Uniform Knots Research Articles

A meshless multi-symplectic local radial basis function collocation scheme for the “good” Boussinesq equation

A novel meshless multi-symplectic scheme is proposed for the “good” Boussinesq equation. The scheme consists of a local radial basis function (RBF) collocation method (LRBFCS) in space and a symplectic integrator in time. The LRBFCS is simple and efficient, since only a sparse banded linear system has to be solved. Moreover, it can avoid the ill-conditioned problem and shape-parameter-sensitivity of the global RBF method. The multi-symplectic LRBFCS (MLRBFCS) is more accurate than traditional methods. Numerical experiments with uniform knots and random knots are designed to verify the effectiveness of the method.

Construction of Cubic Trigonometric Curves with an Application of Curve Modelling

This paper introduces new trigonometric basis functions (TBF) in polynomial and rational form with two shape parameters (SPs). Some classical characteristics, such as the partition of unity, positivity, symmetry, CHP, local control and invariance under affine transformation properties are proven mathematically and graphically. In addition, different continuity conditions at uniform knots (UK) are proven. Some open and closed curves from TBS and trigonometric rational B-spline (TRBS) are generated to test the applicability of the suggested technique, and the influence of the shape parameter is also noted. Furthermore, various objects, such as designing an alphabet, star, butterfly, leaf and 3D cube.

Meshless symplectic and multi-symplectic local RBF collocation methods for nonlinear Schrödinger equation

This study formulates a novel meshless symplectic and multi-symplectic local radial basis function (RBF) collocation method (LRBFCM) for the nonlinear Schrödiner equation. The LRBFCM is employed for spatial discretization and then symplectic integrator is conducted for time discretization. The properties of the space differentiation matrix of LRBFCM are discussed in detail. The conservativeness of the proposed method is discussed and the accuracy is assessed. The LRBFCM is simple and efficient, since it can avoid the ill-conditioned problem and shape-parameter-sensitivity of global RBF method. Numerical tests with uniform knots and random knots are designed to show the effectiveness of the method.

Generating bicubic B-spline surfaces by a sixth order PDE

As the solutions of partial differential equations (PDEs), PDE surfaces provide an effective way for physical-based surface design in surface modeling. The bicubic B-spline surface is a useful tool for surface modeling in computer aided geometric design (CAGD). In this paper, we present a method for generating bicubic B-spline surfaces with the uniform knots and the quasi-uniform knots from the sixth order PDEs. From the given boundary condition, based on the cubic B-spline basis representation and the PDE mask, the resulting bicubic B-spline surface can be generated uniquely. The boundary condition is more flexible and can be applied for curvature-continuous surface design, surface blending and hole filling. Some representative examples show the effectiveness of the presented method.

A Method for the Construction of LocalParabolic Splines with Additional Knots

We propose a general method for the construction of local parabolic splines with an arbitrary arrangement of knots for functions given on grid subsets of the number axis or its segment. Special cases of this scheme are Yu. N. Subbotin’s and B. I. Kvasov’s splines. For Kvasov’s splines, we consider boundary conditions different from those suggested by Kvasov. We study the approximating and smoothing properties of these splines in the case of uniform knots. In particular, we find two-sided estimates for the error of approximation of the function classes $$W_{\infty}^{2}$$ and $$W_{\infty}^{3}$$ by these splines in the uniform metric and calculate the exact uniform Lebesgue constants and the norms of the second derivatives on the class $$W_{\infty}^{2}$$ . These properties are compared with the corresponding properties of Subbotin’s splines.

New rational cubic trigonometric B-spline curves with two shape parameters

Trigonometric B-spline curves have gained a remarkable attention in computer-aided geometric design (CAGD). This paper presents the cubic and rational cubic trigonometric B-spline curves using new trigonometric functions and shape parameter $$ \eta \in (1/2,2].$$ The proposed curves inherit the basic properties of classical B-spline and have been proved. For uniform knots, both curves are $$C^2$$ continuous. On non-uniform knots, cubic trigonometric curves are $$C^3$$ and $$C^5$$ continuous, whereas rational trigonometric curves are $$C^3$$ continuous and have been derived. The applicability of proposed curves has been checked by constructing open and closed curves. Different models like glass, kettle, human hand, and vase have been designed by both schemes and compared.

