On the randomized solution of initial value problems (Q544122)

The paper focuses on the numerical solution of initial value problems for systems of ordinary differential equations: \[ \begin{cases} y'(x)=f(x,y(x)), &x\in[a,b],\\ y(a)=y_0,\end{cases} \] where \(y_0\in{\mathbb R}^d,\) \(-\infty<a<b<\infty,\) \(f:[a,b]\times{\mathbb R}^d\rightarrow{\mathbb R}^d,\) presenting an order optimal randomized algorithm for the class of \(\gamma\)-smooth functions that uses only values of \(f\) (\(\gamma=1+\rho:\) the \(r\)-th derivatives of \(f\) satisfy a \(\rho\)-Hölder condition). The first part is an introduction in nature. In the second part the author presents the basic notations, defines the IBC framework and introduces some necessary definitions. The third part defines the conditions for the family of deterministic algorithms and the randomized algorithm itself. The fourth part is dedicated to the analysis of the algorithm, proving the optimal order of algorithm for admissible information in the deterministic and in the randomized case. The last section presents the numerical results obtained by performing three tests, by comparison with a 3-stage Runge-Kutta method and Euler method, respectively.