Parallel multigrid method for solving inverse problems.

We considered in this work the linear operator equation and used the Landweber iterative method as an iterative solver. After that, we used the multigrid method as an optimization method for obtaining an approximation solution with a highly accurate and fast process. A new parallel algorithm for the multigrid process has been developed. The proposed algorithm is based on a V-cycle mixed with the two-grid method. This modification of the V-cycle provides for parallel computing for each level. A coarse grid operator with a residual right-hand side vector for each coarse grid is provided. This parallel algorithm is used to accelerate and enhance computation for the solution of the iteration method in solving the inverse ill-posed problems. The necessary cost-time computation for all stages and processes for the parallel V-cycle algorithm has been done. A numerical experiment on solving the IVP (initial value problem) for the heat equation showed that the new parallel algorithm is much more efficient than the sequential method.•The study of iteration algorithms and mathematical experiments reveals a slow rate of convergence.•The Multigrid method is often used to speed up the rate of convergence of iterative methods, which is an effective method of solving large systems of linear algebra equations.•The approximation solution for the linear algebra equations was found by using the parallel method with the multigrid method.