A splitting preconditioner for saddle point problems

Abstract

For large sparse systems of linear equations iterative techniques are attractive. In this paper, we study a splitting method for an important class of symmetric and indefinite system. Theoretical analyses show that this method converges to the unique solution of the system of linear equations for all t>0 (t is the parameter). Moreover, all the eigenvalues of the iteration matrix are real and nonnegative and the spectral radius of the iteration matrix is decreasing with respect to the parameter t. Besides, a preconditioning strategy based on the splitting of the symmetric and indefinite coefficient matrices is proposed. The eigensolution of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomials for the preconditioned matrix is obtained. Numerical experiments of a model Stokes problem and a least-squares problem with linear constraints presented to illustrate the effectiveness of the method.

A preconditioned Multiple Shooting Shadowing algorithm for the sensitivity analysis of chaotic systems

“…A preconditioner based on the partial Krylov-Schur decomposition of a matrix (see [30] for details) was also used in [31] to accelerate the computation of limit cycles for thermoacoustic systems. The preconditioner had a form very similar to (27), the difference being that instead of the diagonal matrix Σ −2 1 in the first term on the right-hand side, the inverse of an upper triangular l × l matrix was taken. The preconditioner reduced the condition number significantly, speeded up the convergence of GMRES, and lead to modest overall cost savings (mainly due to the cost of converging the eigenvalues used to form the preconditioner).…”

“…Obtained for ρ = 80 with T = 50 and ∆T = 0.5 (K = 100 segments).values σ(Φ i ) approximate σ(A) better for the larger ∆T value. This is because the diagonal blocks of matrix A become more dominant for ∆T = 1 and the off-diagonal identity matrices can be neglected (refer to(30)) without impairing the accuracy of the large singular values.Next, we investigate the spectrum of the preconditioned system, starting with the exact preconditioner M (l)(27). The l largest singular values and vectors of S used to form M (l) have been obtained iteratively until convergence.Figure (4a) shows the eigenvalues of µ(S) and µ(M (l) S).For the case examined, the matrix S has approximately K = 100 eigenvalues µ(S) > 1 (blue line), corresponding to the local positive Lyapunov exponent in each segment.…”

Variants of the accelerated parameterized inexact Uzawa method for saddle-point problems

“…For example, the matrix splitting iterative methods [7,11,18,36,40,45], Uzawa-type methods [12,17,21,22,25], HSS method and its variants [2,[4][5][6]8,9,28,29,38], Krylov subspace methods [1,10,[33][34][35]42], null space methods [24] and so on. When the saddle-point problem (1) is singular, there are also many relaxation iteration methods which have been established, e.g., the Uzawa-type methods [30,46,48,49] and the HSS-like methods [3,20,39].…”

Preconditioning optimal control of the unsteady Burgers equations withH1regularized term

“…However, it is never used without the application of appropriate preconditioning, which is chosen to enhance the convergence behavior for the iterative method. In the last years, many efficient preconditioners have been introduced, e.g., see [3][4][5][6]8,9,13,21,25].…”