A special generalized HSS method for discrete ill-posed problems
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A special generalized HSS method for discrete ill-posed problems H. Aminikhah1 · M. Yousefi1
Received: 2 October 2016 / Revised: 18 November 2016 / Accepted: 5 December 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract In this paper, we use a new special generalized Hermitian and skew-Hermitian splitting (SGHSS) method for solving ill-posed inverse problems. Based on an augmented system formulation, we apply a new splitting for the Hermitian part of the coefficient matrix and discuss on the convergence of this iterative method. Moreover, the optimal parameter minimizing the spectral radius of iteration matrix is derived. Finally, we present some numerical examples to demonstrate the performance of the presented method. Keywords Generalized Hermitian and skew-Hermitian splitting (GHSS) · Ill-posed problem · Tikhonov regularization · Image restoration Mathematics Subject Classification 65F10 · 65D18 · 65N20
1 Introduction Discrete ill-posed problems arise from the discretization of ill-posed problems such as Fredholm integral equations of the first kind (Groetsch 1984; Hansen 1997). Linear system of equations 2 2 2 A f = g, A ∈ Rn ×n , f, g ∈ Rn , (1.1) is often referred to as linear discrete ill-posed problem, where the matrix A is typically ill-conditioned. As it is known, in applications the right-hand side vector g might be contam-
Communicated by Andres Barrea.
B
H. Aminikhah [email protected]; [email protected] M. Yousefi [email protected]
1
Department of Applied Mathematics and Computer Sciences, Faculty of Mathematical Sciences, University of Guilan, P. O. Box 1914, Rasht, Iran
123
H. Aminikhah, M. Yousefi 2
inated by an error e ∈ Rn . Note that, the approximate solution of the system (1.1) is possibly not meaningful, due to the error e and ill-conditioning of matrix A. Thus, to determine a meaningful approximate solution, one typically replaces the linear system (1.1) by a nearby system that is less sensitive to the perturbations of the right-hand side. Therefore, the solution of this system is considered as an approximation of the solution of error-free linear system. This replacement is commonly referred to as regularization (Hansen 2010). Tikhonov regularization is one of the most popular regularization methods, see Phillips (1962), Tikhonov and Arsenin (1977), Tikhonov (1963a, b) for detailed discussions. In simplest form, Tikhonov regularization replaces the solution of the linear system (1.1) by the minimization problem min A f − g22 + μ2 f 22 , f
(1.2)
where μ > 0 is called the regularization parameter. We remark that for any μ > 0, the minimization problem (1.2) has a unique solution. It can be easily seen that the problem (1.2) is mathematically equivalent to solving the following equation (AT A + μ2 ) f = AT g. Moreover, the following 2n 2 -by-2n 2 augmented system, I A e g = , f 0 −AT μ2 I
(1.3)
(1.4)
is considered as a different approach to solve (1.3), where the auxiliary variable e represents the ad
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