CAREER: Synergistic Inverse Wave Analysis and Computation

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. 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.