Preconditioning complex symmetric linear systems

A new polynomial preconditioner for symmetric complex linear systems based on Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear systems is herein presented. It applies to Conjugate Orthogonal Conjugate Gradient (COCG) or Conjugate Orthogonal Conjugate Residual (COCR) iterative solvers and does not require any estimation of the spectrum of the coefficient matrix. An upper bound of the condition number of the preconditioned linear system is provided. Moreover, to reduce the computational cost, an inexact variant based on incomplete Cholesky decomposition or orthogonal polynomials is proposed. Numerical results show that the present preconditioner and its inexact variant are efficient and robust solvers for this class of linear systems. A stability analysis of the method completes the description of the preconditioner.Comment: 26 pages, 4 figures, 4 table

Matrix-equation-based strategies for convection-diffusion equations

We are interested in the numerical solution of nonsymmetric linear systems arising from the discretization of convection-diffusion partial differential equations with separable coefficients and dominant convection. Preconditioners based on the matrix equation formulation of the problem are proposed, which naturally approximate the original discretized problem. For certain types of convection coefficients, we show that the explicit solution of the matrix equation can effectively replace the linear system solution. Numerical experiments with data stemming from two and three dimensional problems are reported, illustrating the potential of the proposed methodology

A DELAYED FREQUENCY PRECONDITIONER APPROACH FOR SPEEDING-UP FREQUENCY RESPONSE COMPUTATION OF STRUCTURAL COMPONENTS

In this work, a delayed frequency preconditioner (DFP) is developed and applied in structural problems for speeding-up the frequency response computation. The challenge of computing the frequency response lies in the computation of the linear system that involves the excitation forces and also the dynamic stiffness which is frequency-dependent. For each frequency, the dynamic stiffness must be updated and a new factorization must be performed, which introduces a high computational cost on the solutions of the linear systems. Alternatively, iterative solver such as GMRES can be applied to avoid the cost of factorization, however they require good preconditioners that are traditionally also frequency-dependent. In the new approach, the dynamic stiffness operator is updated with the frequency whereas the preconditioner is kept constant for a range of frequencies serving as a low-cost preconditioner for the iterative solver. This technique saves computation time because a new factorization is avoided for each frequency point. On the other hand, the effectiveness of the delayed preconditioner is destroyed when the frequency of the dynamic operator is too far away from each other. Therefore, we propose a heuristic approach to update the preconditioner when it is underperforming. The algorithm is tested on structural problems and the results show that this approach can drastically reduce the number of iterations for the computation of the frequency response