Stability issues in some biomedical, financial, and geophysical inverse problems

该研究项目将开发有效的数值方法来解决经济、地球物理、国家安全、医学等领域的重要逆问题。一些具体的例子包括根据重力数据定位地下洞穴和隧道，通过电磁勘探包括不同的电导率，地下地震勘探，以及监测金融市场的波动。其中一个目标是使用最小的重力、电磁和金融数据来开发廉价、快速、可靠和安全的诊断和勘探技术。另一个目标是通过更高频率的（声波和电磁波）波进行勘探时提高分辨率，而不受传感器位置的传统凸性假设的限制。PI将参加即将在威奇托州立大学举行的工业数学诊所，作为与当地工业公司的联系，并将继续指导（女）研究生为当代科学和技术的国家人力资源作出贡献。PI旨在研究设计有效数值方法时的稳定性这一基本问题。由于（几乎）静止场（电磁或引力）的勘探是严重病态的，他计划找出源、介质或障碍物的多少参数可以以稳定的方式识别，以及所需的最小数据量是多少。在高频平稳波勘探时，目标是在未知物体和传感器位置不存在凸性条件的情况下获得较好的稳定性，并利用所需的解析先验约束作为惩罚/正则化项来设计有效的数值方法。一个特定的目标是通过以下方式来改进电阻抗层析成像：1)找到低频时的最佳分辨率以及获得该分辨率的传感器的最佳数量和位置；2)通过使用完整的麦克斯韦系统确定该分辨率如何在更高频率下提高，以及衰减带来的限制是什么。另一个研究计划是考虑更复杂、更现实的一篮子期权，设计更快、更可靠的方法，从市场数据中发现波动性。为了更好地理解和正确地使用稳定性，人们需要对现有的方法和新思想进行实质性的修改。其中一个挑战是如何从最小边界数据中证明重力测量或电阻抗层析中包裹体的独特性和利普希茨稳定性。另一个挑战是在没有标准几何（凸性）假设的情况下，显示出更好的声源或电磁源的稳定性。最后，PI期望从许多内部源生成的边界数据中设计出与时间无关的一般（各向异性）双曲二阶方程系数的稳定恢复。复变量理论、能量（卡尔曼）估计、傅立叶分析和势理论将被用作工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project will develop efficient numerical methods to solve important inverse problems in economics, geophysics, national security, and medicine. Some specific examples include locating underground caves and tunnels from gravimetric data, inclusion of different conductivity by electromagnetic prospecting, underground seismic prospecting, and monitoring the volatility of financial markets. One of the goals is to use minimal gravimetric, electromagnetic, and financial data to develop cheap, fast, reliable, and safe diagnostic and exploratory techniques. Another goal is to increase the resolution when prospecting by (acoustic and electromagnetic) waves of higher frequencies without restrictive traditional assumptions of convexity on location of sensors. The PI will participate in the forthcoming Industrial Mathematical Clinic at Wichita State University as an outreach to local industrial companies and will continue to direct (female) graduate students to contribute to national human resources for contemporary science and technology.The PI intends to study the fundamental issue of stability in the design of efficient numerical methods. Since prospecting by (almost) stationary fields (electromagnetic or gravitational) is severely ill-posed, he plans to find how many parameters of a source, a medium, or an obstacle can be identified in a stable way and what is the minimal amount of the data needed. When prospecting by stationary waves of higher frequencies, a goal is to achieve a better stability without convexity conditions on the locations of unknown objects and sensors and to use needed analytically a priori constraints as penalty/regularizing terms to design effective numerical methods. A particular goal is to improve the electrical impedance tomography by 1) finding what is the best resolution at low frequencies and the optimal number and location of sensors to get this resolution and 2) determining how this resolution improves with higher frequency by using the complete Maxwell system and what are limitations due to the attenuation. Another research plan is to consider more complicated and realistic basket options and to design faster and more reliable methods for finding volatility from the market data. To better understand and properly use stability one needs a substantial modification of available methods and new ideas. One of the challenges is to demonstrate the uniqueness and Lipschitz stability of an inclusion in gravimetry or in the electrical impedance tomography from a minimal boundary data. Another challenge is to show better stability for acoustic or electromagnetic sources without standard geometrical (convexity) assumptions. Finally, the PI expects to design a stable recovery of time independent coefficients of general (anisotropic) hyperbolic second order equations from the boundary data generated by many interior sources. Complex variable theory, energy (Carleman) estimates, Fourier analysis, and potential theory will be used as tools.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.