We describe a block matrix iterative algorithm for solving a linearquadratic parabolic optimal control problem (OCP) on a finite time interval. We derive a reduced symmetric indefinite linear system involving the control variables and auxiliary variables, and solve it using a preconditioned MINRES iteration, with a symmetric positive definite block diagonal preconditioner based on the parareal ...

Saddle-point systems, i.e., structured linear systems with symmetric matrices are considered. A modified implementation of (preconditioned) MINRES is derived which allows subvectors of the residual to be monitored individually. Compared to the implementation from the textbook of [Elman, Silvester and Wathen, Oxford University Press, 2014], our method requires one extra vector of storage and no ...

Abstract In this paper we present general-purpose preconditioners for regularized augmented systems, and their corresponding normal equations, arising from optimization problems. We discuss positive definite preconditioners, suitable CG MINRES. consider “sparsifications" which avoid situations in eigenvalues of the preconditioned matrix may become complex. Special attention is given to systems ...

In this paper we consider the computation of an eigenvalue and corresponding eigenvector of a large sparse Hermitian positive definite matrix using inexact inverse iteration with a fixed shift. For such problems the large sparse linear systems arising at each iteration are often solved approximately by means of symmetrically preconditioned MINRES. We consider preconditioners based on the incomp...

We consider deflation and augmentation techniques for accelerating the convergence of Krylov subspace methods for the solution of nonsingular linear algebraic systems. Despite some formal similarity, the two techniques are conceptually different from preconditioning. Deflation (in the sense the term is used here) “removes” certain parts from the operator making it singular, while augmentation a...

For MinRes and SymmLQ it is essential to compute the QR decompositions of tridiagonal coefficient matrices gained in the Lanczos process. Likewise, for GMRes one has to find those of Hessenberg matrices. These QR decompositions are computed by an update scheme where in every step a single Givens rotation is constructed. Generalizing this approach we introduce a block-wise update scheme for the ...

Incomplete LDL factorizations sometimes produce an indefinite preconditioner evenwhen the input matrix is Hermitian positive definite. The two most popular iterative solvers for symmetric systems, CG and MINRES, cannot use such preconditioners; they require a positive definite preconditioner. One approach, that has been extensively studied to address this problem is to force positive definitene...

This work is devoted to fast and parameter-robust iterative solvers for frequency domain finite element equations, approximating the eddy current problem with harmonic or multiharmonic excitations in time. We construct a preconditioned MinRes solver for the frequency domain equations, that is robust with respect to the discretization parameters as well as all involved “bad” parameters like the ...

This work is devoted to distributed optimal control problems for time-periodic eddy current problems. We apply the multiharmonic approach to the optimality system and construct a new preconditioned MinRes solver for the system of frequency domain equations. We show that this solver is robust with respect to the space discretization and time discretization parameters as well as all involved “bad...