Studies on iterative algorithms for modeling of frequency-domain wave equation based on multi-grid precondition

Frequency-domain full waveform inversion (FWI) is an important method for seismic imaging, and frequency-domain modeling is the basis of frequency-domain FWI. For large-scale problems, the LU-decomposition-based direct method is no longer applicable due to limitations of storage and computational time. Instead, iterative methods are employed. The bi-conjugate-gradient-stabilized method, which is based on multi-grid precondition, is an important iterative method. The preconditioner based on a heavily damped wave equation is approximately solved with one multi-grid cycle. The multi-grid method is implemented with a weighted Jacobi smoothing, a standard full-weighting coarsening, a linear interpolation, and a matrix-free implementation. To make it convergent for complex model, local model analysis is used to obtain the relaxation factor in the implementation of weighted Jacobi smoothing. Numerical experiments reveal: (1) for a generally-chosen relaxation factor, the levels of the multi-grid decrease as the complexity of the model increases, and, accordingly, the method becomes less practical; (2) for complex models, the relaxation factor obtained by local mode analysis increases the levels and reduces the number of iterations for each single frequency. The bi-conjugate-gradient-stabilized method based on multi-grid precondition obtains its efficiency and precision by using one multi-grid cycle for the approximate inversion of the preconditioner. To obtain reasonably fast convergence of the multi-grid method for complex model, local model analysis is applied in the relaxation factor selection. Compared with a generally-chosen relaxation factor, the relaxation factor obtained by local mode analysis increases the levels and guarantees the convergence and practicality of the multi-grid method. These conclusions are of great significance for application of the multi-grid-precondition-based iterative algorithm.