Geodesic
Effective algorithms for computing global and local posterior error estimates of solutions to linear ill-posed problems
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2020), pp. 29-38

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We consider the problems of calculating global and local a-posteriori error estimates of approximate solutions to ill-posed inverse problems, introduced and investigated earlier by the author. For linear inverse problems in Hilbert spaces, they consist in maximizing a quadratic functional with two quadratic constraints. The article shows how under certain conditions these problems can be reduced to a problem of maximizing a special (written analytically) differentiable functional with one constraint. New algorithms for calculating global and local a-posteriori error estimates based on the solution of these problems are proposed. Their effectiveness is illustrated by numerical experiments on a-posteriori error estimation of solutions to the model two-dimensional inverse problem of potential continuation. Experiments show that the proposed algorithms give a-posteriori error estimates close to the true error values. Proposed algorithms for global a-posteriori error estimation turn out to be more rapid (3 to 5 times) than the previously known algorithms.
Keywords: linear ill-posed problems, regularizing algorithms, a-posteriori error estimates. 
@article{IVM_2020_2_a3,
     author = {A. S. Leonov},
     title = {Effective algorithms for computing global and local posterior error estimates of solutions to linear ill-posed problems},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {29--38},
     publisher = {mathdoc},
     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2020_2_a3/}
}
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A. S. Leonov. Effective algorithms for computing global and local posterior error estimates of solutions to linear ill-posed problems. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2020), pp. 29-38. http://geodesic.mathdoc.fr/item/IVM_2020_2_a3/