CAREER: Integrated Approaches for Fast and Accurate Large-Scale Inversion

在各种科学应用中(例如，用于癌症诊断或地下矿山的裂缝检测)，计算逆问题的解的能力是必不可少的，但对于包含物理或数据信息约束的大型非线性问题的实时解的计算，使用当前的反演算法是不可行的。此外，随着反问题的数值解越来越多地被用于数据分析和辅助决策，这些计算限制在不确定性量化(例如，估计解的方差)的算法中构成了重大瓶颈。该项目的首要目标是显著降低数值反演的成本，并使统计工具能够帮助科学家做出明智的决定。这些发展将导致许多重要领域的科学进步。例如，现有的与生物医学和采矿工程师的合作将确保拟议的研究能够通过先进的医疗保健成像技术改善医疗诊断，由于改进地下矿山的地面控制监测而减少伤害，以及用于生理系统实时分析的先进信号估计。此外，国际学生联合会将继续积极参与鼓励来自历史上代表性不足群体的学生的活动。这项研究将通过开发更快的方法和更强大的框架来设计、计算和分析逆问题的解，从而促进计算逆问题领域的知识进步。将采用一个综合框架，其中的主要研究重点是(I)开发新的正则化方法和实现来处理特定应用的约束，同时纳入稳健的参数选择方法；(Ii)促进实时计算大型非线性逆问题的解的技术(例如，通过整合随机方法和更新方法)；以及(Iii)通过开发有效的误差估计方法，实现对复杂的、非线性系统的关键但以前无法获得的定量诊断。

The ability to compute solutions to inverse problems is essential in various scientific applications (e.g., for cancer diagnosis or for crack detection in underground mines), but computing real-time solutions to large nonlinear problems that incorporate physics- or data-informed constraints is not feasible with current inversion algorithms. Moreover, as numerical solutions to inverse problems are increasingly being used for data analysis and to aid in decision-making, these computational limitations pose significant bottlenecks in algorithms for uncertainty quantification (e.g., for estimating solution variances). The overarching goal of this project is to significantly reduce the costs of numerical inversion and to enable statistical tools to aid scientists in making informed decisions. These developments will lead to scientific advancement in many important fields. For example, existing collaborations with biomedical and mining engineers will ensure that the proposed research can result in improved medical diagnosis via advanced point-of-care imaging technologies, fewer injuries due to improved ground control monitoring of underground mines, and advanced signal estimation for real-time analysis of physiological systems. Moreover, the PI will continue to actively engage in activities that encourage students from historically under-represented groups. The PI's focus on upper elementary to high school girls and on outreach that will feed back into the greater research and teaching communities (e.g., K-12 teachers) will contribute to the recruitment, training, and retention of a diverse next generation of computational scientists.This research will advance knowledge in the field of computational inverse problems by developing faster methodologies and more robust frameworks for the design, computation, and analysis of solutions to inverse problems. An integrated framework will be adopted, where the main research thrusts are (i) to develop novel regularization methods and implementations to handle application-specific constraints, while simultaneously incorporating robust parameter selection methods; (ii) to advance technologies for real-time computation of solutions to large, nonlinear inverse problems (e.g., by integrating stochastic methods and update approaches); and (iii) to enable critical, yet previously unobtainable, quantitative diagnostics for complex, nonlinear systems by developing efficient error estimation methods.