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

在日常生活中，人们普遍需要将肉眼无法看到的区域可视化。例如，在医学中，需要组织的精确可视化来诊断和治疗肿瘤。成像技术中的关键步骤需要解决逆问题，以便将测量数据转换为感兴趣的量的详细图像重建。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）尚处于起步阶段，因此，对研究贡献的影响潜力很高。用于成像的UQ在计算上具有挑战性，因为除了初始反演之外还需要数千次反演来生成不确定性的准确统计。目前的方法UQ是不够的，因为他们要么无法提供解决方案，在一个合理的计算时间，或者他们缺乏广泛的成像technology.The项目的适用性，在大规模的反问题，适用于广泛的成像technology.The项目是UQ的快速算法的发展。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）推进图像重建和UQ技术，以基于分数偏微分方程（PDE）和贝叶斯水平集方法纳入先验信息;（2）使用随机和Krylov子空间方法，为贝叶斯逆问题中的UQ开发数据驱动维度技术的新算法和分析。这里开发的算法将严格分析和验证几个模型的问题和应用，包括扩散光学和光声层析成像（生物医学）和液压层析成像和卫星数据融合（地球科学）。这里开发的算法也适用于其他成像为基础的反问题在生物医学，生物物理学，材料科学等成像应用程序之外，这些数学进步将感兴趣的科学家在计算科学的许多领域，例如，分数偏微分方程（PDE），模型简化，张量分解和主成分分析。最后，PI的教育和推广活动将使UQ和成像技术更加模块化，更容易获得，更容易为职前和早期职业K-12教育工作者，本科生和研究生理解。具体而言，该项目的教育计划将：（1）通过教师培训研讨会和为职前和早期职业K-12教师提供的实践研究经验，加强STEM教育，这将产生可复制的教学模块，用于K-12教育;以及（2）通过创建无障碍的研讨会讲座和新的课程内容，增强北卡罗来纳州州立大学的本科生和研究生课程。反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.