Domain Decomposition-Based Preconditioners and Neural Networks

Attend this seminar to explore how domain decomposition methods bridge classical numerical computing with modern scientific machine learning. Learn about the evolution of domain decomposition techniques from Schwarz's 19th-century foundations through Lions' 1980s algorithmic developments to their current applications in high-performance computing. Discover how these methods decompose global computational domains into smaller subdomains, enabling efficient parallel processing of large-scale problems. Examine modern overlapping Schwarz preconditioners as implemented in the FROSch package of the Trilinos library, which demonstrate robustness and scalability across challenging applications on both CPU and GPU architectures. Explore innovative applications of domain decomposition beyond traditional partial differential equation solvers, including their use in localizing neural networks and operator-learning architectures to introduce sparsity, improve scalability, and enhance training in multiscale settings. See these algorithmic concepts validated through representative test problems covering diffusion, wave propagation, and flow problems, demonstrating how domain decomposition provides a unifying framework that connects large-scale numerical simulation with contemporary scientific machine learning approaches.