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

逆散射理论在超声、探地雷达和微波成像等技术中起着关键作用。它被广泛应用于医学成像、地球物理勘探和未爆弹药探测等领域。逆散射理论中的反演算法旨在通过间接和非侵入性测量散射体与入射波之间的相互作用来重建被称为散射体的物体的几何和材料特性。研究者将开发和实现新的鲁棒和计算高效的反演算法。该项目将为本科生和研究生提供数学和科学计算方面的研究指导和培训机会。该项目将开发新的分裂一次性反演算法，用于利用与标量波动方程相关的多频率或多源散射数据重建散射体的材料属性。这些算法将确定散射体的材料属性和状态变量，这是波动方程的解，通过最小化目标泛函，其变量包括散射体的材料属性和状态变量。交替最小化方法将用于确定散射体的材料属性和状态变量的替代程序。该项目有两个主要目标。首先，将前向散射模型的算子分解为较小维数空间中的算子，降低目标泛函变量的维数。第二个目标是通过引入Carleman加权目标泛函来建立前向散射模型非线性情况下的凸目标泛函。理论研究将集中在Carleman估计，目标泛函的凸性，所提出的反演算法的全局收敛性和收敛率。本文将研究两个目标中所开发算法的并行实现和准确性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.