Taming Nonlinear Inverse Problems: Theory and Algorithms

虽然大规模传感和成像模式的现代发展为发现新的科学现象和提高生活质量带来了很大的前提，但以有效和准确的方式理解所感测的数据需要可扩展和有效的优化方法的变革性设计，以解决超越经典线性模型的逆问题。有一个显着的需要，以推进非线性反问题的理论，算法和应用程序，其中收集的数据表现出非线性关系的未知正在寻求后。该项目专注于驯服非线性逆问题，将与教育，推广和传播活动紧密结合，包括指导具有不同背景的研究生和本科生，开发数据科学中非线性逆问题的课程和专著，并在合适的会议地点组织特别会议。该项目的智力目标是发展解决非线性问题的理论和算法基础，逆问题，包括设计和分析具有可证明保证的有效算法，表征资源之间的基本权衡（样本，计算和内存复杂性，信噪比等）和性能（统计错误率、分辨率等），并在适用时验证真实的数据。该项目旨在利用多个测量的多样性和算法设计中数据表示的不变性，以最大限度地降低复杂性并提高性能。该项目中开发的工具和技术将进一步促进信号处理、逆问题、优化理论和机器学习领域的相互交流。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

While modern developments in large-scale sensing and imaging modalities bring great premise in discovering novel scientific phenomena and improving the quality-of-life, making sense of the sensed data in an efficient and accurate manner require transformative designs of scalable and effective optimization methods for solving inverse problems that go beyond classical linear models. There is a significant need to advance the theory, algorithms, and applications of nonlinear inverse problems, where the collected data exhibit a nonlinear relationship with respect to the unknowns being sought after. Focused on taming nonlinear inverse problems, this project will be tightly integrated with education, outreach and dissemination activities including mentoring both graduate and undergraduate students with diverse backgrounds, developing courses and monographs on nonlinear inverse problems in data science, and organizing special sessions at suitable conference venues.The intellectual goal of this project is to develop theoretical and algorithmic foundations for solving nonlinear inverse problems, including the design and analysis of efficient algorithms with provable guarantees, characterization of fundamental trade-offs between resources (sample, computational and memory complexities, signal-to-noise ratio, etc.) and performance (statistical error rates, resolution, etc.), and validations on real data whenever applicable. The project seeks to leverage the diversity of multiple measurements and the invariance of data representation in the algorithm designs to minimize complexity and improve performance. The tools and techniques developed in this project will lead to further cross fertilization among the fields of signal processing, inverse problems, optimization theory, and machine learning.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.