TY - JOUR

T1 - Application of Runge–Kutta method with non-standard finite difference for the construction of preconditioned gradient methods

AU - Кривовичев, Герасим Владимирович

AU - Семина, Ксения

PY - 2026/4/15

Y1 - 2026/4/15

N2 - This paper is devoted to the construction and analysis of gradient methods based on the explicit second-order Runge–Kutta method with a non-standard finite difference. The application of this type of finite difference provides the possibility to use the function of step instead of the step value (like in standard gradient descent) and add new parameters for the improvement of stability and convergence rate. The preconditioned gradient descent method and its version with momentum (heavy ball method) are constructed. Theorems on convergence conditions for a strongly convex quadratic function and its perturbation are formulated. The numerical approaches for obtaining optimal parameters for two methods are proposed. Optimal convergence rates are compared with the optimal rates for well-known first-order methods with constant parameters (standard gradient descent, Polyak's heavy ball method, and Nesterov's accelerated methods). Theoretical analysis is supported by the numerical experiments carried out on problems from different applications (2D and 3D Dirichlet problems for the Poisson equation, minimization of integral functional, problem for nonlinear integro-differential equation, and regularized logistic regression). It is demonstrated that the preconditioned method with momentum performs better than well-known methods with constant parameters, used for comparison.

AB - This paper is devoted to the construction and analysis of gradient methods based on the explicit second-order Runge–Kutta method with a non-standard finite difference. The application of this type of finite difference provides the possibility to use the function of step instead of the step value (like in standard gradient descent) and add new parameters for the improvement of stability and convergence rate. The preconditioned gradient descent method and its version with momentum (heavy ball method) are constructed. Theorems on convergence conditions for a strongly convex quadratic function and its perturbation are formulated. The numerical approaches for obtaining optimal parameters for two methods are proposed. Optimal convergence rates are compared with the optimal rates for well-known first-order methods with constant parameters (standard gradient descent, Polyak's heavy ball method, and Nesterov's accelerated methods). Theoretical analysis is supported by the numerical experiments carried out on problems from different applications (2D and 3D Dirichlet problems for the Poisson equation, minimization of integral functional, problem for nonlinear integro-differential equation, and regularized logistic regression). It is demonstrated that the preconditioned method with momentum performs better than well-known methods with constant parameters, used for comparison.

KW - Iterative methods

KW - Non-standard finite differences

KW - Runge – Kutta methods

KW - Unconstrained optimization

UR - https://www.mendeley.com/catalogue/d60d2e5b-48b8-3ecf-a322-3c2c9e0b32fc/

U2 - 10.1016/j.amc.2025.129854

DO - 10.1016/j.amc.2025.129854

M3 - Article

VL - 515

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

M1 - 129854

ER -