Найкраще рівномірне наближення сплайнами з використанням диференціальної еволюції

Abstract

It is considered a problem of the best uniform approximation of functions by polynomial splines with fixed knots. It is proposed an approach based on evolutionary algorithms — a powerful class of stochastic search optimization methods — for its solution. To find a spline of the best uniform approximation, a differential evolution algorithm is adapted. It is one of the best evolutionary algorithms that consistently finds a global optimum of a target function (optimization criterion) in a minimal time. An evolutionary process in the algorithm begins with a generation of random vectors, coordinates of which are possible values of spline coefficients. Further, the vectors are constantly modified by mutation, crossover and selection operations in order to reduce a value of the target function (spline approximation error). The algorithm is completed if a specified maximum number of populations is exhausted or a stagnation of the evolutionary process takes place. The differential evolution algorithm is simple in program realization and using (it contains few varied parameters that need to be selected). It is easily paralleled. Recommendations for choosing optimal values of main parameters of the algorithm such as a population size, a mutation factor, a crossover probability are developed. A comparison of the approximation errors obtained by the stochastic differential evolution algorithm and by other (deterministic) algorithms is made for a series of test functions. Results of the comparison showed that an accuracy of the functions approximation by splines using the differential evolution is not worse than using much more complicated deterministic algorithms of the best uniform approximation. This testifies about the effectiveness of the differential evolution algorithm. It can be used as an alternative for known deterministic algorithms of spline approximation