Algebraic Multigrid method of Kaczmarz method

We are concerned with the algebraic multigrid (AMG) method for least square problem arisen from image restorations. We employ the Kaczmarz’s method as the smoothers for the AMG and prove the corresponding smoothing property.Abstract vii 1 Introduction 1 2 CT and Radon transform 3 2.1 CT and Radon transform . . . . . . . . . . . . . . . . . . . . . 3 2.2 Discrete Radon transform . . . . . . . . . . . . . . . . . . . . 4 3 Kaczmarz’s Method and SOR method 7 3.1 Kaczmarz’s Method . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 SOR method . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.3 Kaczmarz method : a variant of SOR . . . . . . . . . . . . . . 10 3.4 Kaczmarz’s method for inverse Radon transform . . . . . . . . 12 4 Smoothing property of Kaczmarz’s method 15 4.1 Smoothing property of SOR method . . . . . . . . . . . . . . 16 4.2 Classical result . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.3 Consistent case . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.4 Inconsistent case . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 Algebraic Multigrid for general inconsistnet linear system 2