Random number generation and quasi-Monte Carlo methods
Random number generation and quasi-Monte Carlo methods
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Error analysis of a randomized numerical method
Numerische Mathematik
A quasi-randomized Runge-Kutta method
Mathematics of Computation
Quasi-randomized numerical methods for systems with coefficients of bounded variation
Mathematics and Computers in Simulation - IMACS sponsored Special issue on the second IMACS seminar on Monte Carlo methods
A random Euler scheme for Carathéodory differential equations
Journal of Computational and Applied Mathematics
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In this paper the quasi-Monte Carlo methods for Runge-Kutta solution techniques of differential equations, which were developed by Stengle, Lécot, Coulibaly and Koudiraty, are extended to delay differential equations of the form y'(t) = f(t,y(t),y(t-τ(t))). The retarded argument is approximated by interpolation, after which the conventional (quasi-)Monte Carlo Runge-Kutta methods can be applied. We give a proof of the convergence of this method and its order in a general form, which does not depend on a specific quasi-Monte Carlo Runge-Kutta method. Finally, a numerical investigation shows that similar to ordinary differential equations, this quasi-randomized method leads to an improvement for heavily oscillating delay differential equations, compared even to high-order Runge-Kutta schemes.