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图像的愿望出现在各种应用中，从检测和表征体内癌症，到量化水、油或地下污染物的分布，以及及时准确地识别拥挤场所的爆炸物。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的信息带来了大量的计算挑战。该项目的研究团队旨在通过改进统计逆理论、数值反演方法、仿真模型和混合成像模型，开发大大降低实时成像成本的计算方法。该项目的重点将测试成像在医学断层扫描、环境修复和机场安全成像方面的应用。这些技术构成了解决与跨大范围空间和时间尺度的光信号反演相关的类似问题的基础。作为该项目的一部分，将开发一门模块化课程，在研究生阶段教授这些新方法。课程材料将在互联网上提供。该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不匹配。在不断增加的大量测量中，必须尽量减少潜在的大量参数，同时一些未知的数据集可能是冗余的。然而，在处理相关的实际问题和决策时，并不总是需要详细的图像。将开发计算技术的组合，使大规模参数相关最小化在计算上可行。此外，将开发新的有效方法来推断关键图像特征，从而避免对图像进行完全重建的需要。该研究建立在最近的方法上，这些方法利用随机化在大大降低计算成本的情况下计算准确的解估计，并建立在更小、更近似、更低阶的数值模型的有效构建上，这些模型对相关参数集是准确的，从而降低了传感物理完全模拟的成本。将开发用于指导图像解释和决策的关键图像特征推断的概率方法。与此方法相关的数学要求这些方法利用本项目中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.