On Regularization of the Solution of Integral Equations of the First Kind Using Quadrature Formulas

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On Regularization of the Solution of Integral Equations of the First Kind Using Quadrature Formulas. / Lebedeva, A. V.; Ryabov, V. M.

в: Vestnik St. Petersburg University: Mathematics, Том 54, № 4, 10.2021, стр. 361-365.

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

BibTeX

@article{ef4777732e874b47852118b315566239,

title = "On Regularization of the Solution of Integral Equations of the First Kind Using Quadrature Formulas",

abstract = "Abstract: Ill-conditioned systems of linear algebraic equations (SLAEs) and integral equations of the first kind belonging to the class of ill-posed problems are considered. This class includes also the problem of inversion of the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to SLAEs with special matrices. To obtain a reliable solution, regularization methods are used. The general strategy is to use the Tikhonov stabilizer or its modifications or to represent the desired solution in the form of the orthogonal sum of two vectors, one of which is determined stably, while, to search for the second vector, it is necessary to use some kind of stabilization procedure. In this paper, the methods of numerical solving an SLAE with a symmetric positive definite matrix or with an oscillatory-type matrix with the use of regularization leading to an SLAE with a reduced condition number are considered. A method of reducing the problem of inversion of the integral Laplace transform to an SLAE with generalized Vandermonde oscillatory-type matrices, the regularization of which reduces the ill-conditioning of the system, is indicated.",

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 = "Publisher Copyright: {\textcopyright} 2021, Pleiades Publishing, Ltd.",

year = "2021",

month = oct,

doi = "10.1134/S1063454121040129",

language = "English",

volume = "54",

pages = "361--365",

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

issn = "1063-4541",

publisher = "Pleiades Publishing",

number = "4",

}

RIS

TY - JOUR

T1 - On Regularization of the Solution of Integral Equations of the First Kind Using Quadrature Formulas

AU - Lebedeva, A. V.

AU - Ryabov, V. M.

PY - 2021/10

Y1 - 2021/10

N2 - Abstract: Ill-conditioned systems of linear algebraic equations (SLAEs) and integral equations of the first kind belonging to the class of ill-posed problems are considered. This class includes also the problem of inversion of the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to SLAEs with special matrices. To obtain a reliable solution, regularization methods are used. The general strategy is to use the Tikhonov stabilizer or its modifications or to represent the desired solution in the form of the orthogonal sum of two vectors, one of which is determined stably, while, to search for the second vector, it is necessary to use some kind of stabilization procedure. In this paper, the methods of numerical solving an SLAE with a symmetric positive definite matrix or with an oscillatory-type matrix with the use of regularization leading to an SLAE with a reduced condition number are considered. A method of reducing the problem of inversion of the integral Laplace transform to an SLAE with generalized Vandermonde oscillatory-type matrices, the regularization of which reduces the ill-conditioning of the system, is indicated.

AB - Abstract: Ill-conditioned systems of linear algebraic equations (SLAEs) and integral equations of the first kind belonging to the class of ill-posed problems are considered. This class includes also the problem of inversion of the integral Laplace transform, which is used to solve a wide class of mathematical problems. Integral equations are reduced to SLAEs with special matrices. To obtain a reliable solution, regularization methods are used. The general strategy is to use the Tikhonov stabilizer or its modifications or to represent the desired solution in the form of the orthogonal sum of two vectors, one of which is determined stably, while, to search for the second vector, it is necessary to use some kind of stabilization procedure. In this paper, the methods of numerical solving an SLAE with a symmetric positive definite matrix or with an oscillatory-type matrix with the use of regularization leading to an SLAE with a reduced condition number are considered. A method of reducing the problem of inversion of the integral Laplace transform to an SLAE with generalized Vandermonde oscillatory-type matrices, the regularization of which reduces the ill-conditioning of the system, is indicated.

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=85121557145&partnerID=8YFLogxK

U2 - 10.1134/S1063454121040129

DO - 10.1134/S1063454121040129

M3 - Article

AN - SCOPUS:85121557145

VL - 54

SP - 361

EP - 365

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 4

ER -

ID: 90622844