Solution of the nonlinear inverse scattering problem by T-matrix completion. I. Theory

Area of Science:

• Physics
• Applied Mathematics

Background:

• Nonlinear inverse scattering problems (ISPs) are crucial in fields like medical imaging and seismology.
• Current methods often struggle with the complexity of determining interaction potentials from scattering data.

Purpose of the Study:

• To develop a new iterative algorithm for solving nonlinear ISPs.
• To leverage the theory of nonlocality to overcome limitations in existing methods.

Main Methods:

• Formulating the ISP to determine an unknown interaction potential (V) from scattering data.
• Allowing nonlocal intermediate steps for V while seeking a diagonally dominated (local) solution.
• Proposing a data-compatible T-matrix completion algorithm.

Main Results:

• The proposed method establishes a one-to-one correspondence between the T-matrix and the interaction potential.
• It relaxes the strict diagonality condition of V, enabling broader applicability.
• The algorithm seeks T-matrices compatible with data and yielding maximally diagonally dominated V.

Conclusions:

• This theoretical framework (Part I) introduces a novel approach for nonlinear ISPs.
• The method is particularly suited for large datasets generated by modern instrumentation.
• Part II will present numerical results for image reconstruction in a nonlinear regime.