Numerical algorithms for solving an elliptic optimal control problem with a power-law nonlinearity

The paper is devoted to the development and analysis of approximation-iteration algorithms based on the method of grids and the method of lines for solving an elliptic optimal control problem with a power-law nonlinearity. For the numerical solution of the main boundary value problem and the adjoint one, the second order of accuracy difference schemes are applied using the implicit method of simple iteration. Computational schemes of the method of lines for solving the above-mentioned elliptic boundary value problems are implemented in combination with the shooting method for the approximate solution of boundary value problems for the corresponding ordinary differential equations systems arising in the considered domain after lattice approximation. To minimize the objective functional, well-known gradient-type methods (gradient projection and conditional gradient methods) of constrained optimization are used. The essence of the proposed approximation-iteration approach consists in replacing the original extremal problem with a sequence of grid problems that approximate it on a set of refining grids, and applying an iterative gradient-type method to each of the "approximate" extremal problems. In this case, we propose to construct only a few approximations to the solution for each of the "approximate" problems and to take the last of these approximations, using piecewise linear interpolation, as the initial approximation in the iterative process for the next "approximate" problem. The sequence of the corresponding piecewise linear interpolants is considered as a sequence of approximations to the solution of the original extremal problem. The paper discusses the theoretical foundations of this combined approach, as well as its advantages over traditional methods using the example of solving a model optimal control problem