Order B-spline Research Articles

Optimal B-spline collocation method for self-adjoint singularly perturbed boundary value problems

We present a B-spline collocation method of higher order for a class of self-adjoint singularly perturbed boundary value problems. The essential idea in this method is to divide the domain of the differential equation into three non-overlapping subdomains and solve the regular problems obtained by transforming the differential equation with respective boundary conditions on these subdomains using the present higher order B-spline collocation method. The boundary conditions at the transition points are obtained by the asymptotic approximation of order zero to the solution of the problem. The convergence analysis is given and the method is shown to have optimal order convergence; by collocating the perturbed differential equation, which is satisfied by a special cubic spline interpolate of the true solution. Numerical experiments are conducted to demonstrate the efficiency of the method.

Pseudo-splines, wavelets and framelets

The first type of pseudo-splines were introduced in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (1) (2003) 1–46; I. Selesnick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2) (2001) 163–181] to construct tight framelets with desired approximation orders via the unitary extension principle of [A. Ron, Z. Shen, Affine systems in L 2 ( R d ) : The analysis of the analysis operator, J. Funct. Anal. 148 (2) (1997) 408–447]. In the spirit of the first type of pseudo-splines, we introduce here a new type (the second type) of pseudo-splines to construct symmetric or antisymmetric tight framelets with desired approximation orders. Pseudo-splines provide a rich family of refinable functions. B-splines are one of the special classes of pseudo-splines; orthogonal refinable functions (whose shifts form an orthonormal system given in [I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988) 909–996]) are another class of pseudo-splines; and so are the interpolatory refinable functions (which are the Lagrange interpolatory functions at Z and were first discussed in [S. Dubuc, Interpolation through an iterative scheme, J. Math. Anal. Appl. 114 (1986) 185–204]). The other pseudo-splines with various orders fill in the gaps between the B-splines and orthogonal refinable functions for the first type and between B-splines and interpolatory refinable functions for the second type. This gives a wide range of choices of refinable functions that meets various demands for balancing the approximation power, the length of the support, and the regularity in applications. This paper will give a regularity analysis of pseudo-splines of the both types and provide various constructions of wavelets and framelets. It is easy to see that the regularity of the first type of pseudo-splines is between B-spline and orthogonal refinable function of the same order. However, there is no precise regularity estimate for pseudo-splines in general. In this paper, an optimal estimate of the decay of the Fourier transform of the pseudo-splines is given. The regularity of pseudo-splines can then be deduced and hence, the regularity of the corresponding wavelets and framelets. The asymptotical regularity analysis, as the order of the pseudo-splines goes to infinity, is also provided. Furthermore, we show that in all tight frame systems constructed from pseudo-splines by methods provided both in [I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (1) (2003) 1–46] and this paper, there is one tight framelet from the generating set of the tight frame system whose dilations and shifts already form a Riesz basis for L 2 ( R ) .

Wavelets with Short Support

This paper is to construct Riesz wavelets with short support. Riesz wavelets with short support are the objective of interest in both theory and application. In theory, it is known that a B-spline of order m has the shortest support among all compactly supported refinable functions with the same regularity. However, it remained open whether a Riesz wavelet with the shortest support and m vanishing moments can be constructed from the multiresolution analysis generated by the B-spline of order m. In various applications, a Riesz wavelet with a short support, a high order of regularity, and vanishing moments is often desirable in signal and image processing, since they have a good time frequency localization and approximation property, as well as fast algorithms. This paper presents a theory for the construction of Riesz wavelets with short support and gives various examples. In particular, from the multiresolution analysis whose underlying refinable function is the B-spline of order m, we are able to construct the shortest supported Riesz wavelet with m vanishing moments. The support of the wavelet functions can be made even shorter by reducing their orders of vanishing moments. The study here also provides a new insight into the structures of the spline tight frame systems constructed in [A. Ron and Z. Shen, J. Funct. Anal., 148 (1997), pp. 408-447, I. Daubechies, B. Han, A. Ron, and Z. Shen, Appl. Comput. Harmon. Anal., 14 (2003), pp. 1-46, B. Han and Q. Mo, Proc. Amer. Math. Soc., 132 (2004), pp. 77-86] and bi-frame systems in [I. Daubechies, B. Han, A. Ron, and Z. Shen, Appl. Comput. Harmon. Anal., 14 (2003), pp. 1-46, I. Daubechies and B. Han, Constr. Approx., 20 (2004), pp. 325-352].

