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Omid Nikan

Publications and source records attributed to Omid Nikan.

2 recordsLinked to original sources
arXiv2023

Numerical Approximation of Nonlinear Stochastic Volterra Integral Equation using Walsh Function

This paper adopts a highly effective numerical approach for approximating non-linear stochastic Volterra integral equations (NLSVIEs) based on the operational matrices of the Walsh function and the collocation method. The method transforms the integral equation into a system of algebraic equations, which allows for the derivation of an approximate solution. Error analysis has been performed, confirming the effectiveness of the proposed method, which results in a linear order of convergence. Numerical examples are provided to illustrate the precision and effectiveness of this proposed method.

math.NA↗
arXiv2023

Numerical approximation based on deep convolutional neural network for high-dimensional fully nonlinear merged PDEs and 2BSDEs

This paper proposes two efficient approximation methods to solve high-dimensional fully nonlinear partial differential equations (NPDEs) and second-order backward stochastic differential equations (2BSDEs), where such high-dimensional fully NPDEs are extremely difficult to solve because the computational cost of standard approximation methods grows exponentially with the number of dimensions. Therefore, we consider the following methods to overcome this difficulty. For the merged fully NPDEs and 2BSDEs system, combined with the time forward discretization and ReLU function, we use multi-scale deep learning fusion and convolutional neural network (CNN) techniques to obtain two numerical approximation schemes, respectively. Finally, three practical high-dimensional test problems involving Allen-Cahn, Black-Scholes-Barentblatt, and Hamiltonian-Jacobi-Bellman equations are given so that the first proposed method exhibits higher efficiency and accuracy than the existing method, while the second proposed method can extend the dimensionality of the completely NPDEs-2BSDEs system over $400$ dimensions, from which the numerical results highlight the effectiveness of proposed methods.

math.NA↗