European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
15:45   MS40: Machine Learning for Numerical Simulation
Chair: Stefan Turek
25 mins
Machine Learning in Adaptive FETI-DP - Reducing the Effort in Sampling
Alexander Heinlein, Axel Klawonn, Martin Lanser, Janine Weber
Abstract: The convergence rate of classic domain decomposition methods in general deteriorates severely for large discontinuities in the coefficient functions of the considered partial differential equation. To retain the robustness for such highly heterogeneous problems, the coarse space can be enriched by additional coarse basis functions. These can be obtained by solving local generalized eigenvalue problems on subdomain edges. In order to reduce the number of eigenvalue problems and thus the computational cost, we use a neural network to predict the geometric location of critical edges, i.e., edges where the eigenvalue problem is indispensable. As input data for the neural network, we use function evaluations of the coefficient function within the two subdomains adjacent to an edge. In the present article, we examine the effect of computing the input data only in a neighborhood of the edge, i.e., on slabs next to the edge. We show numerical results for both the training data as well as for a concrete test problem in form of a microsection subsection for linear elasticity problems. We observe that computing the sampling points only in one half or one quarter of each subdomain still provides robust algorithms.
25 mins
Basic Machine Learning Approaches for the Acceleration of PDE Simulations and Realization in the FEAT3 Software
Hannes Ruelmann, Markus Geveler, Stefan Turek, Peter Zajac, Dirk Ribbrock
Abstract: In this paper we present a holistic software approach based on the FEAT3 software for solving multidimensional PDEs with the Finite Element Method that is built for a maximum of performance, scalability, maintainability and extensibilty. We introduce basic paradigms how modern computational hardware architectures such as GPUs are exploited in a numerically scalable fashion. We show, how the framework is extended to make even the most recent advances on the hardware market accessible to the framework, exemplified by the ubiquitous trend to customize chips for Machine Learning. We can demonstrate that for a numerically challenging model problem, artificial neural networks can be used while preserving a classical simulation solution pipeline through the incorporation of a neural network preconditioner in the linear solver.
25 mins
Optimizing Geometric Multigrid with Evolutionary Computation
Jonas Schmitt
Abstract: In many cases the construction of an efficient multigrid solver is a difficult task which requires a high degree of expertise in numerical mathematics. Here we present how evolutionary computation, a subfield of artificial intelligence, can be used to optimize geometric multigrid methods in a fully automatic way. A multigrid solver is represented in form of mathematical expressions which we generate based on a tailored grammar. The quality of each solver is evaluated in terms of convergence and compute performance using automated Local Fourier Analysis (LFA) and performance modeling, respectively, which forms the basis for a multi-objective optimization with strongly typed genetic programming. To target concrete applications scalable implementations of an evolved solver can be automatically generated with the ExaStencils code generation framework. We demonstrate our approach by constructing multigrid solvers that outperform well-known methods in a number of test cases.
25 mins
Logistic Regression in Potential Modeling
Samuel Kost, Oliver Rheinbach, Helmut Schaeben
Abstract: Logistic regression for the targeting of ressources (potential modeling) is described. In comparison with neural networks, regression methods can have the advantage of better interpretability of the results and a better understanding of the problem, since explicit models are used. A search strategy is also described to find suitable explicit models.