Algebraic Domain Decomposition Solvers for Large-Scale Problems Using Graph Techniques
Speaker: Alexander Heinlein
Abstract: Domain decomposition methods (DDMs) solve science and engineering problems by decomposing them into smaller subproblems defined on an overlapping or non-overlapping decomposition of the computational domain. Overlapping Schwarz DDMs are particularly effective for complex problems. However, achieving fast convergence in large-scale problems with large numbers of subdomains requires incorporating additional coarse solver components.
The FROSch (Fast and Robust Overlapping Schwarz) package, part of the Trilinos library, implements algebraic domain decomposition methods (DDMs). These methods require only the fully assembled system matrix and minimal geometric information, making them versatile for various applications. The construction of FROSch solvers begins with a non-overlapping domain decomposition obtained via graph partitioning, expanding into overlapping subdomains based on the system matrix’s sparsity pattern. The coarse scale components are constructed from local solutions of the non-overlapping domain decomposition.
This talk covers FROSch’s features and recent updates, including the use of inexact local solvers on GPUs and CPUs for faster computation, extending the framework to multiscale and multiphysics problems, and employing machine learning techniques, such as graph neural networks, to improve coarse grid construction. Performance is evaluated on various problems, from simple models to real-world applications.
Accelerating Simulations of Materials using Graph Neural Networks with Embedded Constitutive Models
Speaker: Joep Storm
Abstract: Simulating the mechanical response of advanced materials can be done more accurately using concurrent multiscale models than with single-scale finite element (FE) simulations. However, the computational costs stand in the way of the practical application of this approach. In this talk, I will present how a Graph Neural Network (GNN) can replace the most expensive part of these simulations, namely solving the microscopic FE models at every macroscopic integration point. By predicting all microscopic quantities, we keep the multiscale nature of the problem and can retain the microscopic constitutive material model. The GNN can extrapolate to large meshes unseen during training and scales favorably in computation time compared to an FE analysis, enabling the acceleration of multiscale simulations.
Modelling Floods with Graph Neural Networks
Speaker: Roberto Bentivoglio