Optimal Diagonal Preconditioning Beyond Worst-Case Conditioning: Theory and Practice of Omega Scaling
The study compares optimal diagonal preconditioning based on two metrics: classical worst-case $\kappa$-condition number and averaging-based $\omega$-condition number. The authors find that while $\kappa$-optimal methods offer stronger theoretical reductions in the worst case, $\omega$-optimal preconditioning is significantly cheaper to compute and yields better practical performance for iterative solvers like PCG and LSQR.