Vol 64, No 12 (2024)

An approximate algorithm for calculating integral operators of the convolution type that arise when evaluating barrier options in Levy models by the Wiener–Hopf method is constructed. Additionally, the question of the possibility of applying machine learning methods (artificial neural networks) to the approximation of a special type of integrals, which are a key element in the construction of approximate formulas for the Wiener–Hopf integral operators under consideration, is investigated. The main idea is to decompose the price function into a Fourier series and transform the integration contour for each term of the Fourier series. As a result, we obtain a set of typical integrals that depend on Wiener–Hopf factors, but do not depend on the price function, while the most computationally expensive part of the numerical method is reduced to calculating these integrals. Since they need to be calculated only once, and not at each iteration, as was the case in standard implementations of the Wiener–Hopf method, this will significantly speed up calculations. Moreover, a neural network can be trained to calculate typical integrals. The proposed approach is especially effective for spectrally one-sided Levy processes, for which explicit Wiener–Hopf factorization formulas are known. In this case, we obtain computationally convenient formulas by integrating along the section. The main advantage of including neural networks in a computational scheme is the ability to perform calculations on an uneven grid. Such a hybrid numerical method will be able to successfully compete with classical methods of computing convolutions in similar tasks using the fastFourier transform. Computational experiments show that neural networks with one hidden layer of 20 neurons are able to effectively cope with the tasks of approximating the auxiliary integrals under consideration.

Functional relations have been obtained that allow us to evaluate the accuracy of approximate solutions in terms of measures significantly different from the energy norms that are usually used for these purposes. In particular, they are applicable to local norms and measures constructed using specially built linear functionals. The need for such precision control tools arises if there is a special interest in the behavior of the solution in some subdomain or in the special properties of the solution. It is shown that a posteriori functional-type estimates, which were previously used for global estimates, can be adapted to solve this problem. Functional identities and estimates are obtained that allow estimating the error of any conformal approximations in terms of a wide class of measures, including local norms and problem-oriented functionals. The theoretical results are verified in a series of examples that confirm the effectiveness of the proposed method.

Quadrature formulas for singular integrals on the integration interval [−1, 1] with certain weight functions

Abstract. Introduced by G. Bateman in 1915 and studied by J. M. Burgers in 1948, the Burgers equation has found wide application in fluid mechanics, nonlinear acoustics and other fields of applied mathematics. The approaches to its solution were very diverse: asymptotic, numerical, and analytical. In this paper, an analytical method for solving a Burgers-type equation in a Banach space is developed. Namely, after artificially introducing a small parameter into the equation, the existence of an analytical solution for this parameter is proved. At the same time, a multidimensional version of the equation is also considered.

The actual problem of modeling nonlinear wave processes in a microwave generator with magnetic isolation is considered. For its numerical analysis, a new computer model is proposed, including Maxwell’s equations and equations of motion of relativistic charged particles, their joint integration by the grid method and the cloud particle method, as well as a parallel software implementation. In numerical experiments, the space-time characteristics of relativistic electron beams and plasma, as well as the parameters of the output radiation of the generator, are obtained. The analysis of the obtained results confirmed the correctness of the developed numerical approach.