A new approach to finite element simulations of general relativity

University of Minnesota Ph.D. dissertation. June 2015. Major: Mathematics. Advisor: Douglas Arnold. 1 computer file (PDF); vi, 105 pages.In order to study gravitational waves, we introduce a new approach to finite element simulation of general relativity. This approach is based on approximating the Weyl curvature directly through new stable mixed finite elements for the Einstein-Bianchi system. We design and analyze these novel finite elements by adapting the recently developed Finite Element Exterior Calculus (FEEC) framework to abstract Hodge wave equations. This framework enables us to borrow key ideas from Reissner-Mindlin plate bending and elasticity with weakly imposed symmetries to maintain stability of the method. The stability of a discretization often relies on deep connections between fundamental branches of mathematics: the FEEC mimics these connections for the numerical method to achieve similar stability to that of the original equations. The recent development of FEEC has had a transformative impact on electromagnetism and related computational problems, and we are expanding it to general relativity

GEANN: Scalable Graph Augmentations for Multi-Horizon Time Series Forecasting

Encoder-decoder deep neural networks have been increasingly studied for multi-horizon time series forecasting, especially in real-world applications. However, to forecast accurately, these sophisticated models typically rely on a large number of time series examples with substantial history. A rapidly growing topic of interest is forecasting time series which lack sufficient historical data -- often referred to as the ``cold start'' problem. In this paper, we introduce a novel yet simple method to address this problem by leveraging graph neural networks (GNNs) as a data augmentation for enhancing the encoder used by such forecasters. These GNN-based features can capture complex inter-series relationships, and their generation process can be optimized end-to-end with the forecasting task. We show that our architecture can use either data-driven or domain knowledge-defined graphs, scaling to incorporate information from multiple very large graphs with millions of nodes. In our target application of demand forecasting for a large e-commerce retailer, we demonstrate on both a small dataset of 100K products and a large dataset with over 2 million products that our method improves overall performance over competitive baseline models. More importantly, we show that it brings substantially more gains to ``cold start'' products such as those newly launched or recently out-of-stock