Conditional optimization of discrete-stochastic numerical procedures with cubic splines applied

In this paper, the discrete-stochastic numerical procedures of the global function approximation are considered. The procedures are built to approximate the solution to an integral equation of the second kind using the Streng–Fix approximation with cubic B-splines as basis functions. In the case of using cubic splines, a discrete component of the approximation error has a higher order with respect to the grid step as compared to the well-studied case of the multi-linear approximation. Moreover, the property of “error concentration in grid nodes” is proved to hold for approximation with the cubic splines as well. This is because the coefficients of the approximation are, in fact, linear combinations of the function values in grid nodes. The above properties provide the upper bounds for the total approximation error. Finally, for the investigated discrete-stochastic procedures, the conditionally-optimal parameters are calculated that minimize computational costs for the procedures with the fixed error upper bounds.