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

这一研究项目将开发有效的数值方法来解决经济学、地球物理学、国家安全和医学中的重要逆问题。一些具体的例子包括根据重力数据定位地下洞穴和隧道，通过电磁勘探包含不同的电导率，地下地震勘探，以及监测金融市场的波动性。其中一个目标是使用最小的重量、电磁和金融数据来开发廉价、快速、可靠和安全的诊断和探测技术。另一个目标是提高用高频(声波和电磁波)勘探时的分辨率，而不需要对传感器位置的凸性进行限制性的传统假设。国际数学协会将参加即将在威奇托州立大学开设的工业数学诊所，作为对当地工业公司的外展活动，并将继续指导(女性)研究生为国家当代科学和技术的人力资源做出贡献。国际数学协会打算研究有效数值方法设计中的稳定性这一基本问题。由于(几乎)固定磁场(电磁或重力)的勘探是严重不适定的，他计划找出一个源、介质或障碍物的多少参数可以稳定地识别出来，以及所需的最小数据量是多少。当用高频驻波勘探时，一个目标是在未知目标和传感器的位置上获得更好的稳定性而不存在凸性条件，并使用所需的解析先验约束作为惩罚/正则项来设计有效的数值方法。一个特别的目标是通过以下方式改进电阻抗断层成像：1)找到低频下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，确定在较高频率下分辨率如何提高，以及由于衰减而造成的限制。另一项研究计划是考虑更复杂和现实的篮子选择，并设计更快、更可靠的方法来从市场数据中发现波动性。为了更好地理解和正确使用稳定性，需要对现有方法和新思想进行实质性修改。其中一个挑战是从最小的边界数据来证明重力测量或电阻抗断层成像中包裹体的唯一性和Lipschitz稳定性。另一个挑战是在没有标准几何(凸性)假设的情况下，对声源或电磁源表现出更好的稳定性。最后，PI期望从许多内部源产生的边界数据中稳定地恢复一般(各向异性)双曲型二阶方程的与时间无关的系数。将使用复变量理论、能量(Carleman)估计、傅立叶分析和势能理论作为工具。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。

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.