V.V. Vasin, V.V. Belyaev. Analysis of a regularization algorithm for a linear operator equation containing a discontinuous component of the solution ... P. 34-44

We study a linear operator equation that does not satisfy the Hadamard well-posedness conditions. It is assumed that the solution of the equation has different smoothness properties on different segments of its domain. More exactly, the solution is representable as the sum of a smooth and discontinuous components. The Tikhonov regularization method is applied for the construction of a stable approximate solution. In this method, the stabilizer is the sum of the Lebesgue norm and the smoothed $BV$-norm. Each of the functionals in the stabilizer depends only on one component and takes into account its properties. Convergence theorems are proved for the regularized solutions and their discrete approximations. It is shown that discrete regularized solutions can be found with the use of the Newton method and nonlinear analogs of $\alpha$-processes.

Keywords: ill-posed problem, regularization method, discontinuous solution, total variation, discrete approximation

Received April 18, 2019

Revised July 8, 2019

Accepted July 15, 2019

Vladimir Vasil’evich Vasin, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620083 Russia, email: vasin@imm.uran.ru

Vladimir Vasil’evich Belyaev, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620083 Russia, email: beliaev_vv@mail.ru

REFERENCES

1.   Gholami A., Hosseini S.M. A balanced combination of Tikhonov and total variation regularization for reconstruction of piecewise-smooth signal. Signal Processing, 2013, vol. 93, no. 7, pp. 1945–1960. doi: 10.1016/j.sigpro.2012.12.008 

2.   Gandes E.J., Romberg J., Tao T. Stable signal recovery from incomplete and inaccurate measurments. Pure Appl. Math., 2006, vol. 59, no. 8, pp. 1207–1223. doi: 10.1002/cpa.20124 

3.   Vasin V.V. Reconstruction of smooth and discontinuous components of solutions to linear ill-posed problems. Dokl. Math., 2013, vol. 87, no. 1, pp. 23–25. doi: 10.1134/S1064562413010146 

4.   Vasin V.V. Regularization of ill-posed problems by using stabilizers in the form of the total variation of a function and its derivatives. J. Inverse Ill-Posed Problem, 2016, vol. 24, no. 2, pp. 149–158. doi: 10.1515/jiip-2015-0050 

5.   Giusti E. Minimal surfaces functions of bounded variation. Basel, Birkh$\ddot{\mathrm{a}}$user, 1984, Ser. Monographs in Mathematics, vol. 80. doi: 10.1007/978-1-4684-9486-0 

6.   Acar R., Vogel C.R. Analysis of bounded variation penalty methods for ill-posed problems. Inverse Problems, 1994, vol. 10, no. 6, pp. 1217–1229. doi: 10.1088/0266-5611/10/6/003 

7.   Vasin V.V., Belyaev V.V. Approximation of solution components for ill-posed problems by the Tikhonov method with total variation. Dokl. Math., 2018, vol. 97, no. 3, pp. 266–270. doi: 10.1134/S1064562418030250 

8.   Vasin V.V., Belyaev V.V. Modification of the Tikhonov method under separate reconstruction of components of solution with various properties. Eurasian J. Math. Comput. Appl., 2017, vol. 5, iss. 2, pp. 66–79.

9.   Vainikko G. Functionalanalysis der Diskretisierungsmethoden. Leipzig: Teugner Verlag, 1976, 136 S. doi: 10.1002/zamm.19780580410 

10.   Grigorieff R.D. Zur Theorie Approximations regularer Operatoren. I; II. Mathematische Nachrichten, 1973, Bd. 55, Nr. 3, S. 233–249; S. 251–263. doi: 10.1002/mana.19730550113 .

11.   Stummel F. Diskrete Konvergentz linearer Operatoren. I Mathematische Annalen. 1970. Bd. 190, Nr. 1. S. 45–92. doi: 10.1007/BF01349967 ; Diskrete Konvergentz linearer Operatoren. II Mathematische Zeitschrift, 1971, Bd. 120, Nr. 3, S. 231–264.

12.   Vasin V.V. Regularization and iterative approximation for linear ill-posed problems in the space of functions of bounded variation. Proc. Steklov Inst. Math., 2002, Suppl. 1, pp. 225–239.

13.   Vasin V.V., Ageev A.L. Ill-posed problems with a priori information, Utrecht, The Netherlsnds, VSP, 1995. 255 p.

14.   Gajewski H., GrЈoger K., Zacharias K. Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Berlin, Akademie-Verlag, 1974. 336 s. Translated to Russian under the title Gaevsky H., GrЈoger K., Zakharias K, Nonlinear operator equations and operator differential equations, Moscow: Mir Publ., 1978. 336 p.

15.   Vasin V.V., Skurydina A.F. Two-stage method of construction of regularizing algorithms for nonlinear ill-posed problems. Proc. Steklov Inst. Math., 2018, vol. 301, suppl. 1, pp. 173–190. doi: 10.1134/S0081543818050152 

Cite this article as: V.V. Vasin, V.V. Belyaev. Analysis of a regularization algorithm for a linear operator equation containing a discontinuous component of the solution, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 3, pp. 34–44.