We present a new approach for reconstructing a smooth surface from a set of scattered points in three-dimensional (3D) space. Our algorithm first decomposes a given point set into a quadtree-like data structure known as a strip tree. The strip tree is used to fit a set of least squares quadratic surfaces to the data points. These quadratic surfaces are then degree-elevated to bi-cubic surfaces ...

We present a new approach to estimate the risk-neutral probability density function (pdf) of the future prices of an underlying asset from the prices of options written on the asset. The estimation is carried out in the space of cubic spline functions, yielding appropriate smoothness. The resulting optimization problem, used to invert the data and determine the corresponding density function, i...

A C curve interpolation scheme for monotonic data has been developed. This scheme uses piecewise rational cubic functions. The two families of parameters, in the description of the rational interpolant, have been constrained to preserve the shape of the data. The monotone rational cubic spline scheme has a unique representation.

In this article a numerical technique is presented for the solution of Fokker-–Planck equation. This method uses the cubic B-spline scaling functions. The method consists of expanding the required approximate solution as the elements of cubic B-spline scaling function. Using the operational matrix of derivative, the problem will be reduced to a set of algebraic equations. Some numerical example...

Spline interpolation has been used in several applications due to its favorable properties regarding smoothness and accuracy of the interpolant. However, when there exists a discontinuity or steep gradient data, some artifacts can appear Gibbs phenomenon. Also, preservation data monotonicity is requirement applications, that property not automatically verified by interpolator. In this paper, we...

The problem of computing best low order approximations of Boolean functions is treated in this paper. We focus on the case of best quadratic approximations of a wide class of cubic functions of arbitrary number of variables, and provide formulas for their efficient calculation. Our methodology is developed upon Shannon’s expansion formula and properties of best affine approximations of quadrati...

There are thousands of articles published on variants of splines, including least squares cubic splines. One of the first least squares articles was de Boor and Rice [1], and a comprehensive explanatory textbook is Dierckx [2]. Unfortunately, every example in the literature and on the web of a least squares cubic spline makes use of B-splines. While B-splines have a certain elegance, they are s...

Both the 4-point and the uniform cubic B-spline subdivisions double the number of vertices of a closed-loop polygon P and produce sequences of vertices fj and bj respectively. We study the J-spline subdivision scheme Js, introduced by Maillot and Stam, which blends these two methods to produce vertices of the form vj=(1–s)fj+sbj. Iterative applications of Js yield a family of limit curves, the ...

Legendre expansion of the ν ¯ ν ⇀ ↽ e + e − kernel: Influence of high order terms. Abstract We calculate the Legendre expansion of the rate of the process ν + ¯ ν ↔ e + + e − up to 3 rd order extending previous results of other authors which only consider the 0 th and 1 st order terms. Using different closure relations for the moment equations of the radiative transfer equation we discuss the p...