Collaborative Research: An Integrated Approach to Convex Optimization Algorithms

图像重建和特征提取是医学磁共振成像（MRI）和合成孔径雷达（SAR）等各种应用的重要方面。然而，这些程序涉及挑战。不同的应用程序可能在数据采集（采样）领域、所需的细节级别和感兴趣的特征的处理领域有所不同。数据采集通常是规定不足和有噪声的。采样域和/或处理域可能不适合潜在的问题。所有这些都使得问题不适定，需要各种正则化技术来将问题形式化为凸优化模型。本项目将开发一个研究此类凸优化模型的集成框架。该项目将为研究生提供通过研究参与培训的机会，并为他们在科学和工程领域的职业生涯做好准备。pi的目标是提出一种系统的方法来评估这些模型中的各种正则化技术，对模型进行严格的数值分析，并开发有效的求解模型的数值算法。具体来说，pi将解决以下技术问题：(1)为了构建对底层函数的数值鲁棒近似，必须对收集的数据施加什么约束？(2)近似收敛的速度有多快，在什么意义上收敛？(3)为保真度和正则化项开发的相应数值算法是否可行？(4)原始数据扰动的容忍度如何？该项目旨在为所有这些问题提供答案。

Image reconstruction and feature extraction have been important aspects in various applications such as medical resonance imaging (MRI) and synthetic aperture radar (SAR). However, these procedures involve challenges. Different applications may vary in data acquisition (sampling) domains, levels of detail required, and processing domains for the features of interest. The data acquisition is usually under-prescribed and noisy. The sampling domains and/or processing domains may not be well suited for the underlying question. All of these make the problems ill-posed, and various regularization techniques are necessary to study the problems by formulating them as convex optimization models. This project will develop an integrated framework of investigating such convex optimization models. The project will provide graduate students with opportunities for training through research involvement and will prepare them for careers in science and engineering. The PIs aim to propose a systematic way of evaluating various regularization techniques in such models, conduct a rigorous numerical analysis of the models, and develop efficient numerical algorithms of solving the models. Specifically, the PIs will address the following technical questions: (1) What constraints must be placed on the collected data in order to construct a numerically robust approximation to the underlying function? (2) How quickly and in what sense does the approximation converge? (3) Are the corresponding numerical algorithms developed for the fidelity and regularization terms viable? (4) How well are perturbations from the original data tolerated? The project aims to provide answers to all of these questions.