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

本研究计画将发展有效的数值方法，以解决经济学、物理学、国家安全与医学等领域的重要反问题。 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导电性，地下地震勘探，以及监测金融市场的波动。目标之一是使用最少的重力，电磁和金融数据来开发廉价，快速，可靠和安全的诊断和探索技术。另一个目标是提高分辨率时，勘探（声波和电磁）波的频率较高，没有限制性的传统假设的凸性的位置传感器。PI将参加即将在威奇托州立大学举办的工业数学诊所，作为对当地工业公司的宣传，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做出贡献。PI打算研究有效数值方法设计中的稳定性基本问题。由于（几乎）稳定场（电磁场或引力场）的勘探是严重不适定的，他计划找出一个源，一个介质或一个障碍物的多少参数可以以稳定的方式确定，以及所需的最小数据量是多少。当通过更高频率的驻波进行勘探时，目标是在未知对象和传感器的位置上实现更好的稳定性而没有凸性条件，并且使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断层成像：1）找到在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，以及2）通过使用完整的麦克斯韦系统来确定该分辨率如何随着较高频率而改进，以及由于衰减而导致的限制是什么。另一个研究计划是考虑更复杂和更现实的一篮子期权，并设计更快，更可靠的方法来发现波动率从市场数据。 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。在重力测量或电阻抗层析成像中，如何从最小边界数据证明包裹体的唯一性和Lipschitz稳定性是一个难题。另一个挑战是在没有标准几何（凸性）假设的情况下显示声学或电磁源的更好稳定性。最后，PI期望从许多内部源产生的边界数据设计一般（各向异性）双曲二阶方程的时间无关系数的稳定恢复。该奖项反映了NSF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.