Nonlinear Electromagnetic Inverse Scattering Imaging Based on IN-LSQR

Algorithm 1

Algorithm flow of IN-LSQR.

Input: number of transmitting antennas N_T; number of receiving antennas N_r; the number of square cells N_d; scattered field data measured E^sca

Output: solution of (3)

Initialization: initial solution

Outer inexact Newton iterations for

Calculate on the basis of (8) and (9), and calculate on the basis of (10), (11), and (12) and stored in sparse compressed form.

1.1 Internal initialization , , , , , . is a nonnegative real number

 Internal iterations for

1.2 Lanczos bidiagonalization of on the basis of (8), gives orthogonal matrix , and bidiagonal matrix .

1.3 Calculate the intermediate variable of the QR decomposition process in (18),

where and are the initial value of the next iteration.

1.4 Update and , according to

1.5 If , then and repeat steps 1.2–1.4. Otherwise, terminate the internal iteration, achieving , the solution to (13).

2 Update the solution to (3) using

3 When the change of in two consecutive iterations is less than the termination condition, terminate the outer inexact Newton iteration, else and return to step 1.