A family of block preconditioners for block systems
SIAM Journal on Scientific and Statistical Computing
FFT-based preconditioners for Toeplitz-block least squares problems
SIAM Journal on Numerical Analysis
A fast algorithm for block Toeplitz systems with tensor structure
Applied Mathematics and Computation
Conjugate Gradient Methods for Toeplitz Systems
SIAM Review
Finite element solution of boundary value problems: theory and computation
Finite element solution of boundary value problems: theory and computation
Digital Image Processing
Preconditioners for Ill-Conditioned Toeplitz Systems Constructed from Positive Kernels
SIAM Journal on Scientific Computing
Cosine transform based preconditioners for total variation deblurring
IEEE Transactions on Image Processing
| Hi-index | 0.48 |
We study the solutions of block Toeplitz systems Tmnx = b by using the preconditioned conjugate gradient (PCG) method. Here Tmn = Tm ⊗ Tn and Ti, i = m, n are Toeplitz matrices. In [X. Jin, Appl. Math. Comput. 73 (1995) 115-124], Jin introduced a fast algorithm for these systems by applying the PCG method. This fast algorithm allows a tensor problem to be reduced to a one-dimensional problem. It was proved that if the mn- by-mn system is well conditioned, then the PCG method converges superlinearly and only O(mn log mn) operations are required in solving the preconditioned system. However, only well-conditioned systems were considered in Jin, 1995. In this paper, we apply this fast algorithm with the {ω}-circulant preconditioners proposed in [D. Potts, G. Steidl, Preconditioners for Ill-Conditioned toeplitz matrices, BIT, to be appeared] to solve the ill-conditioned systems. Numerical results are included to illustrate the effectiveness of the algorithm for solving the preconditioned systems by using the PCG method. An application in image restoration is also given.