Abstract

In this work, we propose an enhanced implementation of balancing Neumann-Neumann (BNN) preconditioning together with a detailed numerical comparison against the balancing domain decomposition by constraints (BDDC) preconditioner. As model problems, we consider the Poisson and linear elasticity problems. On one hand, we propose a novel way to deal with singular matrices and pseudo-inverses appearing in local solvers. It is based on a kernel identication strategy that allows us to eciently compute the action of the pseudo-inverse via local indenite solvers. We further show how, identifying a minimum set of degrees of freedom to be xed, an equivalent denite system can be solved instead, even in the elastic case. On the other hand, we propose a simple modication of the preconditioned conjugate gradient (PCG) algorithm that reduces the number of Dirichlet solvers to only one per iteration, leading to similar computational cost as additive methods. After these improvements of the BNN PCG algorithm, we compare its performance against that of the BDDC preconditioners on a pair of large-scale distributed-memory platforms. The enhanced BNN method is a competitive preconditioner for three-dimensional Poisson and elasticity problems, and outperforms the BDDC method in many cases.

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In this work, we analyze the scalability of inexact two-level balancing domain decomposition by constraints (BDDC) preconditioners for Krylov subspace iterative solvers, when using a highly scalable asynchronous parallel implementation where fine and coarse correction computations are overlapped in time. This way, the coarse-grid problem can be fully overlapped by fine-grid computations (which are embarrassingly parallel) in a wide range of cases. Further, we consider inexact solvers to reduce the computational cost/complexity and memory consumption of coarse and local problems and boost the scalability of the solver. Out of our numerical experimentation, we conclude that the BDDC preconditioner is quite insensitive to inexact solvers. In particular, one cycle of algebraic multigrid (AMG) is enough to attain algorithmic scalability. Further, the clear reduction of computing time and memory requirements of inexact solvers compared to sparse direct ones makes possible to scale far beyond state-of-the-art BDDC implementations. Excellent weak scalability results have been obtained with the proposed inexact/overlapped implementation of the two-level BDDC preconditioner, up to 93,312 cores and 20 billion unknowns on JUQUEEN. Further, we have also applied the proposed setting to unstructured meshes and partitions for the pressure Poisson solver in the backward-facing step benchmark domain.

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As data science becomes increasingly mainstream, there will be an ever-growing demand for data science tools that are more accessible, flexible, and scalable. In response to this demand, automated machine learning (AutoML) researchers have begun building systems that automate the process of designing and optimizing machine learning pipelines. In this chapter we present TPOT v0.3, an open source genetic programming-based AutoML system that optimizes a series of feature preprocessors and machine learning models with the goal of maximizing classification accuracy on a supervised classification task. We benchmark TPOT on a series of 150 supervised classification tasks and find that it significantly outperforms a basic machine learning analysis in 21 of them, while experiencing minimal degradation in accuracy on 4 of the benchmarks—all without any domain knowledge nor human input. As such, genetic programming-based AutoML systems show considerable promise in the AutoML domain. Document type: Part of book or chapter of book

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