Paola F. Antonietti, Professor, Politecnico di Milano

Chensheng Zhang, Professor

Multigrid solvers for high-order DG discrete systems

16:00-17:00 January 15(Wednesday), Zoom Meeting: 995 2238 9833

This presentation addresses multigrid solvers specifically designed for algebraic systems stemming from high-order Discontinuous Galerkin (DG) discretisations. Due to their flexibility and accuracy, high-order DG methods represent a robust approach for discretising partial differential equations, particularly in complex geometries and heterogeneous media scenarios. Conversely, the computational costs of solving the resultant large, ill-conditioned systems remain a significant challenge.

In the first part of the talk, we discuss the design and analysis of geometric multigrid strategies for linear systems arising from high-order DG methods. These solvers utilise coarsening spaces constructed on sequences of (nested or non-nested) polytopal agglomerated meshes obtained through machine-learning techniques. The performance of these multigrid solvers is rigorously analysed in terms of convergence rates and computational efficiency, demonstrating their robustness, scalability, and efficacy.  

In the second part of the talk, we present algebraic multigrid methods for high-order DG methods. Standard AMG approaches cannot be applied here because the degrees of freedom associated with the same grid point are redundant. We introduce new aggregation procedures and test them through extensive numerical experiments, demonstrating that the proposed AMG method is uniformly convergent with respect to all discretisation parameters, specifically the mesh size and the polynomial approximation degree.

If time permits, in the final part of the talk, we introduce a novel deep learning-based algorithm to accelerate—utilising Artificial Neural Networks (ANNs)—the convergence of AMG methods. We demonstrate that ANNs can be effectively employed to predict the optimal strong connection parameter that underpins the sequence of increasingly smaller matrix problems forming the basis of the AMG algorithm, thereby maximising the corresponding AMG convergence factor. To showcase the practical capabilities of the proposed algorithm, which we term AMG-ANN, we consider a wide range of differential problems with highly heterogeneous coefficients.

Bio: Paola F. Antonietti is a Professor of Numerical Analysis at Politecnico di Milano and Head of the Laboratory for Modeling and Scientific Computing (MOX). She holds a Ph.D. in Mathematics and Statistics from the University of Pavia and has held academic positions at the University of Nottingham and Politecnico di Milano. Her research focuses on numerical methods for partial differential equations, including domain decomposition methods, with applications in scientific computing, neurodegenerative disease modeling, and seismic wave simulation. She has received several prestigious awards, including the Jacques-Louis Lions Award (ECCOMAS, 2020) and an ERC Synergy Grant (2023).