Residual-based Chebyshev filtered subspace iteration for sparse Hermitian eigenvalue problems tolerant to inexact matrix-vector products
Nikhil Kodali, Kartick Ramakrishnan, Phani Motamarri
TL;DR
This work introduces R-ChFSI, a residual-based reformulation of Chebyshev Filtered Subspace Iteration that remains convergent under inexact matrix-vector products and supports approximate inverses, enabling efficient generalized Hermitian eigenproblem solves A x = λ B x in large-scale, FE-discretized contexts such as Kohn–Sham DFT. The authors provide a theoretical framework showing contraction of the subspace angle under residual-based filtering, and demonstrate robustness to low-precision arithmetic (e.g., FP32, TF32) and diagonal B^{-1} approximations. Extensive benchmarks on real symmetric and complex Hermitian problems with millions of degrees of freedom reveal that R-ChFSI achieves residuals orders of magnitude smaller than standard ChFSI under inexact computations, with substantial speedups on GPU-accelerated hardware. The results highlight the method’s practicality for high-performance computing applications where exact inverses are prohibitively expensive and mixed-precision computation is desirable.
Abstract
Chebyshev Filtered Subspace Iteration (ChFSI) has emerged as a robust alternative to Krylov eigensolvers for extracting a small subset of extremal eigenpairs from large sparse matrices, particularly in situations where these eigenpairs must be computed repeatedly as the system matrix evolves within an outer iteration. In this work, we propose R-ChFSI, a residual based reformulation of ChFSI designed to exhibit strong convergence properties even when the matrix-vector products are computed inexactly. We derive convergence guarantees under matrix-vector product approximations, providing a rigorous foundation for the method in large-scale eigenvalue computations. The tolerance of R-ChFSI to inexact matrix-vector products enables an efficient treatment of generalized Hermitian definite eigenproblems of the form $\textbf{A} \textbf{x} = λ\textbf{B} \textbf{x}$ where exact factorizations or high-accuracy iterative solves for evaluating $\textbf{B}^{-1}$ are often prohibitively expensive. Moreover, R-ChFSI naturally accommodates low-precision arithmetic for both standard and generalized eigenproblems, making it well-suited for modern hardware accelerators optimised for mixed-precision computation. To demonstrate the effectiveness of the approach, extensive numerical experiments are conducted on finite-element discretized eigenproblems with millions of degrees of freedom, solving for thousands of eigenpairs arising in \emph{ab initio} material modelling using Kohn-Sham density functional theory. For generalized eigenproblems employing approximate $\textbf{B}^{-1}$, R-ChFSI achieves desired residual norms orders of magnitude smaller than those obtained with standard ChFSI. In addition, R-ChFSI reliably reaches target residual tolerances (e.g., 10$^{-8}$) even with FP32 and TF32 arithmetic, significantly outperforming standard ChFSI in similar settings.