CAREER: Synergistic Inverse Wave Analysis and Computation

非破坏性探测技术利用波来探测无法直接观察到的目标。它们在用于肿瘤检测的医学成像、用于石油勘探的地震成像以及用于检测隐藏威胁的军事场景中得到了广泛的应用。现代无损探测技术的成功在很大程度上依赖于揭示观测和目标之间隐藏联系的数学原理。对这些原则的深入理解不仅将揭示这些技术的优势和局限性，而且还将指明未来增强和改进的可能方向。该项目旨在从理论和计算的角度促进对非破坏性探测技术中数学原理的理解。理论观点将开发基本的数学工具和方法，以适应解决其他数学分支的挑战。计算视角将把这一理论转化为创新的成像算法，使无损探测技术能够获得增强的可视化。该项目的实施将与许多教育活动相结合，使K-12、STEM专业的本科生和研究生，特别是那些来自教育资源有限、代表性不足的群体的学生参与进来。学生将接受严格的数学和编程技能的科研培训，为他们成为下一代能够在未来几十年继续发展科学技术的STEM劳动力做准备。该项目旨在研究波动方程反问题的基本数学理论和计算方法。这将通过发展计算边界控制方法及其基本的数学理论来实现。该方法的一个特点是数学分析和基于波的科学计算之间的协同。从理论上讲，跨学科的研究将加强偏微分方程分析、波动物理、几何学和科学计算之间的相互作用。从计算的角度来看，这项研究将启发新的计算范例，利用波传播的性质来实现非破坏性探测技术的最佳、理论上合理的算法。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。

Non-destructive probing techniques utilize waves to probe targets that are not directly observable. They have found broad applications in medical imaging for tumor detection, in seismic imaging for oil prospecting, and in military scenarios for hidden threat detection. The success of modern non-destructive probing techniques relies in an essential way on the mathematical principles that reveal the hidden connection between the observation and the targets. In-depth understanding of such principles will not just reveal the strengths and limitations of the techniques, but also indicate possible directions for future enhancement and improvement. This project aims to promote understanding of the mathematical principles in non-destructive probing techniques from both theoretical and computational perspectives. The theoretical perspective will develop fundamental mathematical tools and methods that are adaptable to resolve challenges in other mathematical branches. The computational perspective will turn the theory into innovative imaging algorithms that empower non-destructive probing techniques to acquire enhanced visualization. Implementation of the project will be integrated with numerous educational activities to involve K-12, undergraduate and graduate students majoring in STEM disciplines, especially those from underrepresented groups with limited educational resources. Students will receive rigorous scientific research training in mathematical and programming skills to prepare them as the next generation of STEM workforce that can continue advancing science and technology in the next few decades.The project aims to investigate fundamental mathematical theory and computational methods for inverse problems for wave equations. This will be achieved by developing the computational boundary control method as well as its underlying mathematical theory. A feature of the approach is the synergy between mathematical analysis and wave-based scientific computing. From the theoretical perspective, the interdisciplinary research will enhance the interplay among PDE analysis, wave physics, geometry, and scientific computing. From the computational perspective, the research will inspire novel computational paradigms that utilize the nature of wave propagation to achieve optimal, theoretically justifiable algorithms for non-destructive probing techniques.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.