The Hierarchical Subspace Iteration Method for Laplace–Beltrami Eigenproblems

A. Nasikun, K.A. Hildebrandt*

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

5 Citations (Scopus)
72 Downloads (Pure)

Abstract

Sparse eigenproblems are important for various applications in computer graphics. The spectrum and eigenfunctions of the Laplace–Beltrami operator, for example, are fundamental for methods in shape analysis and mesh processing. The Subspace Iteration Method is a robust solver for these problems. In practice, however, Lanczos schemes are often faster. In this article, we introduce the Hierarchical Subspace Iteration Method (HSIM), a novel solver for sparse eigenproblems that operates on a hierarchy of nested vector spaces. The hierarchy is constructed such that on the coarsest space all eigenpairs can be computed with a dense eigensolver. HSIM uses these eigenpairs as initialization and iterates from coarse to fine over the hierarchy. On each level, subspace iterations, initialized with the solution from the previous level, are used to approximate the eigenpairs. This approach substantially reduces the number of iterations needed on the finest grid compared to the non-hierarchical Subspace Iteration Method. Our experiments show that HSIM can solve Laplace–Beltrami eigenproblems on meshes faster than state-of-the-art methods based on Lanczos iterations, preconditioned conjugate gradients, and subspace iterations.
Original languageEnglish
Article number17
Pages (from-to)1-14
Number of pages14
JournalACM Transactions on Graphics
Volume41
Issue number2
DOIs
Publication statusPublished - 2022

Funding

Lembaga Pengelola Dana Pendidikan (LPDP), Indonesia

Keywords

  • Laplace--Beltrami operator
  • Laplace matrix
  • spectral methods
  • multigrid
  • eigensolver
  • subspace iteration method

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