Collaborative Research: CDS&E-MSS: Local Approximation for Large Scale Spatial Modeling

计算机模拟作为研究应用科学中复杂动力学的一种手段正在发展。 它曾经是工业工程和计算物理学的专用工具，但在生物学、化学和经济学中越来越普遍。 平衡动力学是合适的，可爱的方程组可以用手求解的日子已经一去不复返了。 由于现代超级计算，计算机实验变得越来越多样化，越来越复杂，规模也越来越大。 我们需要一个新的先锋建模工具，可以科普现代计算机实验的需要，特别是它们不断增加的规模（大数据），以及随着模型变得更加复杂而迅速演变和完善的性质，超级计算环境接近exa规模。 这项资助的研究目标是扩展和应用一种新的灵活和快速的响应面方法，即所谓的局部近似高斯过程（laGP）。 我们的激励应用程序主要来自计算机实验和不确定性量化中的问题，并从地质统计学和机器学习的相关文献中借用了一些想法，并将成为这些文献的重要扩展。 高斯过程（GP）模型在空间建模环境中很受欢迎，如地质统计学或计算机实验，其中响应面相当光滑，但几乎没有其他假设。GP模型提供了准确的预测，但越来越多地施加计算瓶颈：大型密集矩阵分解阻碍努力跟上现代趋势的数据采集。为了快速近似，一场混乱正在上演。两个常见的主题是稀疏性，允许快速矩阵分解，以及超级计算，允许分布式计算。但这些进展都已饱和。快速扩展的移动终端网络、高分辨率卫星图像（和GPS）以及超级计算机模拟生成的数据规模不断增加。这项资助的研究集中在本地近似GP（laGP）模型上，作为使强大的GP空间建模框架能够解决现代大数据问题的一种手段。最初的实现显示出希望，将数据大小能力扩展了几个数量级。然而，还有很多工作要做，以确保laGP方法可以取代传统的GPS在不同的空间建模环境。在这里，我们提出了一些方法上的改进，许多涉及捷径，有证据表明，对LAGP性能的影响最小。我们的动机是两个大数据计算机模型仿真应用：一个涉及卫星定位，另一个涉及太阳能发电。然而，我们注意到，为了使我们的努力产生影响，必须始终考虑到更广泛的空间建模背景。

Computer simulation is growing as a means of studying complex dynamics in applied science. Once a tool exclusive to industrial engineering and computational physics, it is increasingly common in biology, chemistry, and economics. Gone are the days when equilibrium dynamics are appropriate and cute systems of equations can be solved by hand. Computer experiments are becoming more diverse, they are becoming more complex and they are growing in size thanks to modern supercomputing. We need a new vanguard of modeling tools that can cope with the needs of modern computer experiments, particularly their increasing size (big data) and rapidly evolving and refining nature as models become more sophisticated, and supercomputing environments approach the exa-scale. This funded research targets extensions and applications of a new breed of flexible and fast response surface methods, the so-called local approximate Gaussian process (laGP). Our motivating applications come primarily from problems in computer experiments and uncertainty quantification, and ideas are borrowed from -- and will represent an important extension to -- the related literatures of geo-statistics and machine learning. The over-arching goal is a modernization of the response surface and surrogate modeling toolkit to better serve future applications across applied science.Gaussian process (GP) models are popular in spatial modeling contexts, like geostatistics or computer experiments, where response surfaces are reasonably smooth but little else can be assumed. GP models provide accurate predictors, but increasingly impose computational bottlenecks: large dense matrix decompositions impede efforts to keep pace with modern trends in data acquisition. A scramble is on for fast approximations. Two common themes are sparsity, allowing fast matrix decompositions, and supercomputing, allowing distributed calculation. But these inroads are at capacity. Rapidly expanding mobile device networks, high-resolution satellite imagery (and GPS), and supercomputer simulation generate data of ever-increasing size. This funded research centers on local approximate GP (laGP) models as a means of enabling the powerful GP spatial modeling framework to address modern big data problems. Initial implementations show promise, expanding data size capabilities by several orders of magnitude. However much work remains to ensure that laGP methods can supplant conventional GPs in diverse spatial modeling contexts. Here we propose several methodological enhancements, many involving shortcuts that have provably minimal impact on laGP performance. We are motivated by two big data computer model emulation applications: one involving satellite positioning and another on solar power generation. Yet we are mindful that for our efforts to have impact, the wider spatial modeling context must always be kept in view.