LEAPS-MPS: Direct methods for data rich inverse problems

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。科学测量通常基于间接推断。一个著名的例子是19世纪40年代根据对天王星轨道摆动的观测预测了当时尚未被发现的行星海王星。也就是说，通过对天王星的仔细观察，可以推断海王星的存在。这一推论属于“逆问题”领域。分布式和遥感(例如来自卫星的数据、人体成像)的持续增长开辟了反问题的新领域，即具有分布式和全场数据的反问题，这需要创造新的解决方法。这个项目将专注于开发一种特殊的计算效率的方法，称为直接方法，从观测现象的微分方程模型中推断各种感兴趣的量。这项研究将在罗切斯特理工学院进行，这是一所主要专注于本科教育的机构，但最近正在通过创建几个学生人数较少的博士项目来扩大其研究项目。该项目旨在通过招收STEM领域代表性较低的群体的学生来扩大一些博士项目。这些学生将接受最先进的建模和分析数据技术的培训，并将通过出版物和在当地和国际研讨会和会议上的演讲来传播他们的发现。这些努力旨在增加STEM工作队伍中代表性不足的个人的数量。该项目侧重于开发直接变分公式，以解决在有完整现场数据的情况下由微分方程模型管理的反问题。具体地说，主要集中在建立标量波反问题的本构方程直接误差公式(DECA)，以求解给定的时间简谐波场观测的材料的波速。目前用于波传播问题的反问题公式要么没有充分利用已有的全场数据，因此计算代价很高，要么对于具有多个观测值和未知参数的波传播问题完全不适用。预计该项目将通过i)检查其适定性，以及ii)确定寻找唯一解所需的波场数目，从而推进对复杂标量波模型的DECA公式的数学分析。DECA公式在医学成像领域将特别有用，在这些领域，全场内部数据越来越多，有效地解决逆问题对于疾病的实际检测和诊断至关重要。这项研究将在各级学生的帮助下进行。将特别努力从STEM领域代表性不足的群体中招收研究生，以帮助罗切斯特理工学院发展数学建模博士项目。该奖项反映了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。

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.