Method of Moments in the Problem of Inversion of the Laplace Transform and Its Regularization

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Method of Moments in the Problem of Inversion of the Laplace Transform and Its Regularization. / Lebedeva, A. V.; Ryabov, V. M.

в: Vestnik St. Petersburg University: Mathematics, Том 55, № 1, 03.2022, стр. 34-38.

Результаты исследований: Научные публикации в периодических изданиях › статья › Рецензирование

BibTeX

@article{d3a09c0d19514d3fb707134590cb3e51,

title = "Method of Moments in the Problem of Inversion of the Laplace Transform and Its Regularization",

abstract = "Abstract: We consider integral equations of the first kind, which are associated with the class of ill-posed problems. This class also includes the problem of inversing the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to ill-conditioned systems of linear algebraic equations (in which unknowns represent the coefficients of expansion in a series in shifted Legendre polynomials of some function that is simply expressed in terms of the sought original; this function is found as a solution of a certain finite moment problem in a Hilbert space). To obtain a reliable solution of 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; this type is focused on an a priori low degree of smoothness of the desired original. The results of numerical experiments are presented; they confirm the efficiency of the proposed inversion algorithm.",

keywords = "condition number, ill-conditioned problems, ill-posed problems, integral equations of the first kind, regularization method, system of linear algebraic equations",

author = "Lebedeva, {A. V.} and Ryabov, {V. M.}",

note = "Lebedeva, A.V., Ryabov, V.M. Method of Moments in the Problem of Inversion of the Laplace Transform and Its Regularization. Vestnik St.Petersb. Univ.Math. 55, 34–38 (2022). https://doi.org/10.1134/S1063454122010071",

year = "2022",

month = mar,

doi = "10.1134/S1063454122010071",

language = "English",

volume = "55",

pages = "34--38",

journal = "Vestnik St. Petersburg University: Mathematics",

issn = "1063-4541",

publisher = "Pleiades Publishing",

number = "1",

}

RIS

TY - JOUR

T1 - Method of Moments in the Problem of Inversion of the Laplace Transform and Its Regularization

AU - Lebedeva, A. V.

AU - Ryabov, V. M.

N1 - Lebedeva, A.V., Ryabov, V.M. Method of Moments in the Problem of Inversion of the Laplace Transform and Its Regularization. Vestnik St.Petersb. Univ.Math. 55, 34–38 (2022). https://doi.org/10.1134/S1063454122010071

PY - 2022/3

Y1 - 2022/3

N2 - Abstract: We consider integral equations of the first kind, which are associated with the class of ill-posed problems. This class also includes the problem of inversing the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to ill-conditioned systems of linear algebraic equations (in which unknowns represent the coefficients of expansion in a series in shifted Legendre polynomials of some function that is simply expressed in terms of the sought original; this function is found as a solution of a certain finite moment problem in a Hilbert space). To obtain a reliable solution of 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; this type is focused on an a priori low degree of smoothness of the desired original. The results of numerical experiments are presented; they confirm the efficiency of the proposed inversion algorithm.

AB - Abstract: We consider integral equations of the first kind, which are associated with the class of ill-posed problems. This class also includes the problem of inversing the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to ill-conditioned systems of linear algebraic equations (in which unknowns represent the coefficients of expansion in a series in shifted Legendre polynomials of some function that is simply expressed in terms of the sought original; this function is found as a solution of a certain finite moment problem in a Hilbert space). To obtain a reliable solution of 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; this type is focused on an a priori low degree of smoothness of the desired original. The results of numerical experiments are presented; they confirm the efficiency of the proposed inversion algorithm.

KW - condition number

KW - ill-conditioned problems

KW - ill-posed problems

KW - integral equations of the first kind

KW - regularization method

KW - system of linear algebraic equations

UR - http://www.scopus.com/inward/record.url?scp=85131880691&partnerID=8YFLogxK

UR - https://dspace.spbu.ru/handle/11701/36153

UR - https://www.mendeley.com/catalogue/b8693397-89e4-38f2-a82c-178ca1c7d49e/

U2 - 10.1134/S1063454122010071

DO - 10.1134/S1063454122010071

M3 - Article

AN - SCOPUS:85131880691

VL - 55

SP - 34

EP - 38

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 1

ER -

ID: 98303092