Research output: Contribution to journal › Article › peer-review
Regularization of the procedure for inverting the Laplace transform using quadrature formulas. / Лебедева, Анастасия Владимировна; Рябов, Виктор Михайлович.
In: Vestnik St. Petersburg University: Mathematics, Vol. 55, No. 4, 01.12.2022, p. 414-418.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Regularization of the procedure for inverting the Laplace transform using quadrature formulas
AU - Лебедева, Анастасия Владимировна
AU - Рябов, Виктор Михайлович
PY - 2022/12/1
Y1 - 2022/12/1
N2 - Abstract: The problem of inversion of the integral Laplace transform, which belongs to the class of ill-posed problems, is considered. Integral equations are reduced to ill-conditioned systems of linear algebraic equations (SLAE), in which the unknowns are either the expansion coefficients in a series in terms of shifted Legendre polynomials, or approximate values of the desired inverse transform at a number of points. The first step of reducing to SLAE is to apply quadrature formulas that provide the minimum values of the condition number of SLAE. Regularization methods are used to obtain a reliable solution of the system. A common strategy is to use the Tikhonov stabilizer or its modifications. A variant of the regularization method for systems with oscillatory-type matrices is presented, which significantly reduces the conditionality of the problem in comparison with the classical Tikhonov scheme. A method is proposed for actually constructing special quadratures leading to problems with oscillation matrices.
AB - Abstract: The problem of inversion of the integral Laplace transform, which belongs to the class of ill-posed problems, is considered. Integral equations are reduced to ill-conditioned systems of linear algebraic equations (SLAE), in which the unknowns are either the expansion coefficients in a series in terms of shifted Legendre polynomials, or approximate values of the desired inverse transform at a number of points. The first step of reducing to SLAE is to apply quadrature formulas that provide the minimum values of the condition number of SLAE. Regularization methods are used to obtain a reliable solution of the system. A common strategy is to use the Tikhonov stabilizer or its modifications. A variant of the regularization method for systems with oscillatory-type matrices is presented, which significantly reduces the conditionality of the problem in comparison with the classical Tikhonov scheme. A method is proposed for actually constructing special quadratures leading to problems with oscillation matrices.
KW - condition number
KW - ill-conditioned problems
KW - ill-posed problems
KW - integral equations of the first kind
KW - oscillation matrices
KW - regularization method
KW - system of linear algebraic equations
UR - https://www.mendeley.com/catalogue/e7b40319-10db-3a17-9806-f33a9027c2a9/
U2 - 10.1134/s1063454122040136
DO - 10.1134/s1063454122040136
M3 - Article
VL - 55
SP - 414
EP - 418
JO - Vestnik St. Petersburg University: Mathematics
JF - Vestnik St. Petersburg University: Mathematics
SN - 1063-4541
IS - 4
ER -
ID: 101223891