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

目前的问题，在科学和工程中的应用程序驱动的近似，制定在非常高的维度。该项目涉及与高维近似相关的不同目标的研究，这些目标出现在大量的应用中，包括中子，断层扫描和磁共振图像重建，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。 在这项工作中使用的方法将导致大幅改善和数学上有充分依据的方法，计算机模拟解决方案，以现实世界的问题。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、会议谈话、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，从有限数量的稀疏和嘈杂的数据近似高维系统。 这项工作的结果将使科学家能够理解恢复整个高维解映射所需的非线性流形的数量实现，并具有最佳近似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的最佳多项式恢复，建立通过改进的估计的限制等距属性;和先进的多指标方法，减轻复杂性和加速收敛的解决方案，通过构建模型层次结构与使用减少的基础技术。

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.