Unrolled Optimized Restricted Additive Schwarz Methods

Large linear systems from PDE discretizations are typically solved iteratively, and their runtime depends heavily on the quality of the preconditioner. Domain-decomposition preconditioners are attractive because they split the global problem into parallel local solves, but their convergence depends on how information is exchanged across subdomain interfaces. Classical optimized Schwarz methods prescribe these interface parameters analytically, using Fourier minimax arguments that are exact only under idealized assumptions. We introduce UORAS, an unrolled optimized Schwarz preconditioner that keeps the classical analytical parameter rule as a backbone and learns only a multiplicative correction. We also introduce a residual-conditioned non-stationary variant that learns an iteration-dependent schedule through a shared neural network. On the theory side, we show that minimax tuning can be strictly suboptimal for finite-depth distributional losses with positive overlap, and we connect this gap to stationary and residual-conditioned UORAS hypothesis classes. Empirically, UORAS improves on classical optimized Schwarz baselines and is designed to retain the benefits of learning-based interface models with far fewer parameters and improved scalability.