Scalable Approximate Selected Inversion Based on Single-Level Incomplete LDLT Factorization and Spectral Corrections for Large Sparse Systems
Abstract
This article introduces four parallel numerical techniques for computing entries of the inverse of large sparse symmetric systems, all grounded in incomplete LDLT (ILDL) factorizations: (1) the selected inversion method (SelInv), which applies the LDLT factorization to recover entries of the matrix inverse within the sparsity pattern of the computed factors; (2) an approximate inversion method based on a truncated Neumann series expansion applied to the inverse of the L factor (NInv), providing an alternative at the cost of reduced accuracy; (3) a Mix approximation that merges the best of both SelInv and NInv; and (4) Mix-SPAI, which applies sparse approximate inverse (SPAI) refinement on the output of the Mix method to improve entry-level accuracy. To further improve accuracy while maintaining a stable sparsity pattern, we additionally employ a low-rank correction based on eigenvector updates, providing an alternative to tightening the drop tolerance. We report the performance of the proposed numerical techniques on a comprehensive collection of sparse matrices from scientific and industrial applications.
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BibTeX
@misc{akter2026scalable,
title = {{Scalable Approximate Selected Inversion Based on Single-Level Incomplete $LDL^T$ Factorization and Spectral Corrections for Large Sparse Systems}},
author = {Tahamina Akter and Matthias Bollhöfer and Olaf Schenk},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11363},
eprint = {2609.11363},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11363},
}
RIS
TY - UNPB
TI - Scalable Approximate Selected Inversion Based on Single-Level Incomplete LDL^T Factorization and Spectral Corrections for Large Sparse Systems
AU - Tahamina Akter
AU - Matthias Bollhöfer
AU - Olaf Schenk
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11363
UR - https://arxiv.org/abs/2609.11363
AB - This article introduces four parallel numerical techniques for computing entries of the inverse of large sparse symmetric systems, all grounded in incomplete LDL^T (ILDL) factorizations: (1) the selected inversion method (SelInv), which applies the LDL^T factorization to recover entries of the matrix inverse within the sparsity pattern of the computed factors; (2) an approximate inversion method based on a truncated Neumann series expansion applied to the inverse of the L factor (NInv), providing an alternative at the cost of reduced accuracy; (3) a Mix approximation that merges the best of both SelInv and NInv; and (4) Mix-SPAI, which applies sparse approximate inverse (SPAI) refinement on the output of the Mix method to improve entry-level accuracy. To further improve accuracy while maintaining a stable sparsity pattern, we additionally employ a low-rank correction based on eigenvector updates, providing an alternative to tightening the drop tolerance. We report the performance of the proposed numerical techniques on a comprehensive collection of sparse matrices from scientific and industrial applications.
ER -