Three-dimensional finite-difference & finite-element frequency-domain wave simulation with multi-level optimized additive Schwarz domain-decomposition preconditioner: A tool for FWI of sparse node datasets.

Efficient frequency-domain Full Waveform Inversion (FWI) of long-offset node data can be designed with a few discrete frequencies hence allowing for compact volume of data to be managed. Moreover, attenuation effects can be straightforwardly implemented in the forward problem without computational overhead. However, 3D frequency-domain seismic modeling is challenging since it requires solving a large and sparse linear indefinite system per frequency with multiple right-hand sides. This linear system can be solved by direct or iterative methods. The former are very efficient to process multiple right-hand sides but may suffer from limited scalability for very large problems. Iterative methods equipped with a domain decomposition preconditioner provide a suitable alternative to process large computational domains for sparse node acquisition. The domain decomposition preconditioner relies on an optimized restricted additive Schwarz (ORAS) method, where a Robin or Perfectly-Matched Layer (PML) condition is implemented at the boundaries between the subdomains. The preconditioned system is solved by a Krylov subspace method, while a block low-rank Lower-Upper (LU) decomposition of the local matrices is performed at a preprocessing stage. Multiple sources are processed in group with a pseudo-block method. The accuracy, the computational cost and the scalability of the ORAS solver are assessed against several realistic benchmarks. For the considered benchmarks, a compact wavelength-adaptive 27-point finite-difference stencil on regular Cartesian grid shows better accuracy and computational efficiency than a P3 finite-element method on h-adaptive tetrahedral mesh, which remains however beneficial to comply with known boundaries such as bathymetry.