A review of recent forward problem developments used for frequency-domain FWI

For efficient frequency-domain full wave inversion (FWI), frequency-domain solutions of the waveequation are required. The wave equation can be solved by direct or iterative solvers in the frequency domain or, alternatively, one can also use time domain integration until the stationary state has been reached. These three strategies have the same time complexitities. Increasing the order of the finite difference stencil may reduce the memory requirements and should optimize the use of available floating point operations per second (flops) on modern computer platforms. Known drawbacks of direct solvers, as poor scalability and significant in-core memory requirement, could be overcome by recent advances such as block low-rank (BLR) approximation. Finally some new algorithm designs for iterative solvers have shown interesting scalability and good convergence properties in heterogeneous media. We illustrate that a parallel conjugate gradient (CARP-CG) is robust and scalable for 2D elastic equations, extendable to 3D geometries. These recent breakthroughs could be integrated in the frequency-domain FWI workflows for real applications.