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

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  • Krivovichev, Gerasim V.
  • Semina, Ksenia I.

Abstract

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.

Suggested Citation

  • Krivovichev, Gerasim V. & Semina, Ksenia I., 2026. "Application of Runge–Kutta method with non-standard finite difference for the construction of preconditioned gradient methods," Applied Mathematics and Computation, Elsevier, vol. 515(C).
    • Handle: RePEc:eee:apmaco:v:515:y:2026:i:c:s009630032500579x
    DOI: 10.1016/j.amc.2025.129854
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