Parameter-Robust Subspace Correction with Multiple Semidefinite Penalties
Abstract
Independently weighted semidefinite penalties arise in augmented-Lagrangian and constrained formulations. This paper characterizes when an exact additive subspace-correction preconditioner remains uniformly effective over all nonnegative penalty weights on a fixed finite-dimensional space. Robustness holds precisely when the correction spaces decompose every joint kernel generated by a nonempty subset of penalties. If one condition fails, a computable constant determines the exact first-order decay of the smallest preconditioned eigenvalue along the associated parameter ray, and the condition number grows linearly; none of the subset conditions can be discarded in general. Filtered decompositions provide computable sufficient bounds on parameter-ordering cones, while distributive kernel lattices permit a single common splitting. Exact-additive computations confirm the characterization and predicted rates. Separate Scott-Vogelius experiments produce stable multilevel iteration counts over the tested weights and mesh levels. The analysis does not establish mesh-uniformity.
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BibTeX
@misc{bhattacharjee2026parameterrobust,
title = {{Parameter-Robust Subspace Correction with Multiple Semidefinite Penalties}},
author = {Subhransu S. Bhattacharjee},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30265},
eprint = {2608.30265},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30265},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Parameter-Robust Subspace Correction with Multiple Semidefinite Penalties
AU - Subhransu S. Bhattacharjee
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30265
UR - https://arxiv.org/abs/2608.30265
AB - Independently weighted semidefinite penalties arise in augmented-Lagrangian and constrained formulations. This paper characterizes when an exact additive subspace-correction preconditioner remains uniformly effective over all nonnegative penalty weights on a fixed finite-dimensional space. Robustness holds precisely when the correction spaces decompose every joint kernel generated by a nonempty subset of penalties. If one condition fails, a computable constant determines the exact first-order decay of the smallest preconditioned eigenvalue along the associated parameter ray, and the condition number grows linearly; none of the subset conditions can be discarded in general. Filtered decompositions provide computable sufficient bounds on parameter-ordering cones, while distributive kernel lattices permit a single common splitting. Exact-additive computations confirm the characterization and predicted rates. Separate Scott-Vogelius experiments produce stable multilevel iteration counts over the tested weights and mesh levels. The analysis does not establish mesh-uniformity.
ER -