Abstract
Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares problems with a severely ill-conditioned matrix A. This method replaces the given problem by a penalized least-squares problem. The present paper discusses measuring the residual error (discrepancy) in Tikhonov regularization with a seminorm that uses a fractional power of the Moore-Penrose pseudoinverse of AA T as weighting matrix. Properties of this regularization method are discussed. Numerical examples illustrate that the proposed scheme for a suitable fractional power may give approximate solutions of higher quality than standard Tikhonov regularization.
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Communicated by Lars Eldén.
Fröberg, Björck, Ruhe: A Golden Braid for 50 Years of BIT.
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Hochstenbach, M.E., Reichel, L. Fractional Tikhonov regularization for linear discrete ill-posed problems. Bit Numer Math 51, 197–215 (2011). https://doi.org/10.1007/s10543-011-0313-9
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DOI: https://doi.org/10.1007/s10543-011-0313-9
Keywords
- Ill-posed problem
- Regularization
- Fractional Tikhonov
- Weighted residual norm
- Filter function
- Discrepancy principle
- Solution norm constraint