Square smoothing regularization matrices with accurate boundary conditions

This paper is concerned with the solution of large-scale linear discrete ill-posed problems. The determination of a meaningful approximate solution of these problems requires regularization. We discuss regularization by the Tikhonov method and by truncated iteration. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed ap- proximate solution. The present paper describes the construction of square reg- ularization matrices from finite difference equations with a focus on the bound- ary conditions. The regularization matrices considered have a structure that makes them easy to apply in iterative methods, including methods based on the Arnoldi process. Numerical examples illustrate the properties and effectiveness of the regularization matrices described.