On Integral Lebesgue Constants of Local Splines with Uniform Knots

We study the stability properties of generalized local splines of the form $$S(x) = S(f,x) = \sum\limits_{j \in \mathbb{Z}} {{y_j}{B_\varphi }\left( {x + \frac{{3h}}{2} - jh} \right),}\;\;\; x \in \mathbb{R},$$ where ϕ ∈ C1[−h, h] for h > 0, ϕ(0) = ϕ′(0) = 0, ϕ(−x) = ϕ(x)for x ∈ [0; h], ϕ(x) is nondecreasing on [0; h], f is an arbitrary function from ℝ to ℝ, yj = f(jh) for j ∈ ℤ, and $${B_\varphi }(x) = m(h)\left\{ {\begin{array}{*{20}{c}} {\varphi (x),}&{x \in [0;h],} \\ {2\varphi (h) - \varphi (x - h) - \varphi (2h - x),}&{x \in [h;2h],} \\ {\varphi (3h - x),}&{x \in [2h;3h],} \\ {0,}&{x \notin [0;3h],} \end{array}} \right.$$ with m(h) > 0. Such splines were constructed by the author earlier. In the present paper, we calculate the exact values of their integral Lebesgue constants (the norms of linear operators from l to L) on the axis ℝ and on any segment of the axis for a certain choice of the boundary conditions and the normalizing factor m(h) of the spline S.

Uniform Approximation of the Curvature of Smooth Plane Curves with the Use of Partial Fourier Sums

An error bound for the approximation of the curvature of graphs of periodic functions from the class Wr for r ≥ 3 in the uniform metric is obtained with the use of the simplest approximation technique for smooth periodic functions, which is approximation by partial sums of their trigonometric Fourier series. From the mathematical point of view, the interest in this problem is connected with the specific nonlinearity of the graph curvature operator on the class of smooth functions Wr on a period or a closed interval for r ≥ 2. There are several papers on curvature approximation for plane curves in the mean-square and Chebyshev norms. In previous works, the approximation was performed by partial sums of trigonometric series (in the L2 norm), interpolation splines with uniform knots, Fejer means of partial sums of trigonometric series, and orthogonal interpolating wavelets based on Meyer wavelets (in the C∞ norm). The technique of this paper, based on the lemma, can possibly be generalized to the Lp metric and other approximation methods.

Optimal Spline Spaces for $$L^2$$ L 2 n-Width Problems with Boundary Conditions

In this paper we show that, with respect to the $L^2$ norm, three classes of functions in $H^r(0,1)$, defined by certain boundary conditions, admit optimal spline spaces of all degrees $\geq r-1$, and all these spline spaces have uniform knots.

A Method for the Construction of Wavelet Analogs by Means of Trigonometric B-Splines

We construct an analog of two-scale relations for basis trigonometric splines with uniform knots corresponding to a linear differential operator of order 2r + 1 with constant coefficients L2r+1(D) = D(D2 + α12 )(D2 + α22 )... (D2 + α r 2 ), where α1, α2,..., α r are arbitrary positive numbers. The properties of nested subspaces of trigonometric splines are analyzed.

Upper bounds for uniform Lebesgue constants of interpolation periodic sourcewise representable splines

Upper bounds for Lebesgue constants (norms of linear operators from C to C) of interpolation periodic sourcewise representable splines with uniform knots are obtained for a wide class of periodic integrable kernels K.

Two-scale relations for B-L-splines with uniform knots

Analogs of scaling relations are constructed for basis exponential splines with uniform knots corresponding to a linear differential operator of arbitrary order with constant coefficients and real pairwise distinct roots of the characteristic polynomial; the construction does not employ techniques from harmonic analysis.

Splines over regular triangulations in numerical simulation

We investigate the use of smooth spline spaces over regular triangulations as a tool in (isogeometric) Galerkin methods. In particular, we focus on box splines over three-directional meshes. Box splines are multivariate generalizations of univariate cardinal B-splines sharing the same properties. Tensor-product B-splines with uniform knots are a special case of box splines. The use of box splines over three-directional meshes has several advantages compared with tensor-product B-splines, including enhanced flexibility in the treatment of the geometry and stiffness matrices with stronger sparsity. Boundary conditions are imposed in a weak form to avoid the construction of special boundary functions. We illustrate the effectiveness of the approach by means of a selection of numerical examples.

