Separate reconstruction of solution components with singularities of various types for linear operator equations of the first kind. (English. Russian original) Zbl 1319.47008
Proc. Steklov Inst. Math. 289, Suppl. 1, S216-S226 (2015); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 20, No. 2, 63-73 (2014).
Summary: A linear operator equation of the first kind is investigated. The solution of this equation contains singularities of various types; namely, along with a smooth background, the solution has sharp bends and jump discontinuities. For the construction of a stable approximated solution, a modified Tikhonov method with a stabilizer in the form of the sum of three functionals is proposed. Each of the functionals accounts for the specific character of the corresponding component of the solution. Convergence theorems are formulated, a general discrete approximation scheme of the regularizing algorithm is justified, and results of numerical experiments are discussed.
MSC:
| 47A52 | Linear operators and ill-posed problems, regularization |
| 65J20 | Numerical solutions of ill-posed problems in abstract spaces; regularization |
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