Combining direct and iterative solvers for improving ef ciency of solving wave equations when considering multi-sources problems

Frequency-domain full-waveform inversion (FWI) has been extensively developed during last decade<br>to build high-resolution velocity models (Pratt, 2004). One advantage of the frequency domain is that<br>inversion of a few frequencies are enough to build velocity models from wide-aperture acquisitions.<br>Multi-source frequency-domain wave modeling requires resolution of a large sparse system of linear<br>equations with multiple right-hand side (RHS). In 3D geometries or for very large 2D problems, the<br>memory requirements of state-of-the-art direct solvers preclude applications involving hundred millions<br>of unknowns. In order to overcome this limitation, we investigate a domain decomposition method based<br>on the Schur complement approach for 2D/3D frequency-domain acoustic wave modeling. The method<br>relies on a hybrid direct-iterative solver. Direct solver is applied to sparse impedance matrices assembled<br>on each subdomain, hence, reducing the memory requirement of the overall simulation. Iterative<br>solver based on a preconditioned Krylov method is used for solving the interface nodes between adjacent<br>domains. A possible drawback of the hybrid approach is that the time complexity of the iterative part<br>linearly increases with the number of RHS, if single-RHS Krylov subspace method is sequentially applied<br>to each RHS. We mention that block-Krylov techniques or de ation techniques can be used in that<br>case to partially overcome this effect. In the following, we introduce the domain decomposition method<br>before illustrating its features with 2D and 3D simulations.