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Optimal discretization of Ill-posed problems

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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

  1. 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).

    Google Scholar 

  2. 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).

    Google Scholar 

  3. J. F. Traub and H. Wozniakowski, A General Theory of Optimal Algorithms, Academic Press, New York 1980.

    MATH  Google Scholar 

  4. J. F. Traub, G. W. Wasilkowski, and H. Wozniakowski, Information. Uncertainty. Complexity, Addison-Wesley, London 1983.

    MATH  Google Scholar 

  5. N. P. Komeichuk, “Complexity of approximation problems,” Ukr. Mat. Zh., 48, No. 12, 1683–1694 (1996).

    Article  Google Scholar 

  6. N. P. Komeichuk, “On complexity of approximation problems,” E. J. Approxim., 3, No. 3, 251–273 (1997).

    Google Scholar 

  7. 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).

    Google Scholar 

  8. 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).

    Article  Google Scholar 

  9. 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).

    Google Scholar 

  10. S. Pcrcverzev and C. Scharipov, “Information complexity of equations of second kind with compact operators in Hilbert space,” J. Complexity, 8, 176–202 (1992).

    Article  MathSciNet  Google Scholar 

  11. S. Heinrich, “Random approximation in numerical analysis,” in: Functional Analysis, New York (1994), pp. 123–171.

  12. 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).

    Article  Google Scholar 

  13. K. Frank, S. Heinrich, and S. Pcreverzev, “Information complexity of multivariate Fredholm integral equations in Sobolev classes,” J. Complexity. 12. 17–34 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  14. K. Frank. Optimal Numerical Solution of Multivariate Integral Equations, Shaker, Aachen 1997.

    Google Scholar 

  15. S. G. Solodkii, “On the information complexity of some classes of operator equations,” Ukr. Mat. Zh., 49, No. 9, 1271–1277 (1997).

    Article  MathSciNet  Google Scholar 

  16. V. N. Temlyakov, “Approximation of functions with bounded mixed derivatives,” Tr. Mat. Inst. Akad. Nauk SSSR, 128, 3–112 (1986).

    Google Scholar 

  17. G. M. Vainikko and A. Yu. Veretcnnikov, Iterative Procedures in Ill-Posed Problems [in Russian] Nauka, Moscow 1986.

    Google Scholar 

  18. V. K. Ivanov, V. V. Vasin, and V. PTanana. Theory of Linear III-Posed Problems and Its Applications (in Russian], Nauka, Moscow (1978).

    Google Scholar 

  19. A. N. Tikhonov and V. Ya. Arsenin, Methods for Solution of III-Posed Problems [in Russian] Nauka, Moscow 1979.

    Google Scholar 

  20. 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).

    Google Scholar 

  21. 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).

    MathSciNet  Google Scholar 

  22. R. Plato and G. Vainikko, “On the rcgularization of projection methods in solving ill-posed problems.” Numer. Math., 57, 63–70 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  23. 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).

    Google Scholar 

  24. 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).

    Article  MathSciNet  Google Scholar 

  25. 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).

    MathSciNet  Google Scholar 

  26. S. G. Solodkii, “Optimization of projective schemes of discretization of ill-posed problems,” Ukr. Mat. Zh., 51, No. 10, 1398–1410 (1999).

    Article  MathSciNet  Google Scholar 

  27. 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).

    Google Scholar 

  28. 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.

    Google Scholar 

  29. 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).

    Article  MathSciNet  Google Scholar 

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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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