Abstract
The numerical solution of linear discrete ill-posed problems typically requires regularization. Two of the most popular regularization methods are due to Tikhonov and Lavrentiev. These methods require the choice of a regularization matrix. Common choices include the identity matrix and finite difference approximations of a derivative operator. It is the purpose of the present paper to explore the use of fractional powers of the matrices \(A^\mathrm{T}\!A\) (for Tikhonov regularization) and A (for Lavrentiev regularization) as regularization matrices, where A is the matrix that defines the linear discrete ill-posed problem. Both small- and large-scale problems are considered.
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Acknowledgments
M. E. Hochstenbach was supported by an NWO Vidi Grant. L. Reichel was supported in part by NSF Grant DMS-1115385.
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Hochstenbach, M.E., Noschese, S. & Reichel, L. Fractional regularization matrices for linear discrete ill-posed problems. J Eng Math 93, 113–129 (2015). https://doi.org/10.1007/s10665-013-9671-4
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DOI: https://doi.org/10.1007/s10665-013-9671-4