# Methods for Solving Ill-Posed Extremum Problems with Optimal and Extra-Optimal Properties

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## Abstract

The notion of the quality of approximate solutions of ill-posed extremum problems is introduced and a posteriori estimates of quality are studied for various solution methods. Several examples of quality functionals which can be used to solve practical extremum problems are given. The new notions of the optimal, optimal-in-order, and extra-optimal qualities of a method for solving extremum problems are defined. A theory of stable methods for solving extremum problems (regularizing algorithms) of optimal-in-order and extra-optimal quality is developed; in particular, this theory studies the consistency property of a quality estimator. Examples of regularizing algorithms of extra-optimal quality for solving extremum problems are given.

## Keywords

ill-posed extremum problems regularizing algorithms quality of approximate solution a posteriori estimate of quality regularizing algorithm of extra-optimal quality## Preview

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## References

- 1.A. N. Tikhonov and V. Ya. Arsenin,
*Methods for Solving Ill-Posed Problems*(Nauka, Moscow, 1974; Wiley, New York, 1977).Google Scholar - 2.A.N. Tikhonov, A.S. Leonov, and A.G. Yagola,
*Nonlinear Ill-Posed Problems*(Nauka, Moscow, 1995; Chapman and Hall, London, 1998), Vols. 1 and 2.zbMATHGoogle Scholar - 3.V. A. Morozov, “Calculation of the lower bounds of functionals from approximate information,” Zh. Vychisl. Mat. i Mat. Fiz.
**13**(4), 1045–1049 (1973) [U. S. S. R. Comput. Math. and Math. Phys.**13**(4), 275–281 (1973)].Google Scholar - 4.A. N. Tikhonov and F. P. Vasil’ev, “Methods for solving ill-posed extremum problems,” in
*Mathematical Models and Numerical Methods*,*Banach Center Publ.*(PWN, Warsaw, 1978), Vol. 3, pp. 297–342.Google Scholar - 5.F. P. Vasil’ev,
*Methods for Solving Extremum Problems. Minimization Problems in function Spaces, Regularization, Approximation*(Nauka, Moscow, 1981) [in Russian].Google Scholar - 6.O. A. Liskovets,
*Variational Methods for Solving Unstable Problems*(Nauka i Tekhnika, Minsk, 1981) [in Russian].Google Scholar - 7.A. S. Leonov, “On an application of the generalized discrepancy principle for the solution of ill-posed extremum problems,” Dokl. Akad. Nauk SSSR
**262**(6), 1306–1310 (1982) [Soviet Math. Dokl.**25**, 227–231 (1982)].MathSciNetGoogle Scholar - 8.M. Yu. Kokurin, “Source conditions and accuracy estimates in Tikhonov’s scheme of solving ill-posed nonconvex optimization problems,” J. Inverse Ill-Posed Probl.
**26**(4), 463–475 (2018).MathSciNetzbMATHCrossRefGoogle Scholar - 9.M. Yu. Kokurin, “Necessary and sufficient conditions for power convergence rate of approximations in Tikhonov’s scheme for solving ill-posed optimization problems,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 6, 60–69 (2017) [Russian Math. (Iz. VUZ)
**61**(6), 51–59 (2017)].Google Scholar - 10.V. K. Ivanov, V. V. Vasin, and V. P. Tanana,
*Theory of Linear Ill-Posed Problems and Its Applications*(Nauka, Moscow, 1978) [in Russian].zbMATHGoogle Scholar - 11.V. P. Tanana,
*Methods for Solving Operator Equations*(Nauka, Moscow, 1981) [in Russian].zbMATHGoogle Scholar - 12.G. M. Vainikko,
