Method of moments in the problem of inversion of the Laplace transform and its regularization
DOI:
https://doi.org/10.21638/spbu01.2022.105Abstract
Integral equations of the first kind are considered, which belong to the class of ill-posed problems. This also includes the problem of inverting the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to illconditioned systems of linear algebraic equations, in which the unknowns are the coefficients of the expansion in a series in the shifted Legendre polynomials of some function that simply expresses in terms of the sought original. This function is found as a solution to a certain finite moment problem in a Hilbert space. To obtain a reliable solution to the system, regularization methods are used. The general strategy is to use the Tikhonov stabilizer or its modifications. A specific type of stabilizer in the regularization method is indicated, focused on a priori low degree of smoothness of the desired original. The results of numerical experiments are presented, confirming the effectiveness of the proposed inversion algorithm.Keywords:
system of linear algebraic equations, integral equations of the first kind, ill-posed problems, ill-conditioned problems, condition number, regularization method
Downloads
Download data is not yet available.
References
Литература
1. Лаврентьев М.А., Шабат Б.В. Методы теории функций комплексного переменного. Москва, Лань (2002).
2. Крылов В.И., Скобля Н.С. Методы приближенного преобразования Фурье и обращения преобразования Лапласа. Москва, Наука (1974).
3. Cohen A.M. Numerical methods for Laplace transform inversion. New York, Springer (2007).
4. Рябов В.М. Численное обращение преобразования Лапласа. Санкт-Петербург, Изд-во С.- Петерб. ун-та (2013).
5. Суетин П.К. Классические ортогональные многочлены. Москва, Наука (1976).
6. Тихонов А.Н., Арсенин В.Я. Методы решения некорректных задач. Москва, Наука (1979).
7. Gautschi W. On the condition of a matrix arising in the numerical inversion of the Laplace transform. Mathematics of computation 23 (105), 109–118 (1969).
8. Иванов В.К., Васин В.В., Танана В.П. Теория линейных некорректных задач и ее прило- жения. Москва, Наука (1978).
9. Кабанихин С.И. Обратные и некорректные задачи. Новосибирск, Сибирское научное изд- во (2009).
10. Даугавет И.К. Теория приближенных методов. Линейные уравнения. Санкт-Петербург, БХВ-Петербург (2006).
11. Лебедева А.В., Рябов В.М. О численном решении систем линейных алгебра- ических уравнений с плохо обусловленными матрицами. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 6 (64), вып. 4, 619–626 (2019). https://doi.org/10.21638/11701/spbu01.2019.407
12. Лебедева А.В., Рябов В.М. О регуляризации решения интегральных уравне- ний первого рода с помощью квадратурных формул. Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия 8 (66), вып. 4, 593–599 (2021). https://doi.org/10.21638/spbu01.2021.404
13. Papoulis A. A new method of inversion of the Laplace transform. Quarterly of applied mathematics 14 (4), 405–414 (1967).
14. Brianzi P., Frontini M. On the regularized inversion of Laplace transform. Inverse problems 7, 355–368 (1991).
15. Крейн М. Г., Нудельман А.А. Проблема моментов Маркова и экстремальные задачи. Москва, Наука (1973).
References
1. Lavrent’ev M.A., Shabat B.V. Methods of the theory of functions of a complex variable. Moscow, Lan’ Publ. (2002). (In Russian)
2. Krylov V. I., Skoblya N. S. Methods of the approximate Fourier transform and the inversion of the Laplace transform. Moscow, Nauka Publ. (1974). (In Russian)
3. Cohen A.M. Numerical methods for Laplace transform inversion. New York, Springer (2007).
4. Ryabov V.M. Numerical inversion of the Laplace transform. St Petersburg, StPetersburg Univ. Press (2013). (In Russian)
5. Suetin P.K. Classical orthogonal polynomials. Moscow, Nauka Publ. (1976). (In Russian)
6. Tikhonov A.N., Arsenin V.Ya. Metody resheniia nekorrektnykh zadach. Moscow, Nauka Publ. (1979). (In Russian) [Eng. transl.: Tikhonov A.N., Arsenin V.Ya. Solutions of Ill-Posed Problems. Winston (1977)].
7. Gautschi W. On the condition of a matrix arising in the numerical inversion of the Laplace transform. Mathematics of computation 23 (105), 109–118 (1969).
8. Ivanov V.K., Vasin V.V., Tanana V.P. Theory of linear ill-posed problems and its applications. Moscow, Nauka Publ. (1978). (In Russian)
9. Kabanikhin S. I. Inverse and ill-posed problems. Novosibirsk, Sibirskoe nauchnoe izdatel’stvo Publ. (2009). (In Russian)
10. Daugavet I.K. The theory of approximate methods. Linear equations. St Petersburg, BHVPetersburg Publ. (2006). (In Russian)
11. Lebedeva A.V., Ryabov V.M. Numerical solution of systems of linear algebraic equations with ill-conditioned matrices. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 6 (64), iss. 4, 619–626 (2019). https://doi.org/10.21638/11701/spbu01.2019.407 (In Russian) [Eng. transl.: Vestnik St Petersburg University, Mathematics 52 (4), 388–393 (2019). https://doi.org/10.1134/S1063454119040058].
12. Lebedeva A.V., Ryabov V.M. On the regularization of the solution of integral equations of the first kind using quadrature formulas. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 8 (66), iss. 4, 593–599 (2021). https://doi.org/10.21638/spbu01.2021.404 (In Russian) [Eng. transl.: Vestnik St Petersburg University, Mathematics 54 (4), 361–365 (2021). https://doi.org/10.1134/S1063454121040129].
13. Papoulis A. A new method of inversion of the Laplace transform. Quarterly of applied mathematics 14 (4), 405–414 (1967).
14. Brianzi P., Frontini M. On the regularized inversion of Laplace transform. Inverse problems 7, 355–368 (1991).
15. Krein M.G., Nudel’man A.A. Problema momentov Markova i ekstremal’nye zadachi. Moscow, Nauka Publ. (1973). (In Russian) [Eng. transl.: Krein M.G., Nudel’man A.A. The Markov moment problem and extremal problems. In Ser.: Translations of Mathematical Monographs, vol. 50. AMS (1977)].
Downloads
Published
2022-04-10
How to Cite
Lebedeva, A. V., & Ryabov, V. M. (2022). Method of moments in the problem of inversion of the Laplace transform and its regularization. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 9(1), 46–52. https://doi.org/10.21638/spbu01.2022.105
Issue
Section
Mathematics
License
Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
