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

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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.