Fast Numerical Integration Techniques for 2.5-Dimensional Inverse Problems

TL;DR

This paper introduces fast numerical integration methods for 2.5-D inverse scattering problems, significantly reducing computational complexity by employing transformations that achieve exponential convergence.

Contribution

The paper proposes novel transformation-based integration techniques that enable efficient computation of 2.5-D Green's functions with exponential convergence, improving inverse scattering analysis.

Findings

Achieves exponential convergence in integration methods.

Reduces computational complexity for 2.5-D inverse problems.

Enables efficient approximation of 3-D scattering with 2-D models.

Abstract

Inverse scattering involving microwave and ultrasound waves require numerical solution of nonlinear optimization problem. To alleviate the computational burden of a full three-dimensional (3-D) inverse problem, it is a common practice to approximate the object as two-dimensional (2-D) and treat the transmitter and receiver sensors as 3-D, through a Fourier integration of 2-D modes of scattering. The resulting integral is singular, and hence requires a prohibitively large number of integration points, where each point corresponds to a 2-D solution. To reduce the computational complexity, this paper proposes fast integration approaches by a set of transformations. We model the object in 2-D but the transmit and receiver pairs as 3-D; hence, we term the solution as a 2.5-D inverse problem. Convergence results indicate that the proposed integration techniques have exponential convergence and hence have a reduces the computational complexity to compute 2.5-D Green's function to solve inverse scattering problems.