A baseline correction algorithm for Raman spectroscopy by adaptive knots B-spline

The Raman spectroscopy technique is a powerful and non-invasive technique for molecular fingerprint detection which has been widely used in many areas, such as food safety, drug safety, and environmental testing. But Raman signals can be easily corrupted by a fluorescent background, therefore we presented a baseline correction algorithm to suppress the fluorescent background in this paper. In this algorithm, the background of the Raman signal was suppressed by fitting a curve called a baseline using a cyclic approximation method. Instead of the traditional polynomial fitting, we used the B-spline as the fitting algorithm due to its advantages of low-order and smoothness, which can avoid under-fitting and over-fitting effectively. In addition, we also presented an automatic adaptive knot generation method to replace traditional uniform knots. This algorithm can obtain the desired performance for most Raman spectra with varying baselines without any user input or preprocessing step. In the simulation, three kinds of fluorescent background lines were introduced to test the effectiveness of the proposed method. We showed that two real Raman spectra (parathion-methyl and colza oil) can be detected and their baselines were also corrected by the proposed method.

Local modification of Pythagorean-hodograph quintic spline curves using the B-spline form

The problems of determining the B–spline form of a C 2 Pythagorean–hodograph (PH) quintic spline curve interpolating given points, and of using this form to make local modifications, are addressed. To achieve the correct order of continuity, a quintic B–spline basis constructed on a knot sequence in which each (interior) knot is of multiplicity 3 is required. C 2 quintic bases on uniform triple knots are constructed for both open and closed C 2 curves, and are used to derive simple explicit formulae for the B–spline control points of C 2 PH quintic spline curves. These B-spline control points are verified, and generalized to the case of non–uniform knots, by applying a knot removal scheme to the Bézier control points of the individual PH quintic spline segments, associated with a set of six–fold knots. Based on the B–spline form, a scheme for the local modification of planar PH quintic splines, in response to a control point displacement, is proposed. Only two contiguous spline segments are modified, but to preserve the PH nature of the modified segments, the continuity between modified and unmodified segments must be relaxed from C 2 to C 1. A number of computed examples are presented, to compare the shape quality of PH quintic and “ordinary” cubic splines subject to control point modifications.

Form preservation under approximation by local exponential splines of an arbitrary order

We continue the study of the properties of local L-splines with uniform knots (such splines were constructed in the authors’ earlier papers) corresponding to a linear differential operator L of order r with constant coefficients and real pairwise different roots of the characteristic polynomial. Sufficient conditions (which are also necessary) are established under which an L-spline locally inherits the property of the generalized k-monotonicity (k ≤ r − 1) of the input data, which are the values of the approximated function at the nodes of a uniform grid shifted with respect to the grid of knots of the L-spline. The parameters of an L-spline that is exact on the kernel of the operator L are written explicitly.

Approximation by local ℒ-splines that are exact on subspaces of the kernel of a differential operator

We construct local ℒ-splines with uniform knots that preserve subsets from the kernel of a linear differential operator ℒ of order r with constant real coefficients and pairwise distinct roots of the characteristic polynomial. Pointwise estimates are found for the error of approximation by the constructed ℒ-splines on classes of functions defined by differential operators of orders smaller than r.

Hybrid spline frames

Using their unitary extension principle, Ron and Shen have constructed a normalized tight frame for L 2 (ℝ) consisting of spline functions with uniform knots. This paper constructs a normalized tight frame for L 2 ((0, oo)) comprising spline functions with knots on a hybrid of uniform and geometric mesh. The construction is motivated by applications in adaptive approximation using spline functions on a hybrid mesh that admits a natural dyadic multiresolution approximation of L 2 ((0,∞)) based on dilation and translation.

Surface fitting based on a feature sensitive parametrization

Most approaches to least squares fitting of a B-spline surface to measurement data require a parametrization of the data point set and the choice of suitable knot vectors. We propose to use uniform knots in connection with a feature sensitive parametrization. This parametrization allocates more parameter space to highly curved feature regions and thus automatically provides more control points where they are needed.

Shape-preserving interpolation by splines using vector subdivision

We give a local convexity preserving interpolation scheme using parametricC2 cubic splines with uniform knots produced by a vector subdivision scheme which simultaneously provides the function and its first and second order derivatives. This is also adapted to give a scheme which is both local convexity and local monotonicity preserving when the data values are strictly increasing in thex-direction.