The Research And Application Of Multigrid Techniques And Preconditoning Techniques

Multigrid techniques and preconditioning techniques are important tools for solvinglarge linear systems. Multigrid techniques can reduce calculation for solving the linearsystems by using a hierarchy of meshes. Multigrid methods for specific physical prob-lems and using multigrid methods as a preconditioner are hot research fields recently.Preconditioning techniques used for accelerating the convergence of iterative methodshave attracted considerable attentions. The development of preconditioning techniquesfor specific issues has become current research topics in the field of numerical algebra.We focus on the application of multigrid techniques and preconditioning techniquesfor image restoration problems. Image deblurring is one of the most classic linear inverseproblems. The linear system obtained by the discretization of the problem model is ill-posed. Because the blurred images are contaminated by the noise, it is difcult to achieverestored images with high quality.Multigrid techniques have been considered for solving image restoration problem-s to improve the regularization property of the iterative method. In2010, Espa n lo andKilmer proposed an efective multilevel approach. Restored images obtained by this al-gorithm with optimal parameters are high-quality. We propose a modified variant forthis algorithm by comparing to the classic multigrid approach. The calculation slightlyincreases by adding a few postsmooth steps to the residual correction step. Numericalexperiments show modified algorithm is not sensitive to the choice of the parameter incoarsest level, so it is easier to choose the parameter. The quality of restored images isalso better.Iterative regularization methods are efcient regularization methods for image restora-tion problems. The IDR(s) and LSMR methods are state-of-arts iterative methods forsolving large linear systems. They have recently attracted considerable attention. Littleis known of them as iterative regularization methods for image restoration. In this pa-per, we study the regularization properties of the IDR(s) and LSMR methods for imagerestoration problems. Comparative numerical experiments show that IDR(s) can givea satisfactory solution with much less computation in some situations than the classic method LSQR when the discrepancy principle (DP) is used as stopping criterion. If thenoise norm is unknown, LSMR more likely produces a more accurate solution by usingthe L-curve method to choose the regularization parameter.