Residual-based Chebyshev filtered subspace iteration for sparse Hermitian eigenvalue problems tolerant to inexact matrix-vector products (2503.22652v3)

Abstract: Chebyshev Filtered Subspace Iteration (ChFSI) has been widely adopted for computing a small subset of extreme eigenvalues in large sparse matrices. This work introduces a residual-based reformulation of ChFSI, referred to as R-ChFSI, designed to accommodate inexact matrix-vector products while maintaining robust convergence properties. By reformulating the traditional Chebyshev recurrence to operate on residuals rather than eigenvector estimates, the R-ChFSI approach effectively suppresses the errors made in matrix-vector products, improving the convergence behaviour for both standard and generalized eigenproblems. This ability of R-ChFSI to be tolerant to inexact matrix-vector products allows one to incorporate approximate inverses for large-scale generalized eigenproblems, making the method particularly attractive where exact matrix factorizations or iterative methods become computationally expensive for evaluating inverses. It also allows us to compute the matrix-vector products in lower-precision arithmetic allowing us to leverage modern hardware accelerators. Through extensive benchmarking, we demonstrate that R-ChFSI achieves desired residual tolerances while leveraging low-precision arithmetic. For problems with millions of degrees of freedom and thousands of eigenvalues, R-ChFSI attains final residual norms in the range of 10$^{-12}$ to 10$^{-14}$, even with FP32 and TF32 arithmetic, significantly outperforming standard ChFSI in similar settings. In generalized eigenproblems, where approximate inverses are used, R-ChFSI achieves residual tolerances up to ten orders of magnitude lower, demonstrating its robustness to approximation errors. Finally, R-ChFSI provides a scalable and computationally efficient alternative for solving large-scale eigenproblems in high-performance computing environments.