Fourier analysis of frequency filtering decomposition preconditioners

Abstract

In this paper, frequency filtering decomposition (FFD) preconditioner is analyzed by the approach of Fourier analysis. The condition number estimation of a preconditioned 2-D model problem is presented. Analysis reveals that condition number of the preconditioned matrix grows like $\mathcal{O}(h^{-1})$, with h be the mesh size. By using the framework of FFD, a stabilized frequency filtering decomposition (SFFD) method is proposed and analyzed by Fourier method. Results show that SFFD preconditioner is superior to FFD preconditioner in the sense that $\kappa (M_{\mathit{SFFD}}^{-1}A)⩽\kappa (M_{\mathit{FFD}}^{-1}A)$. Numerical tests are performed to illustrate the theoretical results and the superiority of SFFD preconditioner.

Introduction

In many applications, we have to iteratively solve large sparse block tridiagonal linear systems of equations. The efficiency of an iterative solver depends strongly on the quality of preconditioners used. One class of preconditioners is based on incomplete factorization of the coefficient matrix. Examples of such kinds of preconditioners include the (block) ILU type preconditioners proposed by Meijerink and van der Vorst [28], Axelsson and co-workers [3], [4], [5], [16], Meurant [26], [27], and the incomplete QR type preconditioners, e.g., IGO explored in [6], [8] and MIQR proposed in [23]. Frequency filtering decomposition (FFD) preconditioner proposed by Wittum [39] is a special kind of incomplete block factorization preconditioner. It satisfies certain filtering condition, which is an important feature of the preconditioner. If a logarithmic sequences of the FFD preconditioners are used in a special combinative way, it is shown in [34], [39] that the FFD preconditioner can lead to h-independent convergence rate for solving a special elliptic problem. The FFD preconditioner has also been shown to be quite efficient for interface problems arising from the mortar element method [21], and it has been parallelized by combining FFD with a Schur complement domain decomposition method [37]. Following the framework of frequency filtering decomposition, several generalized and optimized frequency filtering type preconditioners have been proposed and widely investigated in [1], [2], [11], [12], [17], [18], [34], [35], [36].

Fourier analysis is an old technique for analyzing iterative methods or preconditioners [32]. It is used by Brandt [10] to estimate the convergence rate of multigrid iterations [9], [38]. The framework of Fourier analysis for iterative methods and preconditioners is established by Chan and Elman [13], and popularized by Chan and co-workers [14], [15], [26], Le Veque and Trefethen [22], and Otto [31]. For linear systems arising from the discretization of Poisson equation by standard second-order finite difference scheme with mesh size h, it has been illustrated that condition number of the preconditioned matrix is with order $\mathcal{O}(h^{-2})$ for ILU preconditioner [28], [33], and $\mathcal{O}(h^{-1})$ for modified ILU preconditioner (with a non-zero lumping term) [4], [20], relaxed ILU preconditioner with optimal relaxation parameter [14], [15] and relaxed nested factorization preconditioner [19]. Recently, Niu et al. proposed an optimal modification of the TFFD preconditioner [1] by the approach of Fourier analysis. With the optimal parameters, condition number of the preconditioned matrix is with order $\mathcal{O}(h^{-\frac{2}{3}})$ [29]. Moreover, for linear systems of equations generated from discretization of second-order self-adjoint elliptic-periodic boundary value problems, Bai et al. [7] derive a combinative preconditioner by combining incomplete Cholesky factorization and Sherman–Morrison–Woodbury update, and they illustrate that the preconditioned matrix has a condition number of order $\mathcal{O}(h^{-1})$.

In present paper, we intend to analyze the FFD preconditioner by the approach of Fourier analysis based on a representative 2-Dimensional Poisson equation. The eigenvalue distribution and condition number estimation of the preconditioned matrix are analyzed by the Fourier method. Results show that eigenvalues of the FFD preconditioned Dirichlet operator are always larger than 1, and the condition number grows like $\mathcal{O}(h^{-1})$. These properties can be predicted by Fourier analysis of an associated periodic problem. Spectrum distribution show that the lower part of Fourier eigenvalues match very well; though upper part of the Fourier eigenvalues are not so close to each other, their dependence on mesh size h is quite similar.

By introducing a lumping term ${\mathit{ch}}^2$ in the recursion formula of frequency filtering decomposition, we further introduce a stabilized frequency filtering decomposition (SFFD), and carry out Fourier analysis for the SFFD preconditioner. Analysis reveals that SFFD preconditioner is superior to FFD preconditioner in the sense that condition number of SFFD preconditioned matrix is smaller. Theoretical results are illustrated by numerical experiments. Performance of SFFD preconditioner is compared with block SSOR, FFD and MILU preconditioner on some problems generated from discretization of certain PDEs with discontinuous coefficients. Comparison results show that SFFD/FFD preconditioner is more robust, and SFFD preconditioner is superior to FFD preconditioner with properly chosen parameters.

In this paper, ${\mathit{ctrid}}_m(\alpha \text{,}\beta \text{,}\gamma )$ and ${\mathit{circ}}_m({\gamma }_1\text{,}\dots \text{,}{\gamma }_m)$ are used to denote circulant tridiagonal matrix and circulant matrix of order m:

$$\begin{array}{cc}\hfill & {\mathit{ctrid}}_m(\alpha \text{,}\beta \text{,}\gamma )=\left[\begin{array}{cccc}\hfill \beta \hfill & \hfill \gamma \hfill & \hfill \hfill & \hfill \alpha \hfill \\ \hfill \alpha \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \hfill \\ \hfill \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \gamma \hfill \\ \hfill \gamma \hfill & \hfill \hfill & \hfill \alpha \hfill & \hfill \beta \hfill \end{array}\right]\text{,}\hfill \\ \hfill & {\mathit{circ}}_m({\gamma }_1\text{,}\dots \text{,}{\gamma }_m)\left[\begin{array}{ccccc}\hfill {\gamma }_1\hfill & \hfill {\gamma }_2\hfill & \hfill \dots \hfill & \hfill {\gamma }_{m-1}\hfill & \hfill {\gamma }_m\hfill \\ \hfill {\gamma }_m\hfill & \hfill {\gamma }_1\hfill & \hfill {\gamma }_2\hfill & \hfill \dots \hfill & \hfill {\gamma }_{m-1}\hfill \\ \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {\gamma }_3\hfill & \hfill \dots \hfill & \hfill {\gamma }_m\hfill & \hfill {\gamma }_1\hfill & \hfill {\gamma }_2\hfill \\ \hfill {\gamma }_2\hfill & \hfill {\gamma }_3\hfill & \hfill \dots \hfill & \hfill {\gamma }_m\hfill & \hfill {\gamma }_1\hfill \end{array}\right]\text{;}\hfill \end{array}$$

and ${\mathit{trid}}_m(\alpha \text{,}\beta \text{,}\gamma )$ is used to denote $m\times m$ tridiagonal matrix:

$${\mathit{trid}}_m(\alpha \text{,}\beta \text{,}\gamma )=\left[\begin{array}{cc}\hfill \beta \hfill & \hfill \gamma \hfill \\ \hfill \alpha \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill \\ \hfill \hfill & \hfill \ddots \hfill & \hfill \ddots \hfill & \hfill \gamma \hfill \\ \hfill \hfill & \hfill \hfill & \hfill \alpha \hfill & \hfill \beta \hfill \end{array}\right]\text{.}$$