Solving the large sparse eigenvalue problem that arises in every self-consistent-field iteration is the dominant cost of a Kohn-Sham DFT calculation. Chebyshev-filtered subspace iteration (ChFSI) is well suited to these repeated eigensolves: a Chebyshev polynomial of the Hamiltonian damps the unwanted part of the spectrum and amplifies the occupied subspace without a full diagonalization. However, standard ChFSI is sensitive to errors in the matrix-vector products it relies on. It can stall when the inverse of the overlap matrix in a generalized eigenproblem is only approximated, or when the filtering is carried out in reduced-precision arithmetic.
We introduced R-ChFSI, which recasts the Chebyshev recurrence in terms of residuals rather than direct eigenvector updates. This change damps the errors introduced by inexact products, and we derived convergence guarantees that hold under such approximations. As a result, R-ChFSI can:
use inexpensive approximate inverses for generalized eigenproblems, where exact factorizations are prohibitively expensive;
carry out filtering in low-precision arithmetic (FP32/TF32) on modern GPUs while still reaching tight double-precision tolerances;
tolerate reduced-precision (BF16) inter-process communication in distributed sparse matrix-vector products.
R-ChFSI is the mathematical foundation for the mixed-precision and compressed-communication strategies used in our exascale DFT work.
A Chebyshev polynomial stays bounded over the unwanted part of the spectrum while growing rapidly over the wanted eigenvalues, so applying it to a subspace amplifies the desired eigencomponents.
References
2026
CPC
Residual-based Chebyshev filtered subspace iteration for Hermitian eigenvalue problems tolerant to inexact matrix-vector products
Nikhil Kodali, Kartick Ramakrishnan, and Phani Motamarri
Chebyshev Filtered Subspace Iteration (ChFSI) is widely used for computing a small subset of extremal eigenpairs from large matrices, particularly when the eigenpairs must be computed repeatedly as the system matrix evolves within an outer nonlinear iteration. In this work, we propose R-ChFSI, a residual-based reformulation that recasts the Chebyshev polynomial recurrence in terms of residuals rather than eigenvector estimates, which achieves robust convergence even when matrix–vector products are computed inexactly. We derive convergence guarantees under such approximations and show that R-ChFSI can naturally leverage (i) the use of inexpensive approximate inverses for generalized eigenproblems of the form Ax = λBx, where exact factorizations of B are prohibitively expensive, (ii) low-precision arithmetic (FP32, TF32) for both standard and generalized eigenproblems, and (iii) reduced-precision (BF16) inter-process communication in distributed sparse matrix–vector products. Controlled experiments on dense random matrices quantitatively verify the convergence bounds derived in this work and confirm the robustness of R-ChFSI to prescribed approximation errors for both standard and generalized eigenproblems. Large-scale experiments on finite-element discretized DFT generalized eigenproblems with up to 85 million grid points and 13,500 eigenpairs demonstrate that R-ChFSI achieves residual norms orders of magnitude below those of standard ChFSI when approximate inverses are employed, and reliably meets target tolerances of 10^-8 even when employing reduced precision, yielding filtering speedups of up to 2.7x (2.1x for the full eigensolver) on GPU accelerators.
@article{kodali2025residualbasedchebyshevfilteredsubspace,title={Residual-based Chebyshev filtered subspace iteration for Hermitian eigenvalue problems tolerant to inexact matrix-vector products},author={Kodali, Nikhil and Ramakrishnan, Kartick and Motamarri, Phani},journal={Computer Physics Communications},volume={327},pages={110239},year={2026},issn={0010-4655},doi={10.1016/j.cpc.2026.110239},url={https://doi.org/10.1016/j.cpc.2026.110239},ownpub={true}}
