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

当前的近似值问题是由科学和工程应用驱动的，通常是在非常高的维度上表述的。该项目涉及研究与高维近似相关的各种问题，这些问题在大量应用中出现，包括中子、层析成像和磁共振图像重建、不确定度量化、最优控制和参数识别，以及能源和材料科学的重要领域。这项工作中使用的方法将大大改进和数学上有充分根据的方法，用于计算机模拟现实世界问题的解决方案。该项目将围绕计算数据科学与工程研究生的跨学科培训展开。获得的结果将通过期刊文章、会议演讲和合作网站进行传播。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噪声数据中近似高维系统。这一努力的结果将使科学家们能够理解需要恢复整个高维解图的非线性流形的数字实现是什么，具有最佳近似保证和最小的计算成本。我们严谨的数学方法包括：新颖的加权凸优化和迭代阈值技术，通过改进的受限等距性质估计来建立最优多项式恢复；先进的多指标方法，通过使用约基技术构建模型层次结构，降低了解的复杂性，加快了解的收敛速度。

Current problems in approximation that are driven by applications in science and engineering, are typically formulated in very high dimensions. This project involves the study of different problems 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, as well as in important areas of energy and material science. 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 be centered around the interdisciplinary training of graduate students in computational data science and engineering. The results obtained will be disseminated through journal articles, conference talks, and a collaborative website. 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.