New Numerical Methods on the Basis of Fuzzy Approximation

The notion of a fuzzy set, a natural extension of a classical set, was defined in 1965 by L.A. Zadeh; see [16]. Since that time, the fuzzy set theory has been deeply developed and it has influenced many fields of applications and therefore we can find results in branches like: fuzzy time series, fuzzy modeling, fuzzy graph theory and finally, the most often one fuzzy control. In general, the fundamental idea behind most of them is hidden in expressing dependencies between variables by conditional sentences of human language. The sentences are called fuzzy rules and together they determine so-called fuzzy rule base. One of the main points of of all ’fuzzy theories’ is to reasonably answer how to interpret a fuzzy rule base. Two main forms, namely DNF and CNF (the disjunctive normal form and the conjunctive normal form), of such interpretations are described; see [7]. It is worth mentioning that the DNF in a combination with the minimum t-norm determines well known Mamdani-Assilian’s approach, perhaps the most often used approach in applications; see [5]. Other results concerning the DNF interpretation can be found in [3, 4]. Some results aiming at the CNF interpretations have been published in [7, 11, 1]. For some reasons (higher computational efforts, discontinuity), the CNF approach is the less favorite one, however, a universal approximation ability has been proved for both approaches; see [3, 7, 11, 1]. Moreover, the CNF approach has deep logical roots since the generalized modus ponens is involved in it; see [2]. A fuzzy approach to an approximation surely brings some advantages, namely robustness, interpretability, transparency, or simplicity. Since any approximation model can replace an approximated function in further numerical computations to decrease computational efforts we can establish numerical methods on the basis of fuzzy approximation models. Here we keep this name proposed by I. Perfilieva, the author of one particular fuzzy approximation technique called the fuzzy transform; see [8, 9]. Numerical methods using this simplified model has been already successfully applied. We refer to applications to ordinary differential equations in [6], partial differential equations [13, 14] and noise removing [12]. Promising results have been achieved in the data compression as well; see [10].