Order Spline Research Articles

Problems and Techniques

The three articles in this edition of Problems and Techniques describe fractional splines and wavelets, matrix factorizations arising in numerical linear algebra and applied statistics/psychometrics, and canonical transformations of Hamiltonian systems. Our first article, by Michael Unser and Thierry Blu, extends the popular polynomial B-splines to fractional orders. Why not? Euler's gamma function provides nonintegral factorials and Liouville's formula generalizes differentiation to fractional orders. The fractional order splines are developed from fractional order truncated power functions and fractional order forward difference operators in a spirit similar to that used for polynomial B-splines. The resulting splines are not polynomials and do not have compact support, but they do have many properties that are similar to B-splines. There are some surprises, though. Our second article, by Lawrence Hubert, Jacqueline Meulman, and Willem Heiser, presents a historical discussion of matrix factorization as it arises in numerical linear algebra (NLA) to factor matrices and in applied statistics/psychometrics (AS/P) to obtain lower-rank approximations of matrices. In particular, the authors compare the rank-1 reductions of Chu, Funderlic, and Golub in NLA with similar reductions of Guttman in AS/P. One could probably guess that there would be uses of singular value decompositions in NLA and AS/P, and some interesting generalized procedures are described. There's lots of history here as well as a different perspective on tasks that most of us perform on a regular basis. The final submission, by J. C. Orum, R. T. Hudspeth, W. Black, and R. B. Guenther, deals with transformations of nonautonomous Hamiltonian systems. The authors describe an extension of a 1932 algorithm due to Herglotz for autonomous systems to the nonautonomous case. The authors illustrate the transformation by applying it to problems in dynamical systems including a nice example involving an oscillating pendulum with a variable length rod. I hope you enjoy reading these articles. I did. Please join me in thanking our authors for their fine contributions.

Shape Optimization To Perform Prescribed Air Lubrication Using Genetic Algorithm

A shape optimization of magnetic head for magnetic recording device with rotating drum was studied. An optimization algorithm was developed based on genetic algorithms in conjunction with finite element method which analyzed interface phenomena between a magnetic tape and a magnetic head. The head shape was defined by a second order spline function. Three reference points and first order derivatives at a origin point were coded by the genetic algorithm. As a result, an applicability of genetic algorithms to the given optimization problem was verified. Presented as a Society of Tribologists and Lubrication Engineers Paper at the ASME/STLE Tribology Conference in Seattle, Washington, October 1–4, 2000

Minimum time on-line joint trajectory generator based on low order spline method for industrial manipulators

In this paper, we propose the use of a low order polynomial method named 3-Cubic method to generate minimum time on-line joint trajectory for an industrial manipulator. A 3-Cubic Spline between two end points consists of a set of three cubic splines, which not only give an analytical solution to minimize travel time with velocity, acceleration and jerk constraints but also provides the continuity till second derivative of position (i.e. acceleration). This results in a position trajectory being dynamically balanced. This leads to the generation of a new type of trajectory named 3-Cubic Linear Segment with Trapezoidal Transition (3CLSTT) trajectory. In this trajectory generation, duration of constant acceleration/deceleration and constant velocity phases are calculated according to the requirements of maximum limiting values of velocity and acceleration thus traversing the trajectory in minimum possible time. Wandering (undershoot/overshoot) is completely eliminated in the resulting trajectories with limited computational effort.

Use of weighted least-squares splines for calibration in analytical chemistry

Weighted least-squares spline functions are discussed and applied for calibration processes in analytical chemistry. Different weighting techniques are also considered, and for the evaluation of the results some quality coefficients are proposed. Depending upon the structure of the data, some weighting procedures may improve the results dramatically. Considering the results obtained in the case of TLC densitometry, it seems that nonlinear weighting procedures based upon the distance to the function are the best ones, with a plus for the y-distance type. It is difficult to give general rules regarding the optimal parameters of the weighted calibration splines-function order (m), number (N) and distribution of knots, and weighting technique. These depend upon the structure of the data. However, higher order splines are not recommended since the result might become extremely unstable. The example used to illustrate the performances of the procedures discussed here involved only a single independent variable. The method is general and extends practically to any number of variables, thus resulting in a multivariate approach.

Estimates, decay properties, and computation of the dual function for Gabor frames

We present a simple proof of Ron and Shen's frame bounds estimates for Gabor frames. The proof is based on the Heil and Walnut's representation of the frame operator and shows that it can be decomposed into a continuous family of infinite matrices. The estimates then follow from a simple application of Gershgorin's theorem to each matrix. Next, we show that, if the window function has exponential decay, also the dual function has some exponential decay. Then, we describe a numerical method to compute the dual function and give an estimate of the error. Finally, we consider the spline of order 2; we investigate numerically the region of the time-frequency plane where it generates a frame and we compute the dual function for some values of the parameters.

