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Reduced basis approximation of parametric eigenvalue problems in presence of clusters and intersections

Daniele Boffi, Abdul Halim, Gopal Priyadarshi

TL;DR

The paper analyzes POD-based reduced order modeling for parametric elliptic eigenvalue problems, with a focus on intersections and clusters of eigenvalues that hinder straightforward generalizations from standard PDE ROMs. It develops an online/offline framework with affine parameter dependence and investigates how the composition of snapshot matrices (including single eigenfunctions, multiple eigenvectors, or linear combinations) affects convergence for the first few eigenpairs, especially in the presence of crossings. Key findings show that including all eigenfunctions associated with the first $n$ eigenvalues in the snapshot matrix yields robust convergence for the first $n$ eigenpairs, while relying on isolated eigenfunctions can misalign higher modes; cheaper strategies such as summing the first $n$ eigenfunctions offer trade-offs between cost and accuracy. The results, demonstrated through extensive one- and two-parameter tests, reveal practical guidelines for snapshot construction and basis size to reliably recover low-lying eigensolutions, informing robust strategies for modal analysis in parametric settings.

Abstract

In this paper we discuss reduced order models for the approximation of parametric eigenvalue problems. In particular, we are interested in the presence of intersections or clusters of eigenvalues. The singularities originating by these phenomena make it hard a straightforward generalization of well known strategies normally used for standards PDEs. We investigate how the known results extend (or not) to higher order frequencies.

Reduced basis approximation of parametric eigenvalue problems in presence of clusters and intersections

TL;DR

The paper analyzes POD-based reduced order modeling for parametric elliptic eigenvalue problems, with a focus on intersections and clusters of eigenvalues that hinder straightforward generalizations from standard PDE ROMs. It develops an online/offline framework with affine parameter dependence and investigates how the composition of snapshot matrices (including single eigenfunctions, multiple eigenvectors, or linear combinations) affects convergence for the first few eigenpairs, especially in the presence of crossings. Key findings show that including all eigenfunctions associated with the first eigenvalues in the snapshot matrix yields robust convergence for the first eigenpairs, while relying on isolated eigenfunctions can misalign higher modes; cheaper strategies such as summing the first eigenfunctions offer trade-offs between cost and accuracy. The results, demonstrated through extensive one- and two-parameter tests, reveal practical guidelines for snapshot construction and basis size to reliably recover low-lying eigensolutions, informing robust strategies for modal analysis in parametric settings.

Abstract

In this paper we discuss reduced order models for the approximation of parametric eigenvalue problems. In particular, we are interested in the presence of intersections or clusters of eigenvalues. The singularities originating by these phenomena make it hard a straightforward generalization of well known strategies normally used for standards PDEs. We investigate how the known results extend (or not) to higher order frequencies.
Paper Structure (37 sections, 1 theorem, 37 equations, 52 figures, 24 tables)

This paper contains 37 sections, 1 theorem, 37 equations, 52 figures, 24 tables.

Table of Contents

  1. Introduction
  2. Problem setting
  3. Parametric elliptic eigenvalue problem
  4. Finite element approximation
  5. Reduced basis approach
  6. Relation between the reduced and the high fidelity systems
  7. Online/offline paradigm
  8. Construction of the POD basis functions
  9. Parameter sampling technique
  10. Overview of the numerical investigations
  11. Numerical results for eigenvalue problems depending on one parameter
  12. Reduced order method to obtain $\\lambda_1$
  13. Reduced order method to obtain $\lambda_{2}$
  14. Results of the EVP considering $u_2$ in the snapshot matrix
  15. Results of the EVP considering $u_1 + u_2$ is in the snapshot matrix
  16. ...and 22 more sections

Key Result

Theorem 3.2

Given a matrix $S \in \mathbb{R}^{N_h \times N_s}$ of rank $r$, the matrix satisfies the optimality property where $\|\cdot\|_F$ is the Frobenius matrix norm.

Figures (52)

  • Figure 1: Uniform sampling of the parameter domain $[0.4,1]^2$ with 25 and 49 points.
  • Figure 2: Sequence of unstructured meshes ($h = 0.1$ and $0.05$)
  • Figure 3: First six sorted eigenvalues when $h = 0.05$ and $\mu = -1.4:0.01:1.4$.
  • Figure 4: First four eigenvectors at $\mu=-1.25,-0.75,-0.5$ using FEM.
  • Figure 5: First four eigenfunctions at $\mu=0.5,0.75,1.25$ using FEM
  • ...and 47 more figures

Theorems & Definitions (3)

  • Remark 3.1
  • Theorem 3.2: Schmidt--Eckart--Young. See Quarteronietal16
  • Remark 3.3