Structured compressed sensing algorithms: design, analysis and applications

科学和工程中的许多问题都需要从一组测量数据中重建对象，例如图像、信号或高维函数。由于时间，成本或其他限制，人们往往受到可以收集的数据量的严重限制，这严重影响了准确恢复未知对象的能力。该研究计划涉及开发，分析和应用新的算法解决这个问题，基于压缩传感（CS）的理论和技术。相关应用的例子包括医学成像、显微镜、物理系统中的不确定性量化、机器学习和偏微分方程的数值解。其总体目标是为这些应用引入新的，计算效率高的数值优化技术，这些应用具有更好的精度和更低的采集时间和成本。 具体目标 1)设计和实现新一代的基于CS的成像算法，在采样和恢复过程中加入额外的结构。通过利用这种结构，这项工作预计将带来对当前最先进算法的实质性改进，在关键成像技术，如MRI，X射线CT和电子和荧光显微镜方面产生切实的好处。 2)开发和研究新的基于CS的高维近似方法，利用结构化光滑稀疏先验和随机采样技术来提高精度。随着数据收集变得越来越容易和广泛，科学和工程领域迫切需要通过近似高维函数来理解日益复杂的现象。这项工作的成果将改进技术，为这个问题，带来好处的重要的实际任务，如不确定性量化的物理（如生物，机械或流体）系统。 3)研究科学计算中基于采样的算法的稳定性和准确性的极限，并设计基于保角映射的新计算方法以达到这些极限。基于采样的算法在科学计算中有多种用途，包括曲面重构、偏微分方程数值方法和数值软件。这项工作将通过更好地理解任何算法可实现的理论极限来增强知识，其好处将是基于这些极限的一组改进的方法。 总的来说，该研究计划的重点是开发新算法，以解决具有挑战性的面向数据的问题。它旨在为国家需要的科学、工程和医学关键领域的一系列应用带来好处。这项研究还将有助于通过对HQP的培训，满足学术界和工业界对这些领域技能的迫切需求。

Many problems in science and engineering require the reconstruction of an object - an image, signal or high-dimensional function, for example - from a collection of measurements. Due to time, cost or other constraints, one is often severely limited by the amount of data that can be collected, which significantly affects ones ability to recover the unknown object accurately. This research program involves the development, analysis and application of new algorithms for this problem, based on the theory and techniques of compressed sensing (CS). Examples of relevant applications include medical imaging, microscopy, uncertainty quantification in physical systems, machine learning and the numerical solution of PDEs. Its overarching objective is to introduce new, computationally-efficient numerical optimization techniques for such applications that possess both better accuracy and lower acquisition time and cost. Specific objectives 1) To design and implement a new generation of CS-based algorithms for imaging that incorporate additional structure in both the sampling and recovery process. By leveraging such structure, this work is expected to bring substantial improvements over current state-of-the-art algorithms, yielding tangible benefits in key imaging technologies such as MRI, X-ray CT and electron and fluorescence microscopy. 2) To develop and study new CS-based methods for high-dimensional approximation that exploit structured smoothness-sparsity priors and randomized sampling techniques to enhance accuracy. As data collection becomes easier and more widespread, there is a pressing need in science and engineering to understand increasingly complex phenomena by approximating high-dimensional functions. The outcomes of this work will be improved techniques for this problem, bringing benefits to important practical tasks such as uncertainty quantification in physical (e.g. biological, mechanical or fluid) systems. 3) To investigate the limits of stability and accuracy for sampling-based algorithms in scientific computing, and to design new computational methods based on conformal mappings that attain such limits. Sampling-based algorithms have a variety of uses in scientific computing, including surface reconstruction, numerical methods for PDEs and numerical software. This work will enhance knowledge through a better understanding of the theoretical limits achievable by any algorithm for this problem, and its benefit will be an improved set of methods based on such limits. Overall, the focus of this research program is the development of new algorithms for challenging data-oriented problems. It aims to bring benefits in a range of applications in key areas of national need in science, engineering and medicine. This research will also contribute to the pressing need for skills in these areas, both in academia and industry, through the training of HQP.