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巨大的前提，但要以高效和准确的方式理解传感数据，需要对可扩展和有效的优化方法进行变革性设计，以解决超越经典线性模型的反问题。非常有必要推进非线性反问题的理论、算法和应用，其中收集的数据显示出相对于所追求的未知数的非线性关系。该项目的重点是驯服非线性逆问题，该项目将与教育、推广和传播活动紧密结合，包括指导不同背景的研究生和本科生，开发关于数据科学的非线性逆问题的课程和专著，以及在合适的会议场所组织专题会议。该项目的智力目标是发展解决非线性逆问题的理论和算法基础，包括设计和分析具有可证明保证的高效算法，表征资源(样本、计算和存储器的复杂性、信噪比等)之间的基本权衡。和性能(统计错误率、分辨率等)，以及对实际数据的验证(只要适用)。该项目寻求在算法设计中利用多个测量的多样性和数据表示的不变性，以最大限度地减少复杂性并提高性能。在这个项目中开发的工具和技术将导致信号处理、反问题、最优化理论和机器学习领域的进一步交叉发展。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.