One Approach to the Comparison of Error Bounds at a Point and on a Set in the Solution of Ill-Posed Problems


The approximate solution of ill-posed problems by the regularization method always involves the issue of estimating the error. It is a common practice to use uniform bounds on the whole class of well-posedness in terms of the modulus of continuity of the inverse operator on this class. Local error bounds, which are also called error bounds at a point, have been studied much less. Since the solution of a real-life ill-posed problem is unique, an error bound obtained on the whole class of well-posedness roughens to a great extent the true error bound. In the present paper, we study the difference between error bounds on the class of well-posedness and error bounds at a point for a special class of ill-posed problems. Assuming that the exact solution is a piecewise smooth function, we prove that an error bound at a point is infinitely smaller than the exact bound on the class of well-posedness.

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  1. 1.

    V. K. Ivanov, V. V. Vasin, and V. P. Tanana, Theory of Linear Ill-Posed Problems and Its Applications (Nauka, Moscow, 1978; VSP, Utrecht, 2002).

    Google Scholar 

  2. 2.

    V. P. Tanana, “On a new approach to error estimation for methods for solving ill-posed problems,” Sib. Zh. Ind. Mat. 5 (4), 150–163 (2002).

    MathSciNet  MATH  Google Scholar 

  3. 3.

    V. P. Tanana, A. B. Bredikhina, and T. S. Kamaltdinova, “On an error estimate for an approximate solution of an inverse problem in the class of piecewise smooth functions,” Trudy Inst. Mat. Mekh. UrO RAN 18 (1), 281–288 (2012).

    Google Scholar 

  4. 4.

    V. P. Tanana and N. M. Yaparova, “An order-optimal method for solving conditionally well-posed problems,” Sib. Zh. Vychisl. Mat. 9 (4), 353–368 (2006).

    MATH  Google Scholar 

  5. 5.

    V. P. Tanana and T. N. Rudakova, “The optimum of the M. M. Lavrent’ev method,” J. Inv. Ill-Posed Probl. 18, 935–944 (2011). doi 10.1515/JIIP.2011.012

    MathSciNet  MATH  Google Scholar 

  6. 6.

    A. B. Bredikhina, “The nonlinear projection regularization method,” Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat. Model. Progr., No. 37, 4–10 (2011).

    Google Scholar 

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Correspondence to V. P. Tanana.

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Original Russian Text © V.P. Tanana, 2017, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2017, Vol. 23, No. 2, pp. 230–238.

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Tanana, V.P. One Approach to the Comparison of Error Bounds at a Point and on a Set in the Solution of Ill-Posed Problems. Proc. Steklov Inst. Math. 301, 155–163 (2018).

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  • ill-posed problem
  • regularization
  • estimation of the error at a point
  • estimation of the error on a set