2025/05/13 by Nicolás A. Barnafi, Barnafi, Nicolás A., Massimiliano Lupo Pasini +1 · 1 citation
Computer Science · Physics and Astronomy · #65B99 #65F10 #65F99 #65K10 #65N12 #65N22 #FOS: Mathematics #Matrix Theory and Algorithms #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2505.08587
openalex publication_date 2025/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We present a novel two-level sketching extension of the Alternating Anderson-Picard (AAP) method for accelerating fixed-point iterations in challenging single- and multi-physics simulations governed by discretized partial differential equations. Our approach combines a static, physics-based projection that reduces the least-squares problem to the most informative field (e.g., via Schur-complement insight) with a dynamic, algebraic sketching stage driven by a backward stability analysis under Lipschitz continuity. We introduce inexpensive estimators for stability thresholds and cache-aware randomized selection strategies to balance computational cost against memory-access overhead. The resulting algorithm solves reduced least-squares systems in place, minimizes memory footprints, and seamlessly alternates between low-cost Picard updates and Anderson mixing. Implemented in Julia, our two-level sketching AAP achieves up to 50% time-to-solution reductions compared to standard Anderson acceleration-without degrading convergence rates-on benchmark problems including Stokes, p-Laplacian, Bidomain, and Navier-Stokes formulations at varying problem sizes. These results demonstrate the method's robustness, scalability, and potential for integration into high-performance scientific computing frameworks. Our implementation is available open-source in the AAP.jl library.