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A fast algorithm for regularized focused 3-D inversion of gravity data using the randomized SVD

A fast algorithm for solving the under-determined 3-D linear gravity inverse problem based on the randomized singular value decomposition (RSVD) is developed. The algorithm combines an iteratively reweighted approach for $L_1$-norm regularization with the RSVD methodology in which the large scale linear system at each iteration is replaced with a much smaller linear system. Although the optimal choice for the low rank approximation of the system matrix with m rows is q=m, acceptable results are achievable with q<<m. In contrast to the use of the LSQR algorithm for the solution of the linear systems at each iteration, the singular values generated using the RSVD yield a good approximation of the dominant singular values of the large scale system matrix. The regularization parameter found for the small system at each iteration is thus dependent on the dominant singular values of the large scale system matrix and appropriately regularizes the dominant singular space of the large scale problem. The results achieved are comparable with those obtained using the LSQR algorithm for solving each linear system, but are obtained at reduced computational cost. The method has been tested on synthetic models along with the real gravity data from the Morro do Engenho complex from central Brazil.