TY - EJOU
AU - Liu, Chein-Shan
TI - Optimally Generalized Regularization Methods for Solving Linear Inverse Problems
T2 - Computers, Materials \& Continua
PY - 2012
VL - 29
IS - 2
SN - 1546-2226
AB - In order to solve ill-posed linear inverse problems, we modify the Tikhonov regularization method by proposing three different preconditioners, such that the resultant linear systems are equivalent to the original one, without dropping out the regularized term on the right-hand side. As a consequence, the new regularization methods can retain both the regularization effect and the accuracy of solution. The preconditioned coefficient matrix is arranged to be **equilibrated** or **diagonally dominated** to derive the **optimal scales** in the introduced preconditioning matrix. Then we apply the iterative scheme to find the solution of ill-posed linear inverse problem. Two theorems are proved that the iterative sequences are monotonically convergent to the true solution. The presently proposed optimally generalized regularization methods are able to overcome the ill-posedness of linear inverse problems, and provide rather accurate numerical solution.
KW - Inverse Problem
KW - Ill-posed linear system
KW - Optimally generalized Tikhonov regularization method (OGTRM)
KW - Tikhonov regularization method (TRM)
KW - modified Tikhonov regularization method (MTRM)
KW - Generalized relaxed steepest descent method (GRSDM)
DO - 10.3970/cmc.2012.029.103