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Methods of solving inverse problems

ДВС.1.07

Вибіркова дисципліна для ОП

First

2023/2024

8 Semester

6

LO 10. To know the methods of choosing rational methods and algorithms for solving mathematical problems of optimization, operations research, optimal management and decision-making, data analysis. PLO 21.1. To know the main sections of applied mathematics and informatics, to the extent necessary for mastering general professional mathematical disciplines, applied disciplines and the use of their methods in the chosen profession.

Full-time form

1. Successful completion of the course: mathematical analysis, functional analysis, linear algebra. 2. Know: basic concepts and facts of mathematical analysis, functional analysis and linear algebra. 3. Be able to: solve typical problems in mathematical analysis, functional analysis and linear algebra. 4. Have basic skills of searching for information on the Internet.

Part 1. Linear problems. 1 Topic 1. Correctness according to Hadamard. 2 Topic 2. Incorrect problems. Examples. Correctness according to Tikhonov. 3 Topic 3. Normal solvability of operator equations. 4 Topic 4. Quasi solutions. 5 Topic 5. Tikhonov's regularization method. 6 Topic 6. Moore-Penrose pseudo-inverse operator. 7 Topic 7. Direct-dual method. 8 Topic 8. Quasi-inversion method for heat conduction equation with inverse time. Part 2. Nonlinear problems 10 Topic 9. Incorrect optimization problems. 11 Topic 10. Two-level optimization problems. 12 Topic 11. The Tikhonov regularization method for optimization problems. 13 Topic 12. Bakushinsky's iterative regularization method. 14 Topic 13. Browder-Tikhonov scheme. 15 Topic 14. The direct dual method of Shambol-Pock. 16 Topic 15. Theorems about typicality.

1. Bauschke H.H., Combettes P.L. Convex Analysis and Monotone Operator Theory in Hilbert Spaces. – Springer, 2011. 2. Kinderlerer D., Stampakk-ia G. Vvedenie v variatsionnye neravenstva i ikh prilozheniia. – Moskva: Mir, 1983. 3. Gaevskii Kh., Greger K., Zakharias K. Nelineinye operatornye uravneniia i operatornye differentsial-nye uravneniia. – M.: Mir, 1978. 4. Sea Zh. Optimizatsiia. Teoriia i algoritmy. – M.: Mir, 1973. 5. Tikhonov A.N., Arsenin V.Ia. Metody resheniia nekorrektnykh zadach. – M.: Nauka, 1979. 6. Bakushinskii A. B., Goncharskii A. V. Nekorrektnye zadachi. Chislennye metody i prilozheniia. – Moskva: Izd-vo MGU, 1989.

Lectures, independent work

The maximum number of points that can be obtained by a student: 100/60 points. - semester assessment: 1. Control work 1: RN 1.1., RN 1.2, RN1.3, RN 2.1, RN3.1 – 30 points/18 points. 2. Control work 2: RN 1.1., RN 1.2, RN1.3, RN 2.1, RN3.1 – 30 points/18 points. - final evaluation (in the form of an exam): - the maximum number of points that can be obtained by a student: 40 points; - learning outcomes that will be evaluated: PH1.1, PH1.2, PH1.3, PH2.1; - form of implementation and types of tasks: written. Types of tasks: 4 written tasks.

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