  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.
