[Paper Review] A Fast Hierarchically Preconditioned Eigensolver Based On Multiresolution Matrix Decomposition

Thomas Y. Hou, De Huang+2 more|arXiv (Cornell University)|Apr 10, 2018

Electromagnetic Scattering and Analysis11 references3 citations

TL;DR

This paper proposes a fast, hierarchically preconditioned eigensolver that leverages multiresolution matrix decomposition (MMD) to efficiently compute a large number of leftmost eigenpairs of sparse symmetric positive definite matrices. By integrating MMD with the Implicitly Restarted Lanczos Method (IRLM) via a spectrum-preserving preconditioner, the algorithm reduces computational complexity from O(m_tar·κ(A)·nnz(A)·log(1/ε)) to O(m_tar·nnz(A)·(log(1/ε)+log n)^C), significantly improving performance on ill-conditioned and large-scale problems.

ABSTRACT

In this paper we propose a new iterative method to hierarchically compute a relatively large number of leftmost eigenpairs of a sparse symmetric positive matrix under the multiresolution operator compression framework. We exploit the well-conditioned property of every decomposition components by integrating the multiresolution framework into the Implicitly restarted Lanczos method. We achieve this combination by proposing an extension-refinement iterative scheme, in which the intrinsic idea is to decompose the target spectrum into several segments such that the corresponding eigenproblem in each segment is well-conditioned. Theoretical analysis and numerical illustration are also reported to illustrate the efficiency and effectiveness of this algorithm.

Motivation & Objective

• To develop an efficient iterative method for computing a large number of leftmost eigenpairs of sparse symmetric positive definite (SPD) matrices.
• To improve the performance of the Implicitly Restarted Lanczos Method (IRLM) by integrating it with the multiresolution operator compression framework.
• To design a spectrum-preserving preconditioner for the Conjugate Gradient method that maintains well-conditioned subproblems across multiresolution levels.
• To reduce the computational complexity of IRLM from dependence on the matrix condition number κ(A) to a polylogarithmic dependence on n, enabling scalability on ill-conditioned problems.
• To validate the algorithm’s efficiency and accuracy through theoretical analysis and numerical experiments on real-world applications such as graph partitioning and electronic structure prediction.

Proposed method

• The method employs a multiresolution matrix decomposition (MMD) to hierarchically decompose the inverse of the matrix A into a sum of projection operators and a remainder, enabling recursive spectral compression.
• An extension-refinement iterative scheme is introduced to segment the target spectrum into well-conditioned subproblems, each solved independently using preconditioned Conjugate Gradient (PCG) methods.
• A novel spectrum-preserving preconditioner is constructed for PCG by exploiting the intrinsic structure of MMD, ensuring that the residual spectrum remains narrow and well-conditioned at each level.
• The algorithm integrates the MMD framework into the IRLM by using hierarchical basis functions that are localized and adaptively refined, preserving spectral properties across scales.
• Local subproblems for computing basis functions are solved independently and efficiently using exponential decay properties, enabling embarrassingly parallel computation.
• The method uses a recursive energy decomposition framework to compress the matrix inverse, allowing for fast and accurate approximation of eigenpairs without full matrix inversion.

Experimental results

Research questions

• RQ1 Can multiresolution matrix decomposition be effectively integrated into the Implicitly Restarted Lanczos Method to accelerate convergence for computing leftmost eigenpairs?
• RQ2 Does a spectrum-preserving preconditioner based on MMD significantly reduce the condition number of subproblems and improve the convergence rate of the Conjugate Gradient method?
• RQ3 Can the proposed hierarchical framework achieve computational complexity independent of the matrix condition number κ(A), particularly for ill-conditioned sparse SPD matrices?
• RQ4 How well do the computed eigenpairs preserve spectral properties for applications such as graph partitioning and electronic structure prediction?
• RQ5 To what extent do localized basis functions derived from MMD reflect intrinsic operator structure, and can they be computed efficiently in parallel?

Key findings

• The proposed algorithm reduces the computational complexity of computing m_tar leftmost eigenpairs from O(m_tar·κ(A)·nnz(A)·log(1/ε)) to O(m_tar·nnz(A)·(log(1/ε)+log n)^C), where C is a small constant independent of m_tar, nnz(A), and κ(A).
• The spectrum-preserving preconditioner maintains a narrow residual spectrum during the Lanczos process, significantly accelerating convergence of the Conjugate Gradient solver at each level.
• Numerical experiments on graph Laplacians (e.g., bunny and brain datasets) show that the compressed operator produces eigenvectors that yield nearly identical graph partitioning results to the original matrix, validating spectral fidelity.
• The localized basis functions (e.g., ψ_cm, ψ_o) derived from the MMD framework exhibit similar profiles across different construction methods, suggesting an intrinsic operator-driven localization mechanism.
• The algorithm enables efficient, embarrassingly parallel computation of basis functions due to exponential decay and localized support, reducing global computation cost.
• The method achieves high accuracy and scalability on large-scale, ill-conditioned problems, outperforming classical IRLM in terms of runtime while maintaining spectral precision.