Regularization Methods for large discrete ill-posed problems

The numerical solution of large-scale ill-posed problems is an interesting and challenging topic which has receive a lot of attention in the past decade. The use of Tikhonov regularization requires the determination of a suitable value of the regularization parameter. When the dimensions of the problem are so large as to make factorization of the matrix impossible or infeasible, the selection of a suitable value of the regularization parameter may require that the original problem is solved over and over again for different test values. We recently proposed that instead of exact quantities, difficult to compute, upper and lower bounds are used to determine suitable values of the regularization parameter.

The technique leads to the construction of “ribbons”' and has been shown to be effective under different initial conditions. Several aspects of this work have been carried out in collaboration with colleagues in USA (D. Calvetti, L.Reichel). Most recently, we have extended the methodology to the case where the solution is required to have nonnegative entries. The proposed methods is more efficient that others recently proposed for the solution of this problem.

Another way of damping the disastrous effects of amplified errors in the data on the computed solution is to use iterative methods equipped with suitable stopping rules. When successful, this approach is typically faster than using Tikhonov regularization and may yield solutions of higher quality.

Iterative methods for the solution of linear systems of equations produce a sequence of approximate solutions. In many applications it is desirable to be able to compute estimates of the norm of the error in the approximate solutions generated and terminate the iterations when the estimates are sufficiently small. The research has focused on new iterative methods based on the Lanczos process for the solution of linear systems of equations as well as on the computation of bounds and estimates of the norm of the error in the approximate solutions. These estimates are determined by evaluating certain Gauss, anti-Gauss, or Gauss-Radau quadrature rules.

These problems are very important in several applications, in particular to image processing.