*Methods for Solving Linear Ill-Posed Problems in Hilbert Spaces*(Izd. Tartu Gos. Univ., Tartu, 1982) [in Russian].Google Scholar - 13.V. A. Morozov,
*Regular Methods of Solving Ill-Posed Problems*(Nauka, Moscow, 1987) [in Russian].Google Scholar - 14.A. B. Bakushinskii and M. Yu. Kokurin,
*Algorithmic Analysis of Irregular Operator Equations*(Lenand, Moscow, 2012) [in Russian].zbMATHGoogle Scholar - 15.H. W. Engl, M. Hanke, and A. Neubauer,
*Regularization of Inverse Problems*(Kluwer Acad. Publ., Dordrecht, 1996).zbMATHCrossRefGoogle Scholar - 16.A. S. Leonov,
*Solution of Ill-Posed Inverse Problems. Outline of Theory, Practical Algorithms, and Demonstrations in MATLAB*(Librokom, Moscow, 2009) [in Russian].Google Scholar - 17.V. A. Vinokurov and Yu. L. Gaponenko, “A posteriori estimates of the solutions of ill-posed inverse problems,” Dokl. Akad. Nauk SSSR
**263**(2), 277–280 (1982) [Soviet Math. Dokl.**25**, 325–328 (1982)].MathSciNetzbMATHGoogle Scholar - 18.K. Yu. Dorofeev, V. N. Titarenko, and A. G. Yagola, “Algorithms for constructing a posteriori errors of solutions to ill-posed problems,” Zh. Vychisl. Mat. i Mat. Fiz.
**43**(1), 12–25 (2003) [Comput. Math. and Math. Phys.**43**(1), 10–23 (2003)].MathSciNetzbMATHGoogle Scholar - 19.A.G. Yagola, N.N. Nikolaeva, and V. N. Titarenko, “Error estimation for a solution to the Abel equation on sets of monotone and convex functions,” Sib. Zh. Vychisl. Mat.
**6**(2), 171–180 (2003).zbMATHGoogle Scholar - 20.A. B. Bakushinskii, “A posteriori error estimates for approximate solutions of irregular operator equations,” Dokl. Ross. Akad. Nauk
**437**(4), 439–440 (2011) [Dokl. Math.**83**(2), 192–193 (2011)].MathSciNetGoogle Scholar - 21.A. B. Bakushinsky, A. Smirnova, and H. Liu, “A posteriori error analysis for unstable models,” J. Inverse Ill-Posed Probl.
**20**(4), 411–428 (2012).MathSciNetzbMATHCrossRefGoogle Scholar - 22.A. B. Bakushinskii, and A. S. Leonov, “New a posteriori error estimates for approximate solutions of irregular operator equations,” Vych. Met. Programmirovanie
**15**(2), 359–369 (2014).Google Scholar - 23.A. S. Leonov, “A posteriori accuracy estimations of solutions to ill-posed inverse problems and extra-optimal regularizing algorithms for their solution,” Sib. Zh. Vychisl. Mat.
**15**(1), 83–100 (2012) [Numer. Anal. Appl.**5**(1), 68–83 (2012)].MathSciNetzbMATHGoogle Scholar - 24.A. S. Leonov, “Extra-optimal methods for solving ill-posed problems,” J. Inverse Ill-Posed Probl.
**20**(5–6), 637–665 (2012).MathSciNetzbMATHGoogle Scholar - 25.A. S. Leonov, “Pointwise extra-optimal regularizing algorithms,” Vych. Met. Programmirovanie
**14**(2), 215–228 (2013).MathSciNetGoogle Scholar - 26.A. S. Leonov, “Regularizing algorithms with optimal and extra-optimal quality,” Sib. Zh. Vychisl. Mat.
**19**(4), 371–383 (2016) [Numer. Anal. Appl.**9**(4), 288–298 (2016)].MathSciNetzbMATHGoogle Scholar - 27.E. Giusti,
*Minimal Surfaces and Functions of Bounded Variation*(Birkhauser Verlag, Basel, 1984).zbMATHCrossRefGoogle Scholar - 28.S. M. Nikol’skii,
*Approximation of Functions of Many Variables and Embedding Theorems*(Nauka, Moscow, 1969) [in Russian].Google Scholar - 29.S. J. Wernecke and L. R. D’Addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput.
**C-26**(4), 351–364 (1977).zbMATHCrossRefGoogle Scholar - 30.S. L. Sobolev,
*Several Applications of Functional Analysis in Mathematical Physics*(Nauka, Moscow, 1988) [in Russian].Google Scholar