V. Dolean

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 $H_{k}$ -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…

Preconditioning of GMRES for Helmholtz problems with quasimodes

Victorita Dolean, Pierre Marchand, Axel Modave +1

Finite element methods are effective for Helmholtz problems involving complex geometries and heterogeneous media. However, the resulting linear systems are often large, indefinite,…

Neural network-driven domain decomposition for efficient solutions to the Helmholtz equation

Victorita Dolean, Daria Hrebenshchykova, Stéphane Lanteri +1

Accurately simulating wave propagation is crucial in fields such as acoustics, electromagnetism, and seismic analysis. Traditional numerical methods, like finite difference and fin…

Achieving wavenumber robustness in domain decomposition for heterogeneous Helmholtz equation: an overview of spectral coarse spaces

Victorita Dolean, Mark Fry, Matthias Langer +2

Solving time-harmonic wave propagation problems in the frequency domain within heterogeneous media poses significant mathematical and computational challenges, particularly in the…

Local Feature Filtering for Scalable and Well-Conditioned Domain-Decomposed Random Feature Methods

Jan Willem van Beek, Victorita Dolean, Ben Moseley

Random Feature Methods (RFMs) and their variants such as extreme learning machine finite-basis physics-informed neural networks (ELM-FBPINNs) offer a scalable approach for solving…

Convergence analysis of GMRES applied to Helmholtz problems near resonances

Victorita Dolean, Pierre Marchand, Axel Modave +1

The finite element solution of Helmholtz problems near resonant or quasi-resonant frequencies poses significant challenges, as iterative solvers typically suffer from severely degr…

Modal analysis of a domain decomposition method for Maxwell's equations in a waveguide

Victorita Dolean, Antoine Tonnoir, Pierre-Henri Tournier

Time-harmonic wave propagation problems, especially those governed by Maxwell's equations, pose significant computational challenges due to the non-self-adjoint nature of the opera…

An Iterative Block Matrix Inversion (IBMI) Algorithm for Symmetric Positive Definite Matrices with Applications to Covariance Matrices

Ann Paterson, Jennifer Pestana, Victorita Dolean

Obtaining the inverse of a large symmetric positive definite matrix $A \in R^{p \times p}$ is a continual challenge across many mathematical disciplines. The comput…

ELM-FBPINNs: An Efficient Multilevel Random Feature Method

Samuel Anderson, Victorita Dolean, Ben Moseley +1

Domain-decomposed variants of physics-informed neural networks (PINNs) such as finite basis PINNs (FBPINNs) mitigate some of PINNs' issues like slow convergence and spectral bias t…

Schwarz preconditioner with $H_{k}$ -GenEO coarse space for the indefinite Helmholtz problem

Victorita Dolean, Mark Fry, Ivan G. Graham +1

GenEO (`Generalised Eigenvalue problems on the Overlap') is a method from the family of spectral coarse spaces that can efficiently rely on local eigensolves in order to build a ro…

Nonlinear Multilevel Solution Strategies for Diffusive Wave Flood Models in Perforated Domains

Miranda Boutilier, Konstantin Brenner, Victorita Dolean

This article investigates the numerical solution of the Diffusive Wave equation posed on domains containing a large number of polygonal perforations, motivated by urban flood model…