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

Rowan University, Glassboro NJ

Investigators

Abstract

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.