adjoint operators, with concrete theoretical guarantees and numerical validation.

Abstract

In this paper we discuss the stability of an alternative pollution-free procedure for computing spectra. The main difference with the Galerkin method lies in the fact that it gives rise to a weak approximate problem which is quadratic in the spectral parameter, instead of linear. Previous accounts on this new procedure can be found in Levitin and Shargorodsky (2002) [math.SP/0212087] and Boulton (2006) [math.SP/0503126].

Stability of Quadratic Projection Methods

TL;DR

This paper addresses spectral pollution that arises when approximating the discrete spectrum of a self-adjoint operator via Galerkin methods. It introduces the pollution-free quadratic projection method, formulating a matrix polynomial whose zeros yield reliable spectral information, with a key stability bound . The authors establish stability results under coefficient perturbations and show convergence of the quadratic method under reasonable subspace approximations (even when by appropriate refinements). A finite-rank perturbation case study demonstrates nonpolluting retrieval of isolated eigenvalues and highlights the practical robustness of structured versus unstructured perturbations, supported by numerical experiments. Overall, the work provides a general, reliable framework for pollution-free spectral approximation applicable to broad self-adjoint operators, with concrete theoretical guarantees and numerical validation.

Abstract

In this paper we discuss the stability of an alternative pollution-free procedure for computing spectra. The main difference with the Galerkin method lies in the fact that it gives rise to a weak approximate problem which is quadratic in the spectral parameter, instead of linear. Previous accounts on this new procedure can be found in Levitin and Shargorodsky (2002) [math.SP/0212087] and Boulton (2006) [math.SP/0503126].

Paper Structure

This paper contains 8 sections, 8 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Spectral Pollution in the Galerkin method
  3. Pollution-free strategies and quadratic methods
  4. Stability of Quadratic Projection Methods
  5. Pollution-free Stability
  6. Stability of Convergence in the Quadratic Method
  7. Case Study
  8. Acknowledgements

Key Result

Lemma 1

For $z\in\mathbb{C}$ and $\delta > 0$, let Then,

Figures (2)

  • Figure 1: Exact solutions to $(\mathrm{Q})$ for $n=50$.
  • Figure 3: Error predicted by Theorem \ref{['pract']} in the approximation of $\lambda^-$. Here we depict $|{\rm Im}\; \zeta_n^-|$ (unperturbed), $|{\rm Im}\; \zeta_n^{\mathrm{u},-}|$ and $|{\rm Im}\; \zeta_n^{\mathrm{s},-}|$ for $n=5:10:100$. We average the two perturbed solutions of $(\tilde{\mathrm{Q}})$ over a sample of $20$ problems with $\varepsilon=10^{-1}$. The scaling is log-log and the horizontal axis shows $2n+1$.

Theorems & Definitions (16)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Remark 4
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • ...and 6 more