José Marín
Preconditioners for nonsymmetric linear systems with low-rank skew-symmetric part

Institut de Matematica Multidisciplinar
Universitat Politecnica de Valencia
Camí de Vera
14
46022 Valencia
Spain
jmarinma@imm.upv.es
Juana Cerdán
Danny Guerrero
José Mas

In this work we study the iterative solution of nonsingular, nonsymmetric linear systems of $n$ equations

$$Ax=b$$ (1)

where the skew-symmetric part of the coefficient matrix $A$ can be approximated by a low-rank matrix. Consider $A=H+K$ where $H$ and $K$ are the symmetric and skew-symmetric parts of $A$ , respectively. It is assumed that the skew-symmetric matrix can be written as $K=PQ^T + E$ for some full rank $P,Q \in \mathbb{R}^{n \times s}$ with $k \ll n$ , and $\Vert E \Vert \ll 1$ .

Different strategies have been proposed when the skew-symmetric part $K$ has exactly rank $s \ll n$ . In [1] it is presented a progressive GMRES method that allows for the short-term computation of an orthogonal Krylov subspace basis. As pointed out in [5], although the method is mathematically equivalent to full GMRES [6] in practice it may suffer from instabilities due to the loss of orthogonality between the vectors of the generated Krylov subspace basis. In the same paper, the authors propose a Schur complement method that also permits the application of short-term formulas. The method obtains an approximate solution by applying the MINRES method $s+1$ times. The authors also suggest that can be successfully applied as a preconditioner for GMRES for the problem considered in this paper.

We study a method based on the framework proposed in [4]. Assuming that the matrix $H+PQ^T$ is nonsingular, our approach computes an approximate LU factorization of the matrix

$$\left[ \begin{array}{cc} H & P\ -Q^T & I \end{array} \right]$$ (2)

with the Balanced Incomplete Factorization (BIF) algorithm [2,3]. Interestingly, the matrix in (2) is similar to the one used in [5] to develop the Schur complement method, but in this work it is used to update a previously computed preconditioner for the symmetric part $H$ . Then, the factorization is used as a preconditioner for the GMRES method. The results of the numerical experiments for different problems will be presented.