OP: Collaborative Research: Novel Feature-Based, Randomized Methods for Large-Scale Inversion

从外部收集的数据形成空间区域的图像的期望出现在从检测和表征体内癌症到量化水、油或地下污染物的分布以及在拥挤场所中及时准确识别爆炸物的应用中。在这些问题中，与信号传播和传感相关的物理学为将原始数据转换为有用信息带来了巨大的计算挑战。该项目的研究小组旨在通过提供统计反演理论、数值反演方法、模拟模型和混合成像模型的改进，开发大大降低真实的时间成像成本的计算方法。 该项目的主要目标将在医疗断层扫描、环境修复和机场安全成像中的成像应用上进行测试。这些技术形成了解决与跨越宽范围的空间和时间尺度的光信号的反转相关联的类似问题的基础。作为该项目的一部分，将开发一个模块课程，在研究生一级教授这些新方法。课程材料将通过互联网提供。这个合作团队解决的大规模成像或逆问题需要最小化参数依赖函数，该函数表示候选图像和实际测量数据的预测测量的失配。潜在的大量参数必须在不断增加的大量测量中被最小化，而同时一些未知的数据集可能是冗余的。 然而，详细的图像并不总是需要解决相关的实际问题和决策。将开发一种计算技术的组合，使大规模的参数依赖的最小化计算可行。 此外，将开发用于推断关键图像特征的新的有效方法，从而避免对图像的完全重建的需要。该研究建立在最近的方法，利用随机化计算的解决方案的准确估计，大大降低了计算成本，并有效地构建更小的，近似的，降阶的数值模型，是准确的相关参数集，从而降低成本的传感物理的全面模拟。将开发用于推断指导图像解释和决策的关键图像特征的概率方法。与这种方法相关的数学要求这些方法利用本项目正在开发的其他新工具。

The desire to form an image of a region of space from externally collected data arises in applications ranging from detecting and characterizing cancers in the body, to quantifying the distribution of water, oil, or subsurface pollutants, and to the timely accurate identification of explosives in crowded venues. The physics associated with signal propagation and sensing in these problems creates substantial computational challenges for transforming raw data into useful information. The research team in this project aims to develop computational methods that greatly reduce the cost of real time imaging by providing improvements in statistical inverse theory, numerical inversion methods, simulation models, and hybrid imaging models. The main thrusts of the project will be tested on imaging applications in medical tomography, environmental remediation, and airport security imaging. The techniques form the basis for addressing analogous problems associated with inversion of optical signals across a wide range of spatial and temporal scales. As part of the project, a modular course will be developed to teach these new methods at the graduate level. The course materials will be made available over the internet.The large-scale imaging, or inverse, problems addressed by this collaborative team require the minimization of a parameter-dependent function that expresses the misfit of predicted measurements for a candidate image and actual measurement data. The potentially large number of parameters must be minimized over an ever-increasing huge number of measurements, while concurrently some unknown set of the data may be redundant. Detailed images, however, are not always needed for addressing relevant, practical questions and decision making. A combination of computational techniques will be developed to make large-scale parameter-dependent minimization computationally feasible. Furthermore, novel efficient approaches for inferring critical image features will be developed, obviating need for complete reconstruction of an image. The research builds on recent methods that exploit randomization to compute accurate estimates of solutions at greatly reduced computational cost, and on the efficient construction of smaller, approximate, reduced order numerical models that are accurate for relevant sets of parameters, and thus reduce the cost of full simulation of the sensing physics. Probabilistic approaches for inference of critical image features that guide image interpretation and decision making will be developed. The mathematics associated with this approach requires these methods to capitalize on other new tools also under development in this project.