We investigate the inverse scattering problem for scalar waves. We report conditions under which the terms in the inverse Born series cancel in pairs, leaving only one term at each order. We refer to the resulting expansion as the reduced inverse Born series. The reduced series can also be derived from a nonperturbative inversion formula. Our results are illustrated by numerical simulations that compare the performance of the reduced series to the full inverse Born series and the Newton–Kantorovich method.
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Second column gives the values of $w$ [defined in Eq. (42)]. Third column is the error of linearized inversion as defined in Eq. (37). Fourth column is the error of nonperturbative solution in Eq. (29). Fifth column indicates whether the NK iterations converge. A question mark means that for this particular value of $\gamma$ we are not certain about convergence.
Table 2.
Summary of the Reconstruction Results for the Setup of Section 4.C and Different Values of the Contrast a
Reconstruction Error of IBS, rIBS, and NK Iterations in the Setup of Section 4.C, for Two Values of as Labeleda
Order
IBS
rIBS
NK
1
2
3
4
5
N/A
1
2
3
4
5
N/A
Diverges
Diverges
The order $\infty$ corresponds to either the analytical summation or converged result after many iterations. N/A in the IBS columns indicates that the series could not be computed to high orders due to computational complexity.
Table 4.
Reconstruction Error of IBS, rIBS, and NK Iterations in the Setup of Section 4.D, for Two Values of as Labeleda
Second column gives the values of $w$ [defined in Eq. (42)]. Third column is the error of linearized inversion as defined in Eq. (37). Fourth column is the error of nonperturbative solution in Eq. (29). Fifth column indicates whether the NK iterations converge. A question mark means that for this particular value of $\gamma$ we are not certain about convergence.
Table 2.
Summary of the Reconstruction Results for the Setup of Section 4.C and Different Values of the Contrast a
Reconstruction Error of IBS, rIBS, and NK Iterations in the Setup of Section 4.C, for Two Values of as Labeleda
Order
IBS
rIBS
NK
1
2
3
4
5
N/A
1
2
3
4
5
N/A
Diverges
Diverges
The order $\infty$ corresponds to either the analytical summation or converged result after many iterations. N/A in the IBS columns indicates that the series could not be computed to high orders due to computational complexity.
Table 4.
Reconstruction Error of IBS, rIBS, and NK Iterations in the Setup of Section 4.D, for Two Values of as Labeleda