Particular solutions of Helmholtz-type operators using higher order polyhrmonic splines

We obtain explicit analytical particular solutions for Helmholtz-type operators, using higher order splines. These results generalize those in Golberg, Chen and Rashed (1998) and Chen and Rashed (1998) for thin plate splines. This enables one to substantially improve the accuracy of algorithms for solving boundary value problems for Helmholtz-type equations.

Optimal order spline methods for nonlinear differential and integro-differential equations

In this work, we analyze new robust spline approximation methods for mth order boundary value problems described by nonlinear ordinary differential and integro-differential equations with m linear boundary conditions. Our main aim is to introduce a cost-effective alternative to the highly successful orthogonal collocation method, and to prove stability and convergence properties similar to the orthogonal collocation method. Our method is a discrete Petrov-Galerkin method, in which we seek a spline approximation u h of order m + r, but with one higher degree of smoothness than for the standard orthogonal collocation approximation: we require u h ∈ C m (in contrast to C m − 1 in the orthogonal collocation case), so that u h ( m) ∈ C. We show that optimal order convergence and superconvergence at break points hold without any mesh restriction, provided only that a certain underlying quadrature rule has degree of precision at least 2 r − 1. There is one respect in which the present method has an advantage over orthogonal collocation. Firstly, because of the increased continuity of u h , the number of unknown variables is reduced (almost halved in certain practical cases), which is a significant reduction for complex nonlinear problems. Orthogonal collocation method itself can be obtained by changing a few parameters in our method. To test our theoretical results and demonstrate the generality of the method, we consider an application to a catalytic combustion model problem involving nonlinear integro-differential equation, compute solutions of a nonlinear ordinary differential equation, and solve a boundary value problem with a thin boundary layer occurring in a nonlinear convection problem.

The annihilator method for computing particular solutions to partial differential equations

We generalize the well-known annihilator method, used to find particular solutions for ordinary differential equations, to partial differential equations. This method is then used to find particular solutions of Helmholtz-type equations when the right hand side is a linear combination of thin plate and higher order splines. These particular solutions are useful in numerical algorithms for solving boundary value problems for a variety of elliptic and parabolic partial differential equations.

MHD Stability Analysis with Higher Order Spline Functions.

The eigenvalue problem of the linearized magnetohydrodynamic (MHD) equation is formulated by using higher order spline functions as the base functions of Ritz-Galerkin approximation. When the displacement vector normal to the magnetic surface (in the magnetic surface) is interpolated by B-spline functions of degree pl (degree p2) which is continuously cl-th (c2-th) differentiable on neighboring finite elements, the sufficient conditions for the good approximation is given by pl ~ p2 + I , cl ~ c2 + I , (cl ~~ I , p2 ;~ c2 ~ O). The influence of the numerical integration upon the convergence of calculated eigenvalues is discussed.

Stability of discrete orthogonal projections for continuous splines

In this paper Lp stability and convergence properties of discrete orthogonal projections on the finite element space Sh of continuous polynomial splines of order r are proved. The discrete inner products are defined by composite quadrature rules with positive weights on a sequence of nonuniform grids. It is assumed that the basic quadrature rule Q has at least r quadrature points in order to resolve Sh, but no accuracy is required. The main results are derived under minimal further assumptions, for example the rule Q is allowed to be non-symmetric, and no quasi-uniformity of the mesh is required. The corresponding stability of the orthogonal L2-projections has been studied by de Boor [1] and by Crouzeix and Thomee [2]. Stability of the first derivative of the projection is also proved, under an assumption (unless p = 1) of local quasi-uniformity of the mesh.

Numerical solutions for system of second order boundary value problems

We investigate some numerical methods for computing approximate solutions of a system of second order boundary value problems associated with obstacle, unilateral and contact problems. We show that cubic spline method gives approximations which are better than that computed by higher order spline and finite difference techniques.

Detrending of non-stationary noise data by spline techniques

An off-line method of detrending non-stationary noise data is given. It uses a least-squares spline approximation of the noise data with equally spaced breakpoints. Subtraction of the spline approximation from the noise signal at each data point gives a residual noise signal. The method acts as a high-pass filter with very sharp frequency cut-off. The cut-off frequency is determined by the breakpoint distance. The steepness of the cut-off is controlled by the spline order.

Achieving Higher-Order Convergence Rates for Density Estimation with Binned Data

A method is proposed for achieving fast—O(n –8/9) and faster—asymptotic mean integrated squared error (AMISE) convergence rates in density estimation using data condensed to standard histogram bin counts and edges. The method involves the two elements of B splines and the correct mass proportions property, in which the probability mass for the histogram bins equals exactly the fraction of the data found in those bins. It is shown that the asymptotic bias of such an estimate using a spline of order p and bin width h will be O(hP ), resulting in optimal AMISE of O(n –2p/2p+1) form. Theoretical and numeric comparisons to kernel-based and other estimators show that the histospline estimator is usually comparable and often superior. An extension to the multivariate density estimation setting is described.

