@Article{IJNAM-20-1,
author = {Muzhinji, K.},
title = {Optimal Block Preconditioner for an Efficient Numerical Solution of the Elliptic Optimal Control Problems Using GMRES Solver},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2023},
volume = {20},
number = {1},
pages = {47--66},
abstract = {<p style="text-align: justify;">Optimal control problems are a class of optimisation problems with partial differential equations as constraints. These problems arise in many application areas of science and
engineering. The finite element method was used to transform the optimal control problems of
an elliptic partial differential equation into a system of linear equations of saddle point form.
The main focus of this paper is to characterise and exploit the structure of the coefficient matrix
of the saddle point system to build an efficient numerical process. These systems are of large
dimension, block, sparse, indefinite and ill conditioned. The numerical solution of saddle point
problems is a computational task since well known numerical schemes perform poorly if they are
not properly preconditioned. The main task of this paper is to construct a preconditioner the
mimic the structure of the system coefficient matrix to accelerate the convergence of the generalised minimal residual method. Explicit expression of the eigenvalue and eigenvectors for the
preconditioned matrix are derived. The main outcome is to achieve optimal convergence results
in a small number of iterations with respect to the decreasing mesh size $h$ and the changes in $δ$ the regularisation problem parameters. The numerical results demonstrate the effectiveness and
performance of the proposed preconditioner compared to the other existing preconditioners and
confirm theoretical results.</p>},
issn = {2617-8710},
doi = {https://doi.org/10.4208/ijnam2023-1003},
url = {https://global-sci.com/article/82885/optimal-block-preconditioner-for-an-efficient-numerical-solution-of-the-elliptic-optimal-control-problems-using-gmres-solver}
}