Iterative method for solving linear operator equation of the first kind.

In this work, we study the iterative method for solving linear operator equation of the first kind. We present a new version of method based on the applied the iterative performance on the modified Lavrentiev method. This method is used to resolve a linear operator problem of the first kind. The suggested iterative can used to compute approximate solutions with high quality than the (standard) modified Lavrentiev regularization method. We also compared the new iterative method (modified Lavrentiev) with Landweber iterative method. The numerical testing shows the efficiency of the new iterative method in its application to resolve the inverse heat equation when trying to find the boundary value function. •Studying of new iteration algorithm and mathematical experimentations show the efficiency of the new iteration method.•Iteration method is depended on decomposed the main linear operator by using polar decomposition in order to obtain unitary operator.•The new unitary operator increases the convergence of iteration.

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.