21 research outputs found
B-spline-like bases for cubics on the Powell-Sabin 12-split
For spaces of constant, linear, and quadratic splines of maximal smoothness
on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently
introduced. These are simplex spline bases with B-spline-like properties on the
12-split of a single triangle, which are tied together across triangles in a
B\'ezier-like manner.
In this paper we give a formal definition of an S-basis in terms of certain
basic properties. We proceed to investigate the existence of S-bases for the
aforementioned spaces and additionally the cubic case, resulting in an
exhaustive list. From their nature as simplex splines, we derive simple
differentiation and recurrence formulas to other S-bases. We establish a
Marsden identity that gives rise to various quasi-interpolants and domain
points forming an intuitive control net, in terms of which conditions for
-, -, and -smoothness are derived
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Multivariate Splines and Algebraic Geometry
Multivariate splines are effective tools in numerical analysis and approximation theory. Despite an extensive literature on the subject, there remain open questions in finding their dimension, constructing local bases, and determining their approximation power. Much of what is currently known was developed by numerical analysts, using classical methods, in particular the so-called Bernstein-B´ezier techniques. Due to their many interesting structural properties, splines have become of keen interest to researchers in commutative and homological algebra and algebraic geometry. Unfortunately, these communities have not collaborated much. The purpose of the half-size workshop is to intensify the interaction between the different groups by bringing them together. This could lead to essential breakthroughs on several of the above problems
Appropriate choice of aggregation operators in fuzzy decision support systems
Fuzzy logic provides a mathematical formalism for a unified treatment of vagueness and imprecision that are ever present in decision support and expert systems in many areas. The choice of aggregation operators is crucial to the behavior of the system that is intended to mimic human decision making. This paper discusses how aggregation operators can be selected and adjusted to fit empirical data—a series of test cases. Both parametric and nonparametric regression are considered and compared. A practical application of the proposed methods to electronic implementation of clinical guidelines is presented<br /
Positive quartic, monotone quintic C2-spline interpolation in one and two dimensions
AbstractThis paper is concerned with shape-preserving interpolation of discrete data by polynomial splines. We show that positivity can be always preserved by quartic C2-splines and monotonicity by quintic C2-splines. This is proved for one-dimensional interpolation as well as for two-dimensional interpolation on rectangular grids
Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates
In this paper we investigate the approximation properties of kernel
interpolants on manifolds. The kernels we consider will be obtained by the
restriction of positive definite kernels on , such as radial basis
functions (RBFs), to a smooth, compact embedded submanifold \M\subset \R^d.
For restricted kernels having finite smoothness, we provide a complete
characterization of the native space on \M. After this and some preliminary
setup, we present Sobolev-type error estimates for the interpolation problem.
Numerical results verifying the theory are also presented for a one-dimensional
curve embedded in and a two-dimensional torus
Range Restricted Interpolation Using Cubic Bézier Triangles.
A range restricted C1 interpolation local scheme to scattered data is derived. Each macro triangle of the
triangulated domain is split into three mini triangles and the interpolating surface on each mini triangle is a cubic
Bézier triangle. Sufficient conditions derived for the non-negativity of these cubic Bézier triangles are expressed
as lower bounds to the Bézier ordinates. The non-negativity preserving interpolation scheme extends to the
construction of a range restricted interpolating surface with lower or upper constraints which are polynomial
surfaces of degree up to three. The scheme is illustrated with graphical examples