Exact pressure rank and global-mode certification

Determine the exact pressure-space rank and certify the individual global pressure modes selected by the randomized Ritz construction for the nested pressure-space approximation.

Background

The prospective nested pressure-space extension uses a randomized Ritz calculation to select a 443-dimensional space from a sampled 544-dimensional space. The maximum Ritz residual is approximately 0.4472, so the selected dimension is explicitly treated as dependent on the sketch.

Because the calculation is randomized and the residuals are nonzero, the reported cutoff does not establish the exact pressure rank or certify each selected global mode.

References

The maximum Ritz residual was approximately $0.4472$, so $443$ is treated as a sketch-dependent active dimension; exact pressure rank and individual global-mode certification remain open.

— TOGEARI: Interaction-Space Preconditioning for Condensed Finite-Element Systems with IPC Contact  (2608.14162 - Liu et al., 14 Aug 2026) in Appendix D, subsection “Randomized Ritz selection”