Numerical Methods for High Dimensional Partial Differential Equations

本世纪最大的科学挑战之一是准确描述复杂的过程，如气候变化、污染物流动、基因组学，甚至社交媒体和金融。人们理解这些过程的主要障碍是它们取决于许多参数或变量。这不仅使它们的数学描述复杂化，而且还阻碍了现代计算工具的使用。这项研究提出了新的数学思想的基础上稀疏性和模型简化，以确定各种参数的重要性，并推导出更简单的模型，仍然准确地描述了潜在的过程。这反过来又导致更准确和成本更低的计算模型，可以实现与今天的计算资源。它还研究如何同化观测数据以改进模式，甚至建议最有效的新数据站点。针对此类高维问题的数值方法的发展面临所谓的维数灾难，即传统方法注定要失败。这导致了分析、计算机科学和数值计算中无数新技术的发展，这些技术基于稀疏性、压缩感知、变量缩减、各向异性平滑、稀疏网格、差异理论、散列、张量近似、简化建模和流形学习等思想。这个项目描述了家庭的参数函数被数值恢复为一个高维流形，然后寻求开发技术，查询流形，导致低维近似的流形描述的减少基地或高维多项式展开。它还试图描述如何高维观测数据的状态的流形可以融合的简化模型，导致更准确的描述的状态。该程序的成功完成将导致数值方法，可以实现在线执行快速查询的流形的任何规定的参数集。

One of the great scientific challenges of this century is to describe accurately complex processes such as climate change, contaminant flow, genomics, and even social media and finance. The main obstacle to one's understanding of these processes is that they depend on many parameters or variables. This not only complicates their mathematical description but also inhibits the use of modern computational tools for their accurate prediction. This research puts forward new mathematical ideas based on sparsity and model reduction to determine the importance of the various parameters and derive simpler models that still accurately describe the underlying process. This in turn leads to more accurate and less costly computational models that can be implemented with today's computational resources. It also studies how to assimilate observational data to improve models and even suggests the most effective new data sites.The development of numerical methods for such high-dimensional problems faces the so-called curse of dimensionality, which says that traditional methods are doomed to fail. This has led to the development of myriad new techniques in analysis, computer science, and numerical computation based on ideas such as sparsity, compressed sensing, variable reduction, anisotropic smoothness, sparse grids, discrepancy theory, hashing, tensor approximation, reduced modeling, and manifold learning. This project describes the family of parametric functions to be numerically recovered as a high-dimensional manifold and then seeks to develop techniques for querying the manifold that lead to low-dimensional approximations of the manifold described either by reduced bases or by high-dimensional polynomial expansions. It also seeks to describe how high-dimensional observational data on a state of the manifold can be fused with the reduced model leading to an even more accurate description of the state. The successful completion of this program will lead to numerical methods which can be implemented on-line for executing a fast query of the manifold for any prescribed set of parameters.