Collaborative Research: Trust-Search Methods for Inverse Problems in Imaging

该奖项的研究目标是开发和实现用于大规模数据生成优化问题的一阶信任搜索方法。 数据生成问题出现在诸如信号和图像处理的应用中。这些问题特别难以解决，因为数据通常是高维的，并且是有噪声的、不完整的和/或不精确的。 本研究将发展一阶拟牛顿信赖搜寻法，以解决大量资料产生的问题。 所使用的方法是信任搜索方法，这是杂交的最基本类型的方法无约束优化：信任区域的方法和线搜索方法。信任搜索方法试图将线搜索策略与信任域理论相结合，以获得更鲁棒的方法。如果成功，本研究的结果将有助于科学家和工程师解决涉及大量损坏数据的优化问题。 在当今世界，科学数据比以往任何时候都更加丰富;此外，项目已经在以更快的速度产生更多的数据。 为了跟上步伐，新兴的“大数据”领域需要复杂、快速、健壮和大规模的数值算法。 这项研究将使用线性代数和优化理论来开发用于处理和分析非常大的数据集的软件。 特别是，这项研究的结果将有助于解决图像处理应用中的重要问题，如医学成像，低光视频监控和夜间生态活动监测，其中生成的数据不仅非常大，而且非常嘈杂。 这些算法将在科学界内外传播使用。 通过这项研究，研究生将接受科学研究和编程方面的培训，并高度鼓励来自代表性不足背景的学生参与。

The research objective of this award is to develop and implement first-order trust-search methods for use in large-scale data-generated optimization problems. Data-generated problems arise in applications such as signal and image processing. These problems are especially difficult to solve since the data are often high dimensional and are noisy, incomplete, and/or inexact. This research will develop first-order quasi-Newton trust-search methods for solving large data-generated problems. The methods to be used are trust-search methods, which are hybridizations of the most fundamental types of methods for unconstrained optimization: trust-region methods and line-search methods. Trust-search methods seek to implement line-search strategies in combination with trust-region theoretics to obtain more robust methods.If successful, the results of this research will help scientists and engineers solve optimization problems involving large volumes of corrupted data. In today's world, scientific data are more abundant than ever before; moreover, projects are already underway to produce even more data at a faster rate. To keep pace, the emerging field of "big data" requires sophisticated, fast, robust, and large-scale numerical algorithms. This research will use linear algebra and optimization theory to develop software for processing and analyzing very large data sets. In particular, the results of this research will help solve important problems in image processing applications such as medical imaging, low-light video surveillance, and nocturnal ecological activity monitoring, where the generated data are not only very large but are very noisy. The algorithms will be disseminated publically for use within and outside the scientific community. Graduate students will be trained in scientific research and programming through this research, and the participation of students from under-represented backgrounds will be highly encouraged.