Acoustic Inverse Problems with Single and Multiple Measurements

该项目解决了无损检测技术中的基本科学挑战。这些技术广泛应用于肿瘤检测的医学成像、石油勘探的地震成像以及武器缺陷检测的军事应用。这个项目通过为基本的数学问题开发创新的理论和算法来促进这些挑战的解决。该理论将促进数学和相关科学领域的领域知识。理论上巩固的算法将为无损检测技术中的科学计算开辟新的途径，从而导致医学成像、地震成像和军事应用方面的创新，从而促进社会福利和国家安全。该项目的实施将与教育使命相结合，使数学和工程专业的本科生和研究生，特别是来自代表性不足群体的本科生和研究生参与进来。学生们将接受严格的数学理论和编程方面的科学培训，为他们成为下一代STEM劳动力做好准备，在未来几十年里，他们可以继续推动科学技术。该项目的总体目标是根据单一或多个边界测量确定和重建多个声学参数。在第一个主要主题中，研究人员将设计新的算法，使用边界控制方法从多个边界测量中重建波速和波势。与传统的迭代算法不同，该算法给出了非线性反问题的直接重构。在第二个主要主题中，调查者将研究从单一边界测量中联合确定源和系数的唯一性和稳定性。新的充分和必要的条件将从微观分析的角度得出。这一奖项反映了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。

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.