NSF Award
2008154
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.<br/><br/>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.<br/><br/>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.
