Multigrid methods for matrices with structure and applications in image processing

Multigrid methods are among the fastest algorithms for the solution of linear systems of equations Ax=b. For many problems the computational efforts for the multigrid solution of the linear system are of the same complexity as the multiplication of a vector with the matrix A. This thesis deals with multigrid algorithms for structured linear systems. Particular focus is put on Toeplitz matrices, i.e. matrices with entries constant along diagonals. For the case of nonnegative generating functions with a finite number of zeros of finite order new multigrid algorithms are proposed and efficiently implemented. It is pointed out why these algorithms are computationally superior to existing approaches. Imaging applications are the most important practical source of Toeplitz systems. For matrices arising from image deblurring a multigrid algorithm employing a natural coarse grid operator is implemented which improves on an existing approach by R.Chan, T.Chan and J.Wan. Finally, the work takes a look at the problem of high-resolution image reconstruction with multisensors. For the arising sparse linear systems a new method of optimal computational complexity is proposed, analyzed and integrated into a powerful software package.