Advances in Scalable Iterative Solvers: Multilevel, Nonlinearly Preconditioned, and Parallel-in-Time

我们生活在一个科学、技术和商业进步由前所未有的计算能力和大量数字数据驱动的时代。该提案的目标是追求在两个主要应用领域推动这一进展的计算方法的实质性进展。 首先，我们将开发新的流体和等离子体流动的模拟方法，旨在通过寻找新的方法来优化利用超大型并行计算机的能力，并通过开发新的有效方法来量化计算模型中的不确定性，以促进科学发现和工程设计，追求真正的预测计算科学。世界上最大的并行计算机现在有数百万个并行处理器内核，我们将通过考虑不仅在空间上而且在时间上使用并行计算的模拟方法来利用这种并行性。通过考虑由计算模型中的多个水平实现的加速采样技术，将量化未知参数，并以新的有效方式推断未知参数。其次，我们将为数据分析应用开发新的高效优化方法，旨在为数据革命带来前所未有的新定量见解的分析技术提供实质性的速度提升。用于电影推荐或神经网络训练等问题的现有优化技术需要多次迭代才能实现准确的预测。拟议的研究将研究新的方法来加速这些数据分析技术的收敛，通过应用最佳外推迭代更新的方法，结合随机化技术，找到一种方法来解决可用数据的巨大性。拟议研究的结果有望为加拿大带来巨大利益：对具有大型数据集的计算密集型问题的算法探索是我们快速发展的信息经济的重要组成部分。本提案中开发的计算方法将直接应用于加拿大的经济和社会，数据知情的科学和工程模拟以及数据分析的优化。此外，拟议的工作将培训具有关键算法和大规模计算技能的高素质人员，这些技能对加拿大的知识经济至关重要。

We live in an era where progress in science, technology and business is driven by unprecedented availability of computational power and large amounts of digital data. The goal of this proposal is to pursue substantial advances in the computational methods that drive this progress in two principal application areas. First, we will develop novel simulation methods for fluid and plasma flows that aim to advance scientific discovery and engineering design by finding new ways to optimally exploit the power of very large parallel computers, and by developing new efficient ways to quantify uncertainties in computational models, in the pursuit of truly predictive computational science. The world's largest parallel computers now have millions of parallel processor cores, and we will exploit this parallelism by considering simulation methods that use parallel computations not only in space but also in time. Uncertainties will be quantified and unknown parameters will be inferred in new and efficient manners by considering accelerated sampling techniques enabled by multiple levels in the computational models. Second, we will develop new efficient optimization methods for applications in data analysis, aiming to provide substantial speed increases for the analysis techniques that generate unprecedented new quantitative insights enabled by the data revolution. Existing optimization techniques used for problems such as movie recommendation or neural network training require many iterations to reach accurate predictions. The proposed research will investigate novel ways to accelerate the convergence of these data analysis techniques, by applying methods that optimally extrapolate iterative updates, combined with randomized techniques that find a way around the vastness of the available data. The results of the proposed research promise to be of great benefit to Canada: algorithmic exploration of compute-intensive problems with large data sets is a crucial building block in our rapidly evolving information economy. The computational methods developed in this proposal will have direct applications in Canada's economy and society, in data-informed science and engineering simulations, and optimization for data analysis. Moreover, the proposed work will train highly qualified personnel with crucial algorithmic and large-scale computing skills that are essential for Canada's knowledge economy.