An efficient truncated SVD of large matrices based on the low-rank approximation for inverse geophysical problems

In this paper, we propose a new algorithm to compute a truncated singular value decomposition (T-SVD) of the Born matrix based on a low-rank arithmetic. This algorithm is tested in the context of acoustic media. Theoretical background to the low-rank SVD method is presented: the Born matrix of an acoustic problem can be approximated by a low-rank approximation derived thanks to a kernel independent multipole expansion. The new algorithm to compute T-SVD approximation consists of four steps, and they are described in detail. The largest singular values and their left and right singular vectors can be approximated numerically without performing any operation with the full matrix. The low-rank approximation is computed due to a dynamic panel strategy of cross approximation (CA) technique.
At the end of the paper, we present a numerical experiment to illustrate the efficiency and precision of the algorithm proposed.