Regularization of the procedure for inverting the Laplace transform using quadrature formulas

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 the approximate values of the desired original at a number of points. The first step of reduction 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 given for actually constructing special quadratures leading to problems with oscillation matrices.