An advanced direct time domain BEM for 3-D wave propagation in acoustic media

This paper presents a new Boundary Element Method (BEM) in the direct time domain for the solution of the wave propagation problem in 3D homogenous inviscid acoustic media. The methodology is based on impulse response techniques. To this end, the fundamental solutions associated with the Boundary Integral Equation are derived for concentrated sources of higher order B-spline time modulation. High order spatial discretization is adopted. The solution is obtained in two major steps. First the B-spline Impulse Response Functions (BIRF) for a system of a specific geometry are calculated. BIRF functions are independent of the actual external excitation and need to be calculated only once for a given geometry. The second step calculates the response of the system to arbitrary excitations through a mere superposition of the BIRF functions. This paper discusses the development of the proposed BEM methodology and its validation. The method is shown to be stable regardless of the selection of the analysis time step, as opposed to the retarded potential techniques reported in the literature. The high level of accuracy and the efficiency of the method are demonstrated.

Numerical simulation of nonuniformly time-sampled pulse propagation in nonlinear fiber

Numerical simulation of pulse propagation in nonlinear optical fiber based on nonlinear Schrodinger equation plays a significant role in the design, analysis, and optimization of optical communication systems. Unconditionally stable operator-splitting techniques such as the split-step Fourier method or the split-step wavelet method have been successfully used for numerical simulation of uniformly time-sampled pulses along nonlinear optical fibers. Even though uniform time sampling is widely used in optical communication systems simulation, nonuniform time sampling is better or even desired for certain applications. For example, a sampling strategy that uses denser sampling points in regions where the signal changes rapidly and sparse sampling in regions where the signal change is gradual would result in a better replica of the signal. In this paper, we report a novel method that extends the standard operator-splitting techniques to handle nonuniformly sampled optical pulse profiles in the time domain. The proposed method relies on using cubic (or higher order) B-splines as a basis set for representing optical pulses in the time domain. We show that resulting operator matrices are banded and sparse due to the compact support of B-splines. Moreover, we use an algorithm based on Krylov subspace to exploit the sparsity of matrices for calculating matrix exponential operators. We present a comprehensive set of analytical and numerical simulation results to demonstrate the validity and accuracy of the proposed method.

Remarks on the Voronoi method in Paley–Wiener space

In Voronoi method, the reconstruction functions χn are not well-behaved, i.e., not continuous, which affect the convergence rate of this algorithm. In this paper, first, we obtain well-behaved reconstruction functions in place of χn; second, when the sampling set {xn:n∈Z} is denser, we obtain a faster convergent reconstruction iterative algorithm using analog to the mth order B-spline reconstruction functions φn(f)(x).

A class of compactly supported symmetric/antisymmetric B-spline wavelets

An algorithm for constructing a class of compactly supported symmetric/antisymmetric B-spline wavelets is presented. For any mth order and kth order cardinal B-spline Nm (x), Nk (x), if m + k is an even integer, the corresponding mth order B-spline wavelets ψ m k (x) can be constructed, which are compactly supported symmetric/antisymmetric. In addition, if ψ m k (x), m > 1 is mth B-spline wavelet associated with two spline functions Nm (x) and Nk (x), then (ψ m k (x))′(x) is m − 1th B-spline wavelet associated with Nm −1(x) and Nk +1(x), i.e. (ψ m k (x))′(x) = 22ψ m −1 k +1(x). Similarly, ∫0 x ψ m k (t)dt, k > 1 is m + 1th B-spline wavelet associated with Nm +1(x) and Nk −1(x). Using this method, we recovered Chui and Wang's spline wavelets. Since a class of B-spline wavelets are symmetric/antisymmetric, their linear phase property is assured. Several examples are also presented.

Biorthogonal spline type wavelets

Let φ be an orthonormal scaling function with approximation degree p − 1, and let B n be the B-spline of order n. Then, spline type scaling functions defined by ƒ n = ƒ * B n ( n = 1, 2, ...) possess higher approximation order, p + n − 1, and compact support. The corresponding biorthogonal wavelet functions are also constructed. This technique is extended to the case of biorthogonal scaling function system. As an application of the method supplied in this paper, one can easily construct a sequence of pairs of biorthogonal spline type scaling functions from one pair of biorthogonal scaling functions or an orthonormal scaling function. In particular, if both the method and the lifting scheme of Sweldens (see [1]) are applied, then all pairs of biorthogonal spline type scaling functions shown in references [2] and [3] can be constructed from the Haar scaling function.

