New preconditioners for systems of linear equations with Toeplitz structure

作者：Shi, Yong-Jie^*; Pi, Xue-Bo

来源：Calcolo, 2014, 51(1): 31-55.

DOI：10.1007/s10092-012-0074-0

• BTTB matrix
• Generating function
• Generalized Jackson kernel
• Preconditioner
• PCG method

In this paper, we consider applying the preconditioned conjugate gradient (PCG) method to solve system of linear equations where is a block Toeplitz matrix with Toeplitz blocks (BTTB). We first consider Level-2 circulant preconditioners based on generalized Jackson kernels. Then, BTTB preconditioners based on a splitting of BTTB matrices are proposed. We show that the BTTB preconditioners based on splitting are special cases of embedding-based BTTB preconditioners, which are also good BTTB preconditioners. As an application, we apply the proposed preconditioners to solve BTTB least squares problems. Our preconditioners work for BTTB systems with nonnegative generating functions. The implementations of the construction of the preconditioners and the relevant matrix-vector multiplications are also presented. Finally, Numerical examples, including image restoration problems, are presented to demonstrate the efficiency of our proposed preconditioners.

• 出版日期2014-3
• 单位汕头大学