A family of range restricted iterative methods for linear discrete ill-posed problems

The solution of large linear systems of equations with a matrix of ill-determined rank and an error-contaminated right-hand side requires the use of specially designed iterative methods in order to avoid severe error propagation. The choice of solution subspace is important for the quality of the computed approximate solution, because this solution typically is determined in a subspace of fairly small dimension. This paper presents a family of range restricted minimal residual iterative methods that modify the well-known GMRES method by allowing other initial vectors for the solution subspace. A comparison of the iterates computed by different range restricted minimal residual methods can be helpful for determining which iterates may provide accurate approximations of the desired solution. Numerical examples illustrate the competitiveness of the new methods and how iterates from different methods can be compared.