On the basis of the subspace-based optimization method (SOM), a twofold SOM (TSOM) and its variation, the FFT-TSOM, are proposed to solve in a more stable and more efficient manner the two-dimensional (2D) and three-dimensional (3D) electromagnetic inverse scattering problems. In the SOM, part of the induced current is found directly from the measured scattered fields while the remaining is searched within a current subspace, which has small contribution to the scattered fields, via optimization. By using the spectral property of the current-to-field operator, the TSOM further shrinks the dimension of the current subspace within which the induced current is optimized. Since the new current subspace is much smaller than the one used in the SOM, the TSOM shows better stability and better robustness against noise compared the SOM. However, in order to obtain the spectral property of the current-to-field operator, the singular value decompostion (SVD) of the operator is involved, and it is computationally burdensome, especially when dealing with problems with a large amount of unknowns. In order to decrease the computational complexity, the FFT-TSOM is proposed. In the FFT-TSOM, the discrete Fourier bases are used to construct a current subspace that is a good approximation to the original current subspace spanned by singular vectors. Such an approximation avoids the SVD and uses the FFT to accomplish the construction of the induced current, which reduces the computational complexity and memory demand of the algorithm compared to the original TSOM. By using the new current subspace approximation, the FFT-TSOM inherits the merits of the TSOM, better stability during the inversion and better robustness against noise compared to the SOM, and meanwhile has much lower computational complexity than the TSOM. Numerical tests for both TSOM and FFT-TSOM will be shown in the seminar.