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Advanced Linear Algebra with Applications - Part I (Numerical linear algebra for PDEs, machine learning, and data assimilation)
Victorita Dolean, Jemima Tabeart
These lecture notes form the first part of a master's-level course on advanced numerical linear algebra. Their aim is not only to present the classical algorithms, but to show why…
Spectral coarse spaces based on indefinite operators: the -GenEO method
Théophile Chaumont-Frelet, Victorita Dolean, Mark Fry +2
GenEO (`Generalised Eigenvalue problems on the Overlap') is a method for constructing coarse spaces used in the preconditioning of iterative solvers for discrete PDEs. This method…
A Guided Tour of Modern Domain Decomposition: From Schwarz Iterations to Robust Preconditioners and HPC Implementations
Victorita Dolean, Pierre Jolivet, Frédéric Nataf +1
Domain decomposition methods (DDMs) provide a unifying framework for the scalable numerical solution of partial differential equations. Originating from Schwarz's alternating metho…
When can a neural operator replace a coarse solve? Architectural principles for two-level preconditioning
Hugo Melchers, Victorita Dolean, Michael Abdelmalik
Neural operators are increasingly used as accelerators inside classical numerical methods, but it is rarely clear which architectural ingredients matter for which application. We a…
Are Deep Learning Based Hybrid PDE Solvers Reliable? Why Training Paradigms and Update Strategies Matter
Yuhan Wu, Jan Willem van Beek, Victorita Dolean +1
Deep learning-based hybrid iterative methods (DL-HIMs) integrate classical numerical solvers with neural operators, utilizing their complementary spectral biases to accelerate conv…