Fuzzy rule interpolation (FRI) is well known for reducing the complexity of fuzzy models and making inference possible in sparse rule-based systems. However, in practical fuzzy applications with inter-connected rule bases, situations may arise when a crucial antecedent of observation is absent, either due to human error or difficulty in obtaining data, while the associated conclusion may be der...

We explore extended B-splines as a stable basis for isogeometric analysis with trimmed parameter spaces. The stabilization is accomplished by an appropriate substitution of Bsplines that may lead to ill-conditioned system matrices. The construction for non-uniform knot vectors is presented. The properties of extended B-splines are examined in the context of interpolation, potential, and linear ...

A boundary value problem for a 4-th order self-adjoint ordinary differential equation is considered in the case where the coefficients of the equation and its right-hand side can be nonsmooth (discontinuous, concentrated or rapidly oscillating functions). Generalized cubic splines of deficiency 1 depending on the major coefficient of the equation are applied. An error analysis of finite element...

A quasi-interpolation method for quadratic piecewise polynomials in three variables is described which can be used for the efficient visualization of gridded volume data. We analyze the smoothness properties of the trivariate splines. We prove that the splines yield nearly optimal approximation order while simultaneously its piecewise derivatives provide optimal approximation of the derivatives...

The algorithms and programs in this handy book deal almost exclusively with cubic splines. A short introduction stresses the maximum "smoothness" of "natural" splines. The next chapter reviews the numerical solution of tridiagonal linear systems by Gauss elimination and of block tridiagonal linear systems by relaxation. Chapter 3 offers programs for cubic spline interpolation with various (sepa...

The general mathematical problem of fuzzy control is an interpolation problem: a list of fuzzy input-output data, usually provided by a list of linguistic control rules, should be realized as argument-value pairs for a suitably chosen fuzzy function. However, contrary to the usual understanding of interpolation, in the actual approaches this interpolation problem is considered as a global one: ...

Fuzzy rule interpolation forms an important approach for performing inference with systems comprising sparse rule bases. Even when a given observation has no overlap with the antecedent values of any existing rules, fuzzy rule interpolation may still derive a useful conclusion. Unfortunately, very little of the existing work on fuzzy rule interpolation can conjunctively handle more than one for...

The “Double Fuzzy Point” rule representation opens a new dimension for expressing changes of fuzziness in fuzzy rule-based systems. In the case of standard “Fuzzy Point” rule representations, it is difficult to describe fuzzy functions in which crisp observations are required to have fuzzy conclusions, or in which an increase in the fuzziness of observations leads to reduced fuzziness in conclu...