LEAPS-MPS: Direct methods for data rich inverse problems

Rochester Institute Of Tech, Rochester NY

Investigators

Abstract

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). Scientific measurement is often based on indirect inference. A famous example is the prediction of the then-undiscovered planet Neptune in the 1840s based on observations of wiggles in the orbit of Uranus. That is, by making close observations of Uranus, the existence of Neptune could be inferred. That inference is the domain of the field of "inverse problems." Continued growth in distributed and remote sensing (e.g. data from satellites, imaging in the human body) opens a new realm of inverse problems, those with distributed and full-field data, that call for the creation of new solution methods. This project will focus on developing a special class of computationally efficient methods, called direct methods, to infer various quantities of interest from differential equation models of observed phenomena. The research will be conducted at Rochester Institute of Technology, an institution focussed mainly on undergraduate education, but with a recent push towards growing its research program by the creation of several PhD programs with small student cohort. This project aims to grow some of the PhD programs by recruiting students from groups underrepresented in the STEM field. These students will be trained in state of the art techniques to model and analyze data, and will disseminate their findings through publications and presentations at local and international workshops and conferences. These efforts aim to increase the number of underrepresented individuals in the STEM work force. The project focuses on developing direct variational formulations to solve inverse problems governed by differential equation models where full field data are available. Specifically, the main focus is on developing the direct error in constitutive equations formulation (DECE) for a scalar wave inverse problem to solve for the wave speed of a material, given observations of time harmonic wave fields. Current inverse problems formulations for wave propagation problems either do not make full use of full field data when it is available, and are hence computationally expensive, or they completely fail for wave propagation problems with multiple observations and unknown parameters. The project is expected to advance the mathematical analysis of the DECE formulation for the complex scalar wave model by i) examining its well-posedness, and ii) determining the number of wave fields needed to find a unique solution. The DECE formulation will be especially useful in medical imaging fields, where full field interior data are increasingly available, and solving inverse problems efficiently is critical for practical detection and diagnosis of disease. The research conducted will be done with the help of students at all levels. Special effort will be made to recruit graduate students from groups underrepresented in the STEM fields to help grow the math modeling PhD program at Rochester Institute of Technology. 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.