Arnoldi-Tikhonov regularization methods
Journal of Computational and Applied Mathematics
Tikhonov---Phillips regularization with operator dependent seminorms
Numerical Algorithms
FGMRES for linear discrete ill-posed problems
Applied Numerical Mathematics
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When GMRES (or a similar minimum-residual algorithm such as RRGMRES, MINRES, or MR-II) is applied to a discrete ill-posed problem with a square matrix, in some cases the iterates can be considered as regularized solutions. We show how to precondition these methods in such a way that the iterations take into account a smoothing norm for the solution. This technique is well established for CGLS, but it does not immediately carry over to minimum-residual methods when the smoothing norm is a seminorm or a Sobolev norm. We develop a new technique which works for any smoothing norm of the form $\|L\,x\|_2$ and which preserves symmetry if the coefficient matrix is symmetric. We also discuss the efficient implementation of our preconditioning technique, and we demonstrate its performance with numerical examples in one and two dimensions.