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Regularization parameter selection in discrete ill-posed problems—The use of the U-curve

Krawczyk-Stańdo D., Rudnicki M.

International Journal of Applied Mathematics and Computer Science

2007

Vol. 17, no 2

157-164

To obtain smooth solutions to ill-posed problems, the standard Tikhonov regularization method is most often used. For the practical choice of the regularization parameter \alfa we can then employ the well-known L-curve criterion, based on the L-curve which is a plot of the norm of the regularized solution versus the norm of the corresponding residual for all valid regularization parameters. This paper proposes a new criterion for choosing the regularization parameter \alfa, based on the so-called U-curve. A comparison of the two methods made on numerical examples is additionally included.

On the Choice of Subspace for Iterative Methods for Linear Discrete Ill-Posed Problems

Calvetti D., Lewis B., Reichel L.

International Journal of Applied Mathematics and Computer Science

2001

Vol. 11, no 5

1069-1092

Many iterative methods for the solution of linear discrete ill-posed problems with a large matrix require the computed approximate solutions to be orthogonal to the null space of the matrix. We show that when the desired solution is not smooth, it may be possible to determine meaningful approximate solutions with less computational work by not imposing this orthogonality condition.