  In several geophysical applications, such as full waveform inversion and data
modelling, we are facing the solution of inhomogeneous Helmholtz equation. The
difficulties of solving the Helmholtz equa- tion are two fold. Firstly, in the
case of large scale problems we cannot calculate the inverse of the Helmholtz
operator directly. Hence, iterative algorithms should be implemented. Secondly,
the Helmholtz operator is non-unitary and non-diagonalizable which in turn
deteriorates the performances of the iterative algorithms (especially for high
wavenumbers). To overcome this issue, we need to im- plement proper
preconditioners for a Krylov subspace method to solve the problem efficiently.
In this paper we incorporated shifted-Laplace operators to precondition the
system of equations and then generalized minimal residual (GMRES) method used
to solve the problem iteratively. The numerical results show the performance of
the preconditioning operator in improving the convergence rate of the GMRES
algorithm for data modelling case. In the companion paper we discussed the
application of preconditioned data modelling algorithm in the context of
frequency domain full waveform inversion. However, the analysis of the degree
of suitability of the preconditioners in the solution of Helmholtz equation is
an ongoing field of study.
