A generalized global Arnoldi method for ill-posed matrix equations

This paper discusses the solution of large-scale linear discrete ill-posed problems with a noise-contaminated right-hand side. Tikhonov regularization is used to reduce the influence of the noise on the computed approximate solution. We consider problems in which the coefficient matrix is the sum of Kronecker products of matrices and present a generalized global Arnoldi method, that respects the structure of the equation, for the solution of the regularized problem. Theoretical properties of the method are shown and applications to image deblurring are described.

[1]  H. Sadok,et al.  Global FOM and GMRES algorithms for matrix equations , 1999 .

[2]  Lothar Reichel,et al.  Tikhonov regularization based on generalized Krylov subspace methods , 2012 .

[3]  K. Jbilou,et al.  Sylvester Tikhonov-regularization methods in image restoration , 2007 .

[4]  N. Pitsianis.  The Kronecker Product in Approximation and Fast Transform Geration , 1997 .

[5]  A. N. Tikhonov,et al.  REGULARIZATION OF INCORRECTLY POSED PROBLEMS , 1963 .

[6]  Per Christian Hansen,et al.  Analysis of Discrete Ill-Posed Problems by Means of the L-Curve , 1992, SIAM Rev..

[7]  C. W. Groetsch,et al.  The theory of Tikhonov regularization for Fredholm equations of the first kind , 1984 .

[8]  C. Loan,et al.  Approximation with Kronecker Products , 1992 .

[9]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[10]  Lothar Reichel,et al.  SIMPLE SQUARE SMOOTHING REGULARIZATION OPERATORS , 2009 .

[11]  G. Rodriguez,et al.  An algorithm for estimating the optimal regularization parameter by the L-curve , 2005 .

[12]  G. Golub,et al.  Good Ridge Parameter , 1979 .

[13]  H. Sadok,et al.  ALGEBRAIC PROPERTIES OF THE BLOCK GMRES AND BLOCK ARNOLDI METHODS , 2009 .

[14]  Qiang Ye,et al.  A Krylov Subspace Method for Quadratic Matrix Polynomials with Application to Constrained Least Squares Problems , 2003, SIAM J. Matrix Anal. Appl..

[15]  J. Nagy,et al.  KRONECKER PRODUCT AND SVD APPROXIMATIONS IN IMAGE RESTORATION , 1998 .

[16]  Dianne P. O'Leary,et al.  The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems , 1993, SIAM J. Sci. Comput..

[17]  Gene H. Golub,et al.  Matrix computations , 1983 .

[18]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .