Numerical optimization for large-scale experimental design of ill-posed inverse problems

反问题在计算机视觉、物理学和医学成像等领域中扮演着重要的角色。拟议的工作重点是不适定的反问题的实验设计;一个领域，还没有得到足够的重视，在反问题的社区，重点通常是反问题的分析数据。显然，这已经对可能的解决方案的质量施加了限制。 另一方面，资助工作的目标是研究一个重要的数据获取前问题：在物理限制和可用资源的情况下，应该如何进行实验以获得最佳数据？这个问题的解决需要从数值优化，统计和反问题理论的技术。特别地，实验设计问题可以被转换为由两个嵌套优化问题组成的双层优化问题。本文的工作是研究设计准则和新的数值算法，以解决由此产生的双层优化问题。 不适定逆问题从地球物理和医学成像到生产更好的视觉系统，任何实际实验或仪器的设计都会出现此类问题。这项工作将制定新的标准，设计和开发新的算法，使其数值实现。研究结果将应用于电磁成像，这是一个在生物物理学和医学物理学中经常使用的领域。它将导致更好的实验，产生更好的图像，因此，将有助于地球科学家和医生的决策。

Inverse problems play a key role in a variety of fields such as computer vision, geophysics and medical imaging. The proposed work focuses on experimental design of ill-posed inverse problems; a field that has not received sufficient attention in the inverse problems community where the focus is usually on the analysis of the inverse problem given the data. Obviously, this already imposes restrictions on the quality of the possible solutions. On the other hand, the objective of the funded work is the study of an important pre-data acquisition question: How should the experiment be conducted to obtain optimal data given the physical constraints and available resources? Solutions to this question require techniques from numerical optimization, statistics and inverse problem theory. In particular, the problem of experimental design can be cast as a bilevel optimization problem that consists of two nested optimization problems. The proposed work is a study of design criteria and new numerical algorithms for the solution of the bilevel optimization problems that arise from them.This work will address the fundamental question of experimental design of ill-posed inverse problems. Such problems arise in the design of any practical experiment or instrumentation from geophysical and medical imaging to the production of better vision systems. This work will develop new criteria for the design and development of new algorithms that will enable its numerical implementation. The results of the research will be applied to electromagnetic imaging, a field that is routinely used in geophysics and medical physics. It will lead to better experiments that yield better images and as such, will assist in the decision making of geoscientists and and physicians.