Acoustic Inverse Problems with Single and Multiple Measurements

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The project addresses fundamental scientific challenges in non-destructive testing techniques. These techniques are widely used in medical imaging for tumor detection, in seismic imaging for oil prospecting, and in military applications for weapon defect inspection. This project facilitates resolution of these challenges by developing innovative theory and algorithms for the underlying mathematical problems. The theory will promote the domain knowledge in mathematics and relevant scientific areas. The theoretically consolidated algorithms will open new avenues for scientific computing in non-destructive testing techniques, consequently leading to innovation in medical imaging, seismic imaging, and military application that will advance the societal welfare and national security. Implementation of the project will be integrated with the educational mission to involve undergraduate and graduate students majoring in mathematics and engineering, especially those from underrepresented groups. Students will receive rigorous scientific training in mathematical theory and programming to prepare them as the next generation of the STEM workforce that can continue to promote science and technology in future decades.The overall goal of the project is determination and reconstruction of multiple acoustic parameters from single or multiple boundary measurements. In the first major topic, the investigator will devise novel algorithms to reconstruct the wave speed and potential from multiple boundary measurements using the boundary control method. The algorithms give direct reconstruction to the nonlinear inverse problems as opposed to traditional iterative algorithms. In the second major topic, the investigator will study uniqueness and stability for joint determination of a source and a coefficient from single boundary measurements. Novel sufficient and necessary conditions will be derived from the viewpoint of microlocal analysis.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.