LEAPS-MPS: Splitting All-At-Once Approach to Inverse Medium Scattering Problems

LEAPS-MPS：解决逆介质散射问题的一次性分裂方法

• 批准号: 2316843
• 负责人: Thanh Nguyen
• 金额: $ 21.29万
• 依托单位: Rowan University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2316843&HistoricalAwards=false
• 关键词: LEAPS; MPS; Splitting; Once; Approach

Inverse scattering theory plays a key role in technologies, such as ultrasounds, ground penetrating radars, and microwave imaging. It is widely used in applications such as medical imaging, geophysical exploration, and detection of unexploded ordnance. Inversion algorithms in the inverse scattering theory aim to reconstruct geometrical and material properties of objects, which are referred to as scatterers, from indirect and noninvasive measurements of the interaction between the scatterers and incoming waves. The investigator will develop and implement new robust and computationally efficient inversion algorithms. The project will provide research mentoring and training opportunities in mathematics and scientific computing for undergraduate and graduate students.This project will develop new splitting all-at-once inversion algorithms for reconstructing a material property of a scatterer using multi-frequency or multi-source scattering data associated with a scalar wave equation. These algorithms will determine both the scatterer’s material property and the state variable, which is the solution of the wave equation, by minimizing an objective functional whose variables include both the scatterer’s material property and the state variable. Alternating minimization methods will be used to determine the scatterer’s material property and the state variable in an alternative procedure. The project has two main objectives. The first is to reduce the dimension of the variable of the objective functional by splitting the operator of the forward scattering model into operators in spaces of smaller dimensions. The second objective aims to create a convex objective functional for the case when the forward scattering model is nonlinear, by introducing a Carleman weighted objective functional. Theoretical investigations will be focused on Carleman estimates, the convexity of the objective functional, the global convergence and convergence rate of the proposed inversion algorithms. Parallel implementation and accuracy of the developed algorithms in both objectives will be investigated.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.

逆散射理论在技术，例如超声，地面穿透雷达和微波成像中起关键作用。它被广泛用于医学成像，地球物理探索和未爆炸法令等应用中。反向散射理论中的反转算法旨在重建物体的几何和物质特性，这些特性被称为散射子，从间接和无创测量散射器与传入波之间的相互作用的测量。研究者将开发和实施新的鲁棒和计算有效的反转算法。该项目将为本科和研究生提供数学和科学计算的研究心理和培训机会。该项目将开发新的分类全面反转算法，以使用多种频率或多重为代理散射数据与标量波方程相关联，以重建散点器的材料属性。这些算法将通过最小化目标函数，其变量既包含散点器的材料属性和状态变量，又可以确定散点子的材料属性和状态变量，即波动方程的解决方案。交替的最小化方法将用于在替代过程中确定散点子的材料属性和状态变量。该项目有两个主要目标。首先是通过将正向散射模型的操作员分配到较小尺寸的空间中的操作员中来降低目标函数变量的尺寸。第二个目标旨在通过引入Carleman加权目标函数来为前向散射模型非线性创建一个凸目标功能。理论研究将集中于卡尔曼估计，目标函数的凸，全球收敛和融合算法的收敛速率。将研究这两个目标中开发算法的并行实施和准确性。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估标准，认为通过评估被认为是宝贵的支持。