Vibration analysis of rectangular composite laminated plates using layerwise B-spline finite strip method

A layerwise B-spline finite strip method is developed for free vibration analysis of truly thick and thin composite laminated plates within the context of a layerwise plate theory proposed by Reddy [Commun. Appl. Numer. Methods 3 (1987) 173]. The development is the extension of an earlier work [Int. J. Mech. Sci. 37 (1995) 645] in which a B k, k−1 -spline finite strip method was developed based on the first-order shear deformable plate theory and B-splines of order k and k−1 were used to avoid shear-locking. In this new method the composite laminated plates are divided into a number of numerical layers in the thickness direction. A linear variation of in-plane displacements are assumed within each numerical layer to represent the through thickness shear warping which can be significant for thick composite laminated plates due to their relatively low through thickness shear moduli. Numerical tests show that the method has the capability to produce accurate predictions for natural frequencies of truly thick and thin composite laminated plates.

A new symmetric fractional B-spline

A new definition of a symmetric fractional B-spline is presented. This generalises the usual integer order B-spline, that becomes a special case of the new one.

The density distribution function renewal process of three variables by means of the polynomial spline based on the b-splines

The question of asymptotic property research of local polynomial spline of three variables on the basis of the second order B-splines is considered in the article. Both the correspondent theorem and arguments as well as implications were formulated, proved and mentioned

Approximate merging of B-spline curves via knot adjustment and constrained optimization

This paper addresses the problem of approximate merging of two adjacent B-spline curves into one B-spline curve. The basic idea of the approach is to find the conditions for precise merging of two B-spline curves, and perturb the control points of the curves by constrained optimization subject to satisfying these conditions. To obtain a merged curve without superfluous knots, we present a new knot adjustment algorithm for adjusting the end k knots of a kth order B-spline curve without changing its shape. The more general problem of merging curves to pass through some target points is also discussed.

CHARMS

Finite element solvers are a basic component of simulation applications; they are common in computer graphics, engineering, and medical simulations. Although adaptive solvers can be of great value in reducing the often high computational cost of simulations they are not employed broadly. Indeed, building adaptive solvers can be a daunting task especially for 3D finite elements. In this paper we are introducing a new approach to produce conforming, hierarchical, adaptive refinement methods (CHARMS). The basic principle of our approach is to refine basis functions, not elements. This removes a number of implementation headaches associated with other approaches and is a general technique independent of domain dimension (here 2D and 3D), element type (e.g., triangle, quad, tetrahedron, hexahedron), and basis function order (piece-wise linear, higher order B-splines, Loop subdivision, etc.). The (un-)refinement algorithms are simple and require little in terms of data structure support. We demonstrate the versatility of our new approach through 2D and 3D examples, including medical applications and thin-shell animations.

Dynamic and Seismic Analysis of Foundations based on Free Field B-Spline Characteristic Response Histories

In this research, a direct time domain formulation for analysis of unbounded media and foundations is developed that treats dynamic excitations and ground motion in a uniform manner. The boundary element method with higher order B-Spline fundamental solutions is employed to compute the characteristic responses of the surface of the elastodynamic domain. Subsequently, time histories of the system response to general excitations are computed by a superposition scheme that accommodates in a uniform manner arbitrary time histories of external loads and/or ground motion. The characteristic responses are computed in the form of time dependent flexibility matrices of the media that are sparse due to the finite duration of the B-Spline excitation signal and characteristics of the wave propagation. The duration of the B-Spline impulse response is limited to a few time steps. Significant savings in computing time and storage requirements are achieved. The proposed approach also greatly reduces the size of the problems under consideration and still fully considers the effects of the free field. The significance of nonrelaxed boundary conditions and correct representation of the free field is established. The method is demonstrated and validated through applications pertaining to the analysis of foundations and inclusions subjected to transient loads and seismic excitations.

Analysis of piezolaminated plates by the spline finite strip method

This paper deals with the development of a family of higher order B-spline finite strip models applied to the static and free vibration analysis of laminated plates, with arbitrary shape and lay-ups, loading and boundary conditions. The lamination scheme can be such that the embedded and/or surface bonded piezoelectric actuating and sensing layers are included. The structure is discretised in a specified number of strips, and the geometry and displacement components of each strip are represented by interpolating functions that are products of linear or cubic B-spline, and linear or quadratic Lagrange functions along the y and x orthonormal directions. The accuracy and relative performance of the proposed discrete models are compared and discussed among the developed and alternative models.

CONSTRUCTION OF SUBOPTIMAL FEEDBACK CONTROL FOR CHAOTIC SYSTEMS USING B-SPLINES WITH OPTIMALLY CHOSEN KNOT POINTS

In this paper we consider a class of optimal control problem involving a chaotic system, where all admissible controls are required to satisfy small boundedness constraints. A numerical approach is developed to seek for an optimal feedback control for the optimal control problem. In this approach, the state space is partitioned into subregions, and the controller is approximated by a linear combination of a modified third order B-spline basis functions. The partition points are also taken as decision variables in this formulation. An algorithm based on this approach is proposed. To show the effectiveness of the proposed method, a control problem involving the Lorenz system is solved by the proposed approach. The numerical results demonstrate that the method is efficient in the construction of a robust, near-optimal control.

MOMS: maximal-order interpolation of minimal support

We consider the problem of interpolating a signal using a linear combination of shifted versions of a compactly-supported basis function phi(x). We first give the expression for the cases of phi's that have minimal support for a given accuracy (also known as "approximation order"). This class of functions, which we call maximal-order-minimal-support functions (MOMS) is made of linear combinations of the B-spline of the same order and of its derivatives. We provide an explicit form of the MOMS that maximizes the approximation accuracy when the step-size is small enough. We compute the sampling gain obtained by using these optimal basis functions over the splines of the same order. We show that it is already substantial for small orders and that it further increases with the approximation order L. When L is large, this sampling gain becomes linear; more specifically, its exact asymptotic expression is 2/(pie)L. Since the optimal functions are continuous, but not differentiable, for even orders, and even only piecewise continuous for odd orders, our result implies that regularity has little to do with approximating performance. These theoretical findings are corroborated by experimental evidence that involves compounded rotations of images.

A new factorization technique of the matrix mask of univariate refinable functions

A univariate compactly supported refinable function $f$ can always be factored into $B_kast f$, with $B_k$ the B-spline of order $k$, $f$ a compactly supported distribution, and $k$ the approximation orders provided by the underlying shift-invariant space $S(f)$. Factorizations of univariate refinable {it vectors} $ff$ were also studied and utilized in the literature. One of the by-products of this article is a rigorous analysis of that factorization notion, including, possibly, the first precise definition of that process. The main goal of this article is the introduction of a special factorization algorithm of refinable vectors that generalizes the scalar case as closely (and unexpectedly) as possible: the original vector $ff$ is shown to be `almost' in the form $B_kastfff$, with $fff$ still compactly supported and refinable, and $k$ the approximation order of $S(ff)$: `almost' in the sense that $ff$ and $B_kastfff$ differ at most in one entry. The algorithm guarantees $fff$ to retain the possible favorable properties of $ff$, such as the stability of the shifts of $ff$ and/or the polynomiality of the mask symbol. At the same time, the theory and the algorithm are derived under relatively mild conditions and, in particular, apply to $ff$ whose shifts are not stable, as well as to refinable vectors which are not compactly supported. The usefulness of this specific factorization for the study of the smoothness of FSI wavelets (known also as `multiwavelets' and `multiple wavelets') is explained. The analysis invokes in an essential way the theory of finitely generated shift-invariant (FSI) spaces, and, in particular, the tool of {it superfunction theory}

Interpolation revisited.

Based on the theory of approximation, this paper presents a unified analysis of interpolation and resampling techniques. An important issue is the choice of adequate basis functions. We show that, contrary to the common belief, those that perform best are not interpolating. By opposition to traditional interpolation, we call their use generalized interpolation; they involve a prefiltering step when correctly applied. We explain why the approximation order inherent in any basis function is important to limit interpolation artifacts. The decomposition theorem states that any basis function endowed with approximation order can be expressed as the convolution of a B-spline of the same order with another function that has none. This motivates the use of splines and spline-based functions as a tunable way to keep artifacts in check without any significant cost penalty. We discuss implementation and performance issues, and we provide experimental evidence to support our claims.

Numerical simulation of scattering from rough surfaces: a wavelet-based approach

In this paper, a preliminary study is carried out to demonstrate the application of wavelets for improving the computation time and reducing computational memory required for evaluating the statistics of the scattered field from rough surfaces using the method of moments (MoM) in conjunction with a Monte Carlo simulation. In specific, Haar and the first order B-spline wavelet basis functions are applied to the MoM formulation of one-dimensional rough surfaces in order to compare the computation time and sparsity for wavelets in the same family but of higher order. Since the scattering coefficient (the second moment of the backscatter field per unit area) is a gentle function of the surface parameters and the radar attributes, it is demonstrated that a relatively high thresholding level can be applied to the impedance matrix, which leads to a sparser impedance matrix and faster computation time. It is also shown that applying a high threshold level the coefficients of the high-order wavelets would increase out of proportion, however, the effect of these current components averages out when computing the scattering coefficients. The resulting sparse impedance matrices are solved efficiently using fast search routines such as the conjugate gradient method. A systematic study is carried out to investigate the effect of different threshold levels on the accuracy versus computing speed criterion. The computed scattering coefficients are compared to previous results computed using a conventional pulse basis function as well as the existing theoretical solutions for rough surfaces. It is shown that wavelet basis functions provide substantial reductions in both memory requirements and computation time.