Open AccessMethod of moments in the problem of inversion of the Laplace transform and its regularization
Author(s) -
A. V. Lebedeva,
V. M. Ryabov
Publication year - 2022
Publication title -
vestnik sankt-peterburgskogo universiteta. matematika. mehanika. astronomiâ/vestnik sankt-peterburgskogo universiteta. seriâ 1, matematika, mehanika, astronomiâ
Language(s) - English
Resource type - Journals
eISSN - 2587-5884
pISSN - 1025-3106
DOI - 10.21638/spbu01.2022.105
Subject(s) - mathematics , tikhonov regularization , laplace transform , legendre polynomials , mathematical analysis , integral equation , regularization (linguistics) , backus–gilbert method , algebraic equation , moment problem , inverse problem , regularization perspectives on support vector machines , computer science , nonlinear system , statistics , physics , quantum mechanics , artificial intelligence , principle of maximum entropy
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.