Some topics concerning the singular value decomposition and generalized singular value decomposition
The contributions of this work include a detailed comparison between standard and divergence-free radial basis approximations, a study of the Lebesgue constants for divergence-free approximations and their dependence on node placement, and an investigation of the flat limit of divergence-free interpolants. Finally, numerical solvers for the incompressible Navier-Stokes equations in primitive variables are implemented using discretizations based on traditional and divergence-free kernels. The numerical results are compared to reference solutions obtained with a spectral
method.
Tikhonov regularization for projected solutions of large-scale ill-posed problems is considered. The Golub{Kahan iterative bidiagonalization is used to project the problem onto a subspace and regularization then applied to nd a subspace approximation to the full problem. Determination of the regularization, parameter for the projected problem by unbiased predictive risk estimation, generalized cross validation, and discrepancy principle techniques is investigated. It is shown that the regularized parameter obtained by the unbiased predictive risk estimator can provide a good estimate which can be used for a full problem that is moderately to severely ill-posed. A similar analysis provides the weight parameter for the weighted generalized cross validation such that the approach is also useful in these cases, and also explains why the generalized cross validation without weighting is not always useful. All results are independent of whether systems are over- or underdetermined. Numerical simulations for standard one-dimensional test problems and two- dimensional data, for both image restoration and tomographic image reconstruction, support the analysis and validate the techniques. The size of the projected problem is found using an extension of a noise revealing function for the projected problem [I. Hn etynkov a, M. Ple singer, and Z. Strako s, BIT Numer. Math., 49 (2009), pp. 669{696]. Furthermore, an iteratively reweighted regularization approach for edge preserving regularization is extended for projected systems, providing stabilization of the solutions of the projected systems and reducing dependence on the determination of the size of the projected subspace.