Inducing and Exploiting Grid Structures for Fast, Adaptive, and Accurate Estimation

计算机模拟现在涵盖了科学研究的大部分领域。在这些模拟上进行的计算机实验是利用有限的计算能力的有效方法。这一研究项目将开发工具，以克服现有的计算限制，以解决在计算机模型发挥关键作用的各种应用中出现的日益复杂的问题。使用高性能计算和智能数据收集方案进行高维大规模实验具有巨大的潜力，这些方案可以产生易于处理的计算分析和精度，在大范围估计问题上比现有方法高出几个数量级。这种方法最吸引人的特点是，它允许并行但灵活的实验和计算合理的推理，并有理论保证。这项研究将促进计算机模型从高性能计算环境的独家用户向广泛的科学家和工程师转移。该项目的总体理念将是对底层网格结构进行实验，与基本网格相比，这些结构明显更灵活和适应性更强。实现快速计算的关键是利用广义网格结构的高斯过程的严格表示结果。利用这些表示法的算法都是快速的，这意味着算法的计算成本很小，而且准确，这意味着次最优性的唯一来源来自机器误差。由于结果将是准确的，这些算法将比现有的依赖于近似的快速方法精确得多。这项研究的重点是为高维估计设计维度和空间自适应，以服务于不确定性量化、优化和校准等下游计算。虽然对这个问题的兴趣主要来自计算机实验，但该方法足够普遍，足以在一般机器学习中提供额外的影响，特别是在使用高斯过程预测方面。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Computer simulation now encompasses most areas of scientific research. Computer experiments conducted on these simulations are an effective way to leverage limited computation capacity. This research project will develop tools to overcome existing computational limits to solve increasingly complex problems arising in various applications where computer models play a critical role. There is a vast potential to enable high-dimensional, large-scale experimentation using high-performance computing with smart data collection schemes that yield tractable computational analysis and accuracy that is orders of magnitude better than existing approaches on a large swath of estimation problems. The most appealing features of this approach are that it enables parallel but flexible experimentation and computationally reasonable inference with theoretical guarantees. This research will facilitate the transfer of computer models from exclusive users of high-performance computing environments to a broad array of scientists and engineers. The general philosophy of the project will be to use experiments with underlying grid structures that are significantly more flexible and adaptive compared to basic grids. The key to enabling fast computation are rigorous representation results for Gaussian processes that exploit the generalized grid structures. The algorithms that exploit these representations are both fast, meaning the algorithms have small computational cost, and exact, meaning the only source of sub-optimality comes from machine error. Because the results will be exact, these algorithms will be considerably more accurate than existing fast methods which rely on approximations. The research focuses on designing dimensional and spatial adaptivity for high-dimensional estimation to service downstream calculations like uncertainty quantification, optimization and calibration. While the interest in this problem comes primarily from computer experiments, the approach is general enough to provide additional impacts in general machine learning, specifically in the use of Gaussian process prediction.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.