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In this paper, we combine two techniques together, i.e., the fast Fourier transform-twofold subspace-based optimization method (FFT-TSOM) and multiplicative regularization (MR) to solve inverse scattering problems. When applying MR to the objective function in the FFT-TSOM, the new method is referred to as MR-FFT-TSOM. In MR-FFT-TSOM, a new stable and effective strategy of regularization has been proposed. MR-FFT-TSOM inherits not only the advantages of the FFT-TSOM, i.e., lower computational complexity than the TSOM, better stability of the inversion procedure, and better robustness against noise compared with the SOM, but also the edge-preserving ability from the MR. In addition, a more relaxed condition of choosing the number of current bases being used in the optimization can be obtained compared with the FFT-TSOM. Particularly, MR-FFT-TSOM has even better robustness against noise compared with the FFT-TSOM and multiplicative regularized contrast source inversion (MR-CSI). Numerical simulations including both inversion of synthetic data and experimental data from the Fresnel data set validate the efficacy of the proposed algorithm. [ABSTRACT FROM PUBLISHER]