A Robust Approximation of the Schur Complement Preconditioner for an Efficient Numerical Solution of the Elliptic Optimal Control Problems

Abstract

In this paper, we consider the numerical solution of the optimal control problems of the elliptic partial differential equation. Numerically tackling these problems using the finite element method produces a large block coupled algebraic system of equations of saddle point form. These systems are of large dimension, block, sparse, indefinite and ill conditioned. The solution of such systems is a major computational task and poses a greater challenge for iterative techniques. Thus they require specialised methods which involve some preconditioning strategies. The preconditioned solvers must have nice convergence properties independent of the changes in discretisation and problem parameters. Most well known preconditioned solvers converge independently of mesh size but not for the decreasing regularisation parameter. This work proposes and extends the work for the formulation of preconditioners which results in the optimal performances of the iterative solvers independent of both the decreasing mesh size and the regulation parameter. In this paper we solve the indefinite system using the preconditioned minimum residual method. The main task in this work was to analyse the 3 × 3 block diagonal preconditioner that is based on the approximation of the Schur complement form obtained from the matrix system. The eigenvalue distribution of both the proposed Schur complement approximate and the preconditioned system will be investigated since the clustering of eigenvalues points to the effectiveness of the preconditioner in accelerating an iterative solver. This is done in order to create fast, efficient solvers for such problems. Numerical experiments demonstrate the effectiveness and performance of the proposed approximation compared to the other approximations and demonstrate that it can be used in practice. The numerical experiments confirm the effectiveness of the proposed preconditioner. The solver used is robust and optimal with respect to the changes in both mesh size and the regularisation parameter.

Optimal Block Preconditioner for an Efficient Numerical Solution of the Elliptic Optimal Control Problems Using GMRES Solver

“…The most important observation is that the eigenvalue distribution of P D (S 2 ) −1 K, Q −1 S3 K and Q −1 1 K are clustered around 1 for smaller regularisation parameter. This agrees with the theoretical findings in [13,15] and the Theorem 3.1 for the eigenvalues of our proposed preconditioner. It is expected that for the decreasing regularisation parameter the inexact preconditioned GMRES by the preconditioners performs extremely well.…”

“…The preconditioners in Figure 1 have been considered because they were developed for that coefficient matric Equation 6. We start by extracting the results from [13] where the MIN-RES solver applied with the block diagonal preconditioners based on the Schur complement forms P BD (S 1 ), P BD (S 2 ), Q S3 for comparison purposes. We now give the numerical experiment results from the GMRES iterative solver preconditioned with the block preconditioners P BD (S 1 ), P BD (S 2 ), Q S3 , P BCT , P N , Q 1 for problems 4.1 and 4.2 to demonstrate that our proposed preconditioner Q 1 is applicable, competitive, robust and cost effective.…”

“…We first present the set of results of the numerical experiments obtained from using the preconditioned GMRES solver for solving Equation ( 6) with inexact approximations of the preconditioners. In this case all the subsystems are solved using a fixed 10 Chebyshev semi iterations [13,14,17,23] for the mass matrix M and two algebraic multigrid iterations with two pre-and post-smoothing steps of the Jacobi method for the matrix involving K.…”

“…In [13] the block diagonal preconditioner was presented with the new approximation of the Schur complement approximation form S 3 = (…”

“…In this paper we focus on the last approach which benefit from the spectral distribution and the block form of the coefficient matrix. There are those preconditioners which follow the same structure of the coefficient matrix but based on the Schur complement form, we refer to [13,15,16,17,19,21] and references therein. The second class of preconditioners is based on the block structure and does not need approximations where the GMRES is applied.…”