The Hierarchical Subspace Iteration Method for Laplace--Beltrami Eigenproblems

TL;DR

The paper introduces HSIM, a hierarchical solver for Laplace--Beltrami eigenproblems that improves efficiency by operating across nested vector spaces, outperforming existing methods like Lanczos in shape analysis applications.

Contribution

The Hierarchical Subspace Iteration Method (HSIM) is a novel hierarchical approach that accelerates eigenproblem solutions on meshes by combining coarse-to-fine subspace iterations.

Findings

HSIM reduces iteration count on fine grids.

HSIM outperforms Lanczos and other state-of-the-art methods.

Faster eigenproblem solutions in shape analysis applications.

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 paper, 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.