Three‐dimensional parallel frequency‐domain visco‐acoustic wave modelling based on a hybrid direct/iterative solver

Abstract

ABSTRACTWe present a parallel domain decomposition method based on a hybrid direct‐iterative solver for 3D frequency‐domain modelling of visco‐acoustic waves. The method is developed as a modelling engine for frequency‐domain full waveform inversion. Frequency‐domain seismic modelling reduces to the solution of a large and sparse system of linear equations, resulting from the discretization of the heterogeneous Helmholtz equation. Our approach to the high‐performance, scalable solution of large sparse linear systems is to combine direct and iterative methods. Such a hybrid approach exploits the advantages of both direct and iterative methods. The iterative component uses a small amount of memory and provides a natural way for parallelization. The direct part has favourable numerical properties for multiple right‐hand side modelling. The domain decomposition is based upon the algebraic Schur complement method, which allows for the iterative solution of a reduced system, the solution of which is the wavefield at the interfaces between the subdomains. Once the interface unknowns have been computed, the wavefield at the interior of each subdomain is efficiently computed by local substitutions. The reduced Schur complement system is solved with the generalized minimum residual method and is preconditioned by an algebraic additive Schwarz preconditioner. A direct solver is used to factorize local impedance matrices defined on each subdomain. Theoretical analysis shows that the time complexity of the hybrid solver is the same as that of iterative solver and time‐domain approaches for single frequency modelling. Simulations are performed in the SEG/EAGE overthrust and the salt models for frequencies up to 12.5 Hz. The number of iterations increases linearly with the number of subdomains for a given computational domain but the elapsed time of the iterative resolution remains almost constant. The number of iterations also increases linearly with frequencies, when the grid interval is adapted to the frequencies and the size of the subdomains is kept constant over frequency. These results make the cost of the hybrid solver of the same order as that of finite‐difference time‐domain modeling for one‐frequency modelling. Although the hybrid approach allows one to tackle larger problems than the direct‐solver approach, further improvements are needed to mitigate the computational burden of the iterative component in the context of multisource modelling. On the numerical side, the use of block iterative solvers and of incremental two‐level deflating preconditioners and on the parallel implementation side, the use of two levels of parallelism in the domain decomposition method should mitigate this computational burden.