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

逆散射理论在超声波、探地雷达、微波成像等技术中起着关键作用。它广泛应用于医学成像、地球物理勘探、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地测量散射体和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属性。研究人员将开发和实现新的稳健和计算效率高的反演算法。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。该项目将开发新的一次分裂所有反演算法，用于使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性质。这些算法将通过最小化目标泛函来确定散射体的材料属性和状态变量，状态变量是波动方程的解，该目标泛函的变量既包括散射体的材料属性，也包括状态变量。在另一种方法中，将使用交替最小化方法来确定散射体的材料属性和状态变量。该项目有两个主要目标。第一种是通过将前向散射模型的算子分解为较小维度空间中的算子来降低目标泛函变量的维度。第二个目标是通过引入Carleman加权目标泛函，为前向散射模型是非线性的情况建立一个凸目标泛函。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和收敛速度上。将对两个目标中开发的算法的并行实施和准确性进行调查。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。

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.