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On the choice of subspace for iterative methods for linear discrete ill-posed problems

100%
Calvetti D., Lewis B., Reichel L.
International Journal of Applied Mathematics and Computer Science
|
2001
|
tom 11
|
nr 5
1069-1092
EN
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.
2
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Regularization parameter selection in discrete ill-posed problems - the use of the U-curve

88%
Krawczyk-Stańdo D., Rudnicki M.
International Journal of Applied Mathematics and Computer Science
|
2007
|
tom 17
|
nr 2
157-164
EN
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 α 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 α, based on the so-called U-curve. A comparison of the two methods made on numerical examples is additionally included.
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A discrepancy principle for Tikhonov regularization with approximately specified data

75%
Thamban Nair M., Schock E.
Annales Polonici Mathematici
|
1998
|
tom 69
|
nr 3
197-205
EN
Many discrepancy principles are known for choosing the parameter α in the regularized operator equation $(T*T + αI)x_α^δ = T*y^δ$, $|y - y^δ| ≤ δ$, in order to approximate the minimal norm least-squares solution of the operator equation Tx = y. We consider a class of discrepancy principles for choosing the regularization parameter when T*T and $T*y^δ$ are approximated by Aₙ and $zₙ^δ$ respectively with Aₙ not necessarily self-adjoint. This procedure generalizes the work of Engl and Neubauer (1985), and particular cases of the results are applicable to the regularized projection method as well as to a degenerate kernel method considered by Groetsch (1990).
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