Abstract
We present a survey of results on the optimal discretization of ill-posed problems obtained in the Institute of Mathematics of the Ukrainian National Academy of Sciences.
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References
N. S. Bakhvalov, “On optimal methods for specifying information in the solution of differential equations,” Zh. Vych. Mat. Mat.Fiz., 2, No. 4, 569–592 (1962).
K. I. Babenko, “On the approximation of periodic functions of many variables by trigonometric polynomials,” Dokl. Akad. Nauk SSSR, 132, No. 2, 247–250 (1960).
J. F. Traub and H. Wozniakowski, A General Theory of Optimal Algorithms, Academic Press, New York 1980.
J. F. Traub, G. W. Wasilkowski, and H. Wozniakowski, Information. Uncertainty. Complexity, Addison-Wesley, London 1983.
N. P. Komeichuk, “Complexity of approximation problems,” Ukr. Mat. Zh., 48, No. 12, 1683–1694 (1996).
N. P. Komeichuk, “On complexity of approximation problems,” E. J. Approxim., 3, No. 3, 251–273 (1997).
S. V. Pcrcverzev, “On the complexity of the problem of solution of Fredholm equations of the second kind with smooth kernels. 1,” Ukr. Mat. Zh., 40, No. 1, 84–91 (1988).
S. V. Pcrcverzev, “On the complexity of the problem of solution of Fredholm equations of the second kind with smooth kernels. II,” Ukr. Mat. Zh., 41, No. 2, 189–193 (1989).
S. V. Pcrcverzev, “Hyperbolic cross and the complexity of approximate solution of integral Fredholm equations of the second kindwith dil’fercntiable kernels,” Sib. Mat. Zh., 32, No. 1, 107–115 (1991).
S. Pcrcverzev and C. Scharipov, “Information complexity of equations of second kind with compact operators in Hilbert space,” J. Complexity, 8, 176–202 (1992).
S. Heinrich, “Random approximation in numerical analysis,” in: Functional Analysis, New York (1994), pp. 123–171.
S. V. Pereverzcv and M. Azizov, “On optimal methods for specifying information in the solution of integral equations with analytickernels,” Ukr. Mat. Zh., 48, No. 5, 656–665 (1996).
K. Frank, S. Heinrich, and S. Pcreverzev, “Information complexity of multivariate Fredholm integral equations in Sobolev classes,” J. Complexity. 12. 17–34 (1996).
K. Frank. Optimal Numerical Solution of Multivariate Integral Equations, Shaker, Aachen 1997.
S. G. Solodkii, “On the information complexity of some classes of operator equations,” Ukr. Mat. Zh., 49, No. 9, 1271–1277 (1997).
V. N. Temlyakov, “Approximation of functions with bounded mixed derivatives,” Tr. Mat. Inst. Akad. Nauk SSSR, 128, 3–112 (1986).
G. M. Vainikko and A. Yu. Veretcnnikov, Iterative Procedures in Ill-Posed Problems [in Russian] Nauka, Moscow 1986.
V. K. Ivanov, V. V. Vasin, and V. PTanana. Theory of Linear III-Posed Problems and Its Applications (in Russian], Nauka, Moscow (1978).
A. N. Tikhonov and V. Ya. Arsenin, Methods for Solution of III-Posed Problems [in Russian] Nauka, Moscow 1979.
A. V. Goncharskii, A. S. Leonov, and A. G. Yagola, “Finite-difference approximation of linear ill-posed problems,” Zh. Vych. Mat.Mat. Fiz., 14, No. 1, 15–24 (1974).
A. B. Bakushinskii, “A general procedure for the construction of regularizing algorithms for linear ill-posed equations in a Hilbertspace,” Zh. Vych. Mat. Mat. Fiz., 7, No. 3, 672–677 (1967).
R. Plato and G. Vainikko, “On the rcgularization of projection methods in solving ill-posed problems.” Numer. Math., 57, 63–70 (1990).
S. V. Pereverzcv and S. G. Solodkii, “The minimal radius of Galerkin information for the Fredholm problem of the first kind,” J. Complexity. 12, 176–202 (1996).
S. G. Solodkii, “Information complexity of projective algorithms for the solution of Fredholm equations of the first kind. II,” Ukr. Mat. Zh., 50, No. 6, 838–844 (1998).
S. G. Solodkii, “Optimization of projective methods for the solution of linear ill-posed problems,” Zh. Vych. Mat. Mat. Fiz., 39, No. 2, 195–203 (1999).
S. G. Solodkii, “Optimization of projective schemes of discretization of ill-posed problems,” Ukr. Mat. Zh., 51, No. 10, 1398–1410 (1999).
V. A. Morozov, “On the choice of a parameter in the solution of functional equations by the regularization method,” Dokl. Akad.Nauk SSSR, 175, No. 6, 1225–1228 (1997).
P. Maass and A. Rieder, “Wavelet-accelerated Tikhonov regularization with applications,” in: H. W. Engl, A. K. Louis, and W. Rundell (editors), Inverse Problems in Med. Imaging, Springer, Vienna (1997), pp. 134–158.
S. V. Pereverzev and S. Prossdorf, “A discretization of Volterra integral equations of the third kind with weakly singular kernels,” J. Inverse Ill-Posed Probl, 5, 565–577 (1998).
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Pereverzev, S.V., Solodkii, S.G. Optimal discretization of Ill-posed problems. Ukr Math J 52, 115–132 (2000). https://doi.org/10.1007/BF02514141
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DOI: https://doi.org/10.1007/BF02514141