Collaborative Research: Mathematical Methods for Optimal Polynomial Recovery of High-Dimensional Systems from Sparse and Noisy Data

由科学和工程中的应用驱动的当前近似问题是以非常高的维度表示的。该项目涉及与高维近似有关的不同目标的研究，这些目标出现在大量应用中，包括中子、层析和磁共振图像重建、不确定性量化、工程和科学应用的最优控制和参数识别，以及重要的能源和材料科学应用。这项工作中使用的方法将导致对现实世界问题的解决方案进行计算机模拟的实质性改进和数学基础良好的方法。该项目还将涉及对研究生进行计算数据科学和工程方面的跨学科培训。所取得的成果将通过期刊文章、会议演讲、合作网站以及初级参与者的研究和培训活动传播。在这项工作中，我们建议开发新的数学技术来从有限数量的稀疏和噪声数据中逼近高维系统。这一努力的结果将使科学家能够了解恢复整个高维解图所需的非线性流形的多少个实现，并且具有最佳逼近保证和最小的计算成本。我们严格的数学方法包括：新的加权凸优化和迭代阈值技术，用于最优多项式恢复，通过对受限等距性质的改进估计而建立；以及先进的多指标方法，通过使用减基技术构建模型层次来降低复杂性并加速解的收敛。

Current problems in approximation that are driven by applications in science and engineering, are formulated in very high dimensions. This project involves the study of different objectives related to high-dimensional approximation, that arise in a large number of applications including neutron, tomographic and magnetic resonance image reconstruction, uncertainty quantification, optimal control and parameter identification for engineering and science applications, as well as important energy and material science applications. The approaches used in this work will result in substantially improved and mathematically well-founded methodologies for computer simulations of solutions to real-world problems. The project will also involve the interdisciplinary training of graduate students on computational data sciences and engineering. The results obtained will be disseminated through journal articles, conference talks, a collaborative website, and by the research and training activities of the junior participants. In this effort we propose to develop novel mathematical techniques for approximation of high-dimensional systems from a limited amount of sparse and noisy data. The results of this effort will enable scientists to understand what are the number realizations of a nonlinear manifold that required to recover the entire high-dimensional solution map, with optimal approximation guarantees and minimal computational cost. Our rigorous mathematical approach includes: Novel weighted convex optimization and iterative thresholding techniques for optimal polynomial recovery, established via an improved estimate of the restricted isometry property; and Advanced multi-index methods that alleviate complexity and accelerate convergence of solutions by constructing model hierarchies with the use of reduced-basis techniques.