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Model Order Reduction for Parametric Hermitian Eigenvalue Problems: Local Acceleration with Taylor-Reduced Basis Method

Benjamin Stamm, Zhuoyao Zeng

Abstract

This paper is concerned with the Taylor-reduced basis method (Taylor-RBM) for the efficient approximation of eigenspaces of large scale parametric Hermitian matrices. The Taylor-RBM is a local model order reduction method, which constructs an approximation space by capturing derivatives information of the spectral projector at a reference point in the parameter domain. We perform a concise error analysis to justify the Taylor-RBM for eigenvalue problems, and we present a computationally efficient procedure to assemble the Taylor-reduced basis space. Since this method is tightly connected to the classical multivariate analytic perturbation theory, we also provide a detailed analysis of the spectral approximation using the truncated power series of the eigenprojector, and compare this with the approximation obtained from the Taylor-RBM.

Model Order Reduction for Parametric Hermitian Eigenvalue Problems: Local Acceleration with Taylor-Reduced Basis Method

Abstract

This paper is concerned with the Taylor-reduced basis method (Taylor-RBM) for the efficient approximation of eigenspaces of large scale parametric Hermitian matrices. The Taylor-RBM is a local model order reduction method, which constructs an approximation space by capturing derivatives information of the spectral projector at a reference point in the parameter domain. We perform a concise error analysis to justify the Taylor-RBM for eigenvalue problems, and we present a computationally efficient procedure to assemble the Taylor-reduced basis space. Since this method is tightly connected to the classical multivariate analytic perturbation theory, we also provide a detailed analysis of the spectral approximation using the truncated power series of the eigenprojector, and compare this with the approximation obtained from the Taylor-RBM.
Paper Structure (26 sections, 15 theorems, 124 equations, 4 figures, 1 table, 1 algorithm)

This paper contains 26 sections, 15 theorems, 124 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature review and state-of-the-art
  3. Contributions
  4. Organisation of the manuscript
  5. General notations
  6. Selected aspects of the subspace method for eigenspace approximation
  7. The total projector as an analytic function and its finite truncation
  8. Useful properties of the unperturbed case
  9. Analyticity of the parametric total eigenprojector
  10. Properties of the total projector series and its coefficients
  11. Truncation of the total projector series
  12. Taylor-Reduced Basis Method
  13. The Taylor-RBM and its convergence analysis
  14. Practical assembly
  15. Recursive formula for the assembly of $\rm{span} \{ \mathrm{ran} \, \mathbf{P}_{\!0}^{(\boldsymbol{\beta})} : |\boldsymbol{\beta}| \leq n \}$
  16. ...and 11 more sections

Key Result

Lemma 2.3

Let $\mathbf{P}$, $\mathbf{Q}$ be two orthogonal projectors of the same rank. Then, it holds that

Figures (4)

  • Figure 1: xxz-model of length $L=15$: Approximation error and error of Ritz-value sum of the total eigenspace of rank $m_1$ at parameters $\boldsymbol{\mu}$ with $\boldsymbol{\mu} - \boldsymbol{\mu}_0 \in [-0.08,\, 0.08]^2$ using Taylor-RBM errors of order $n$, where $\boldsymbol{\mu}_0 =(1,1)^{\!\top}$ is marked in red cross and has $m_1 = 1$.
  • Figure 2: xxz-model of length $L=15$: Sections of the error curves of the Taylor-RBM with increasing derivative order $n$ along the parameter path from $(1,1)^{\!\top}$ to $(1.08, 1)^{\!\top}$.
  • Figure 3: xxz-model of length $L=10$: Approximation errors of the total eigenspace of rank $m_1$ at parameters $\boldsymbol{\mu}$ with $\boldsymbol{\mu} - \boldsymbol{\mu}_0 \in [-0.03,\, 0.03]^2$ using Taylor-RBM and PT of order $n \leq 3$ concerning the smallest eigenvalue cluster at $\boldsymbol{\mu}_0 = (-1,0)^{\!\top}$, which is marked in red and has degeneracy $m_1 = 11$.
  • Figure 4: xxz-model of length $L=10$: Sections of the error curves of the Taylor-RBM with increasing derivative order $n$ along the parameter path from $(-1,0)^{\!\top}$ to $(-0.97, 0)^{\!\top}$. "PT approx. error" refers to $\| {\mathbf{Q}_{\rm{pt}}}(\boldsymbol{\mu}) - \mathbf{P}(\boldsymbol{\mu}) \|$, and "truncation error" is $\| \mathbf{P}^{[n]}(\boldsymbol{\mu}) - \mathbf{P}(\boldsymbol{\mu}) \|$, cf. the proof of \ref{['lem:closedness_test_space']}. By "Taylor-RBM approx. error" and "projection error", we mean $\| {\mathbf{Q}_{\rm{rb}}}(\boldsymbol{\mu}) - \mathbf{P}(\boldsymbol{\mu}) \|$ and $\|{\boldsymbol{\Pi}}_n^\perp \mathbf{P}(\boldsymbol{\mu})\|$, respectively, cf. \ref{['thm:Taylor_rbm_projector_error']}.

Theorems & Definitions (39)

  • Example 2.1: Pathological case of subspace method for eigenspace approximation
  • Remark 2.2: Variational approximation
  • Lemma 2.3: Equivalence of subspace approximation error
  • Theorem 2.4: Bounding eigenspace approximation error by eigenspace projection error
  • proof
  • Lemma 2.5: Squaring effect of Ritz-value difference
  • proof
  • Remark 3.1: Connection to classical perturbation results
  • Lemma 3.2: Upper bound on the rank of the coefficient matrices
  • Lemma 3.3: Hermitian property of the coefficient matrices
  • ...and 29 more