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.