Order Spline Research Articles

Two-Sided Estimates of Some Coordinate Splines

Two-sided estimates for the continuously differentiable coordinate splines of the second order are established, and sufficient conditions of their nonnegativity are provided. The results obtained are applied to trigonometric splines.

Sharp Bernstein Type Inequalities for Splines in the Mean Square Metrics

We give an elementary proof of the sharp Bernstein type inequality $$ {\left\Vert {f}^{(s)}\right\Vert}_2\le \frac{n^s}{2^s}{\left(\frac{\kappa_{2r+1-2s}}{\kappa_{2r+1}}\right)}^{1/2}{\left\Vert {\updelta}_{\frac{\uppi}{n}}^sf\right\Vert}_2. $$ Here n, r, s ∈ ℕ, f is a 2π-periodic spline of order r and of minimal defect with nodes $$ \frac{\mathrm{j}\uppi}{n} $$ , j ∈ Z, δ is the difference operator of order s with step h, and the K m are the Favard constants. A similar inequality for the space L 2(ℝ) is also established. Bibliography: 5 titles.

Real-time dynamics acquisition from irregular samples — With application to anesthesia evaluation

Although most digital representations of information sources are obtained by uniform sampling of some continuous function representations, there are many important events for which only irregular data samples are available, including trading data of the financial market and various clinical data, such as the respiration signals hidden in ECG measurements. For such digital information sources, the only available effective smooth function interpolation scheme for digital-to-analog (D/A) conversion algorithms are mainly for offline applications. Hence, in order to adapt the powerful continuous-function mathematical approaches for real-time applications, it is necessary to introduce an effective D/A conversion scheme as well as to modify the desired continuous-function mathematical method for online implementation. The powerful signal processing tool to be discussed in this paper is the synchrosqueezed continuous wavelet transform (SST), which requires computation of the continuous wavelet transform (CWT), as well as its derivative, of the analog signal of interest. An important application of this transform is to extract information, such as the underlying dynamics, hidden in the signal representation. The first objective of this paper is to introduce a unified approach to remove the two main obstacles for adapting the SST approach to irregular data samples in order to allow online computation. Firstly, for D/A conversion, a real-time algorithm, based on spline functions of arbitrarily desired order, is proposed to interpolate the irregular data samples, while preserving all polynomials of the same spline order, with assured maximum order of approximation. Secondly, for real-time dynamic information extraction from an oscillatory signal via SST, a family of vanishing-moment and minimum-supported spline-wavelets (to be called VM wavelets) are introduced for online computation of the CWT and its derivative. The second objective of this paper is to apply the proposed real-time algorithm and VM wavelets to clinical applications, particularly to the study of the “anesthetic depth” of a patient during surgery, with emphasis on analyzing two dynamic quantities: the “instantaneous frequencies” and the “non-rhythmic to rhythmic ratios” of the patient’s respiration, based on a one-lead electrocardiogram (ECG) signal. Indeed, the “R-peaks” of the ECG signal, which constitute a waveform landmark for clinical evaluation, are non-uniform samples of the respiratory signal. It is envisioned that the proposed algorithm and VM wavelets should enable real-time monitoring of “anesthetic depth”, during surgery, from the respiration signal via ECG measurement.

Two-Level Spline Approximations for Two-Dimensional Navier–Stokes Equations

Abstract The C 1 $C^{1}$ spline spaces with degree d ≥ 5 $d\geq 5$ over given triangulations are implemented in the framework of multi-variate spline theory. Based on this approach, two-level methods are proposed by using various order spline spaces for the steady state Navier–Stokes equations in the stream function formulation. The proposed method can be reduced to solving a linear equation in the high-order spline space and the nonlinear equations in the low-order spline space. The convergence analysis is given based on the Newton iteration. Besides, the matrix forms of the two-level scheme are also presented. We finally tabulate the numerical results to validate and show the efficiency of the proposed two-level spline methods.

On Stability Preservation under Sampling and Approximation of Feedbacks for Retarded Systems

It is standard worldwide, in both industrial and academic activities, to design in continuous time a stabilizing state feedback for a nonlinear system and then implement it by digital devices using sampling and zero-order holding. It is well known, from practice, that stability is rather preserved if sampling is performed at suitable high frequency. This fact has been proved in the literature, also from a theoretical point of view, for delay-free nonlinear systems. An analogous result has not been proved, from the theoretical point of view, for nonlinear retarded systems. Moreover, in the case of retarded systems, implementation by means of digital devices often requires some further approximation due to nonavailability in the buffer of the value of the system variables at some past times. In order to cope with this problem, we make use here of the well-known approximation scheme based on first order splines. We introduce the standard hypothesis that there exists a stabilizing continuous-time state feedback and that there exists a related suitable Lyapunov--Krasovskii functional (in a large class already introduced in the literature) for the continuous-time, globally asymptotically stable, closed-loop system. We show, for fully nonlinear retarded systems, that, by sampling at suitable high frequency the system (finite dimensional) variable, stabilization in the sample-and-hold sense is guaranteed, when the held input is obtained as a feedback of the (first order) spline approximation of the (infinite dimensional) system state, whose entries are available at sampling times. In case of nonuniform sampling, the current sampling period (time elapsed between the latest two samplings) has to be measured (for instance by a clock device set to zero at each sampling time). A sampled-data, event-based control law is also studied.

On approximations by polynomial and nonpolynomial integro-differential splines

Abstract The present article is one of a number of articles in which the authors study the properties of integro-differential splines. The paper deals with the construction of integro-differential polynomial and nonpolynomial splines of the fifth order. The order of approximation with integrodifferential polynomial and nonpolynomial splines of the fifth order are given. We use the tensor product of polynomial and non-polynomial splines constructed in this paper for the approximation of functions of two variables. The results of these numerical experiments are given.

Polygonal Spline Spaces and the Numerical Solution of the Poisson Equation

It is known that generalized barycentric coordinates can be used to form Bernstein polynomial-like functions over a polygon with any number of sides. We propose to use these functions to form a space of continuous polygonal splines (piecewise defined functions) of order $d$ over a partition consisting of polygons which is able to reproduce all polynomials of degree $d$. Locