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

Abstract

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