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CAREER: Fast and Accurate Algorithms for Uncertainty Quantification in Large-Scale Inverse Problems

职业：大规模反问题中不确定性量化的快速准确算法

基本信息

批准号：
1845406
负责人：
Arvind Saibaba
金额：
$ 40万
依托单位：
North Carolina State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1845406&HistoricalAwards=false
关键词：
CAREER Fast Accurate Algorithms Uncertainty

项目摘要

The need to visualize regions that are impossible to see with the naked eye is pervasive in everyday life. For example, in medicine, accurate visualization of tissue is needed to diagnose and treat tumors. A key step in imaging technologies requires one to solve an inverse problem in order to transform measured data into detailed image reconstructions of the quantities of interest. However, image reconstruction is inherently uncertain, in part, due to noisy measurements from sensors. Ignoring the uncertainty in the imaging process can lead to undesirable outcomes, such as misjudging the location and spread of a suspected tumor. Uncertainty Quantification (UQ) in imaging is in its infancy and hence, the potential for impact in research contributions is high. UQ for imaging is computationally challenging since thousands of inversions are needed beyond the initial inversion to generate accurate statistics of the uncertainty. Current approaches for UQ are inadequate because they either fail to deliver solutions in a reasonable computational time or they lack the applicability across a broad range of imaging technologies.The project is on the development of fast algorithms for UQ in large-scale inverse problems that are applicable to a broad range of imaging technologies. These algorithms are expected to bring down the computational cost by at least an order of magnitude while maintaining the accuracy of the solutions. More specifically, the project will (1) Advance image reconstruction and UQ techniques for incorporating prior information based on fractional partial differential equation (PDE) and Bayesian level set approaches; and (2) Develop new algorithms and analysis for data-driven dimensionality techniques for UQ in Bayesian inverse problems, using randomized and Krylov subspace methods. The algorithms developed here will be rigorously analyzed and validated on several model problems and applications, including diffuse optical and photoacoustic tomography (in biomedicine) and hydraulic tomography and satellite data fusion (in geoscience). The algorithms developed here are also applicable to other imaging-based inverse problems in biomedicine, geophysics, materials science, etc. Outside of imaging applications, these mathematical advances will be of interest to scientists working in many areas of computational science, for example, fractional partial differential equations (PDEs), model reduction, tensor decompositions, and principal component analysis. Lastly, the PI's education and outreach activities will make UQ and imaging technologies more modular, accessible, and easier to understand for pre-service and early career K-12 educators, undergraduate students, and graduate students. Specifically, the educational program of this project will: (1) Strengthen STEM education through teacher training workshops and practical research experiences for pre-service and early career K-12 teachers, which will result in reproducible teaching modules for use in K-12 education; and (2) Enhance undergraduate and graduate curriculum at North Carolina State University by creating accessible seminar talks and new course content.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.

在日常生活中，对肉眼无法看到的区域进行可视化的需求非常普遍。例如，在医学上，诊断和治疗肿瘤需要准确的组织可视化。成像技术中的一个关键步骤需要解决逆问题，以便将测量数据转换为感兴趣数量的详细图像重建。然而，图像重建本质上是不确定的，部分原因是来自传感器的噪声测量。忽视成像过程中的不确定性可能会导致不良结果，例如误判疑似肿瘤的位置和扩散。成像中的不确定性量化(UQ)还处于初级阶段，因此，对研究贡献产生影响的可能性很高。成像的UQ在计算上具有挑战性，因为除了最初的反演之外，还需要数千次反演才能生成准确的不确定性统计数据。现有的UQ方法不能在合理的计算时间内提供解，或者缺乏对多种成像技术的适用性。该项目致力于开发适用于广泛成像技术的大规模反问题的快速算法。这些算法有望在保持解的准确性的同时，将计算成本降低至少一个数量级。更具体地说，该项目将(1)推进基于分数偏微分方程(PDE)和贝叶斯水平集方法的图像重建和UQ技术；以及(2)利用随机化和Krylov子空间方法，为贝叶斯反问题中UQ的数据驱动维度技术开发新的算法和分析。本文开发的算法将在几个模型问题和应用上进行严格的分析和验证，包括扩散光学和光声层析成像(在生物医学中)和水力层析成像和卫星数据融合(在地球科学中)。这里开发的算法也适用于生物医学、地球物理、材料科学等领域中其他基于成像的反问题。除了成像应用之外，这些数学进展将引起许多计算科学领域的科学家的兴趣，例如，分数阶偏微分方程组(PDE)、模型降阶、张量分解和主成分分析。最后，PI的教育和推广活动将使UQ和成像技术更模块化、更容易获得，更容易为任职前和职业生涯早期的K-12教育工作者、本科生和研究生理解。具体地说，该项目的教育计划将：(1)通过为任职前和职业早期的K-12教师举办教师培训讲习班和实践研究经验来加强STEM教育，这将导致可重复使用的教学模块用于K-12教育；以及(2)通过创建无障碍研讨会演讲和新的课程内容来增强北卡罗来纳州立大学的本科生和研究生课程。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。