# Fast Numerical Integration Techniques for 2.5-Dimensional Inverse Problems

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.

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