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 -