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Alexander Heinlein: Scalability via Localization: Partition of Unity Functions in Overlapping Domain Decomposition Methods
Schwarz methods are the earliest class of domain decomposition methods. They originated in nineteenth-century analysis of partial differential equations and were later developed into practical iterative algorithms in the pioneering work of Lions. The classical Schwarz framework is based on overlapping decompositions of the computational domain. It provides a flexible basis for constructing efficient, robust, and scalable solvers for partial differential equations. This talk highlights how partition of unity functions can be used in this framework to enable localization, both in classical algorithms and in modern learning architectures. In scientific computing, extension-based partition of unity functions can yield coarse spaces and algebraic multilevel preconditioners that are effective for a wide range of applications, including heterogeneous multiscale and coupled multiphysics problems. In scientific machine learning, the same ideas can motivate domain decomposition-based neural-network architectures. Their localized structure mitigates spectral bias, that is, the tendency to learn low-frequency components before high-frequency ones. It also improves the approximation of local and high-frequency features and naturally supports multilevel approaches to multiscale and wave problems. Through a range of examples, the talk illustrates the breadth of this approach and the practical benefits of partition-of-unity-based localization.
