Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems

Abstract

The governing dynamics of fluid flow is stated as a system of partial differential equations referred to as the Navier-Stokes system. In industrial and scientific applications, fluid flow control becomes an optimization problem where the governing partial differential equations of the fluid flow are stated as constraints. When discretized, the optimal control of the Navier-Stokes equations leads to large sparse saddle point systems in two levels.In this paper we consider distributed optimal control for the Stokes system and test the particular case when the arising linear system can be compressed after eliminating the control function. In that case, a system arises in a form which enables the application of an efficient block matrix preconditioner that previously has been applied to solve complex-valued systems in real arithmetic. Under certain conditions the condition number of the so preconditioned matrix is bounded by 2. The numerical and computational efficiency of the method in terms of number of iterations and execution time is favorably compared with other published methods.

A robust structured preconditioner for time-harmonic parabolic optimal control problems

“…Remark 47 Analogously to the discussions in [3], utilizing the symmetric positive definiteness of S, we have seen that S −1 S II can be symmetrized by the similarity transformation S −1 S II = S str-II A II is a normal matrix and diagonalizable, which is beneficial in the Krylov subspace iteration framework.…”

“…For this case, there have been several efficient developments on preconditioned iterative techniques using Krylov subspace methods, see [3,4,27,28] and the reference therein. In this paper, we deal with the case ω = 0.…”

“…Recently, some special preconditioning and iterative techniques have appeared in the literature to solve the linear system (1.8), see [21,33,34]. In this paper, based on the works in [3,9], we construct a new structured two-by-two block preconditioner, which is called P str-II , for the linear system (1.9). The implementation of P str-II is straightforward and the computational complexity is of optimal order.…”

“…In this paper, for A II in (1.9), based on the works in [3,9], we consider the following structured two-by-two block preconditioner…”

“…Thus, it is diagonalizable, which is beneficial to the solution of preconditioned Krylov subspace methods. When ω = 0, the numerical experiments in [3,9] show that the performance of the preconditioner P str-I within a nonsymmetric solver such as GMRES is robust and totally parameter independent. However, when ω = 0, the matrices of the involved systems arising in the implementation of P str-I become complex, which increase the computational burden to implement P str-I .…”