Achieving Higher-Order Convergence Rates for Density Estimation with Binned Data

Abstract A method is proposed for achieving fast—O(n –8/9) and faster—asymptotic mean integrated squared error (AMISE) convergence rates in density estimation using data condensed to standard histogram bin counts and edges. The method involves the two elements of B splines and the correct mass proportions property, in which the probability mass for the histogram bins equals exactly the fraction of the data found in those bins. It is shown that the asymptotic bias of such an estimate using a spline of order p and bin width h will be O(hP ), resulting in optimal AMISE of O(n –2p/2p+1) form. Theoretical and numeric comparisons to kernel-based and other estimators show that the histospline estimator is usually comparable and often superior. An extension to the multivariate density estimation setting is described.

High-quality image resizing using oblique projection operators

The standard interpolation approach to image resizing is to fit the original picture with a continuous model and resample the function at the desired rate. However, one can obtain more accurate results if one applies a filter prior to sampling, a fact well known from sampling theory. The optimal solution corresponds to an orthogonal projection onto the underlying continuous signal space. Unfortunately, the optimal projection prefilter is difficult to implement when sine or high order spline functions are used. We propose to resize the image using an oblique rather than an orthogonal projection operator in order to make use of faster, simpler, and more general algorithms. We show that we can achieve almost the same result as with the orthogonal projection provided that we use the same approximation space. The main advantage is that it becomes perfectly feasible to use higher order models (e.g. splines of degree n=or>3). We develop the theoretical background and present a simple and practical implementation procedure using B-splines. Our experiments show that the proposed algorithm consistently outperforms the standard interpolation methods and that it provides essentially the same performance as the optimal procedure (least squares solution) with considerably fewer computations. The method works for arbitrary scaling factors and is applicable to both image enlargement and reduction.

High order convergence for collocation of second kind boundary integral equations on polygons

We propose collocation methods with smoothest splines to solve the integral equation of the second kind on a plane polygon. They are based on the bijectivity of the double layer potential between spaces of Sobolev type with arbitrary high regularity and involving the singular functions generated by the corners. If splines of order \(2m-1\) are used, we get quasi-optimal estimates in \(H^m\)-norm and optimal order convergence for the \(H^k\)-norm if \(0\le k\le m\). Numerical experiments are presented.

Proportional odds regression and sieve maximum likelihood estimation

This paper introduces, under a censoring mechanism, an estimation procedure for the baseline function and the regression parameters in the proportional odds model based on the sieve maximum likelihood method. The procedure uses monotone splines of variable orders and knots as approximations for the baseline function and an approximately unbiased estimator of the comparative Kullback-Leibler risk to select the best approximation. The procedure adapts to various unknown structures of the baseline function such as spatial structures and smoothness of an unknown degree. The proposed procedure is implemented for right-censored and so-called Case 2 interval-censored data. The estimated regression parameters are shown to be asymptotically normal and efficient. The small sample operating characteristics of the proposed procedure are examined via simulations and are illustrated on a dataset from a study of breast cancer patients.

A new family of convex splines for data interpolation

This paper develops a new family of convexity-preserving splines of order n, hereby entitled the CP n-spline, that preserves convexity when derivatives at the data points satisfy some reasonable conditions. The spline comprises four components: a constant term, a first order term, and two nth order binomials. A slope-averaging-method is proposed for the general implementation of the new spline. Numerical results that allow for an assessment of the new spline are provided. In particular, a comparative analysis of the CP n-spline, the cubic spline, and of the Carnicer '92 spline is performed. By varying two parameters, the spline shape can be controlled at the local level, while other conventional means can be used to control the shape at the global level. The CP n-spline has no singularities in the case where inflection points are present. Additionally, a less general form of the CP n-spline that applies to most practical cases can be implemented with extreme ease.

Multivariate regression splines

Multivariate regression splines of arbitrary order assuming known knots using the additional function are developed. Model description and possible parameter tests for obtaining regression splines are also stated in detail for the triplicate case. A simulation study of drawn data from the Lubricant nonlinear regression model, where “Lubricant” is referred to lubricant data that we use in application, to compare the mean squares errors for the multivariate regression spline models and the multivariate polynomial models show the need of employing the multivariate regression splines in the approximation of the nonlinear regression models.

Design of non-uniformly spaced pseudo-QMF banks based on the spline function

A computationally less complex technique using an optimal order spline as the transition function is proposed to design non-uniformly spaced pseudoquadrature mirror filters (pseudo-QMF) having linear phase and flat overall passband response. The proposed technique permits control of the transition bandwidth of filters along with minimization of integral squared error without generating Gibb's phenomenon. Sharp transition bands with reasonably large stopband attenuation can be obtained at relatively lower filter lengths, which can be very useful for subband coding of speech and image signals in which some bands may be required to be coarsely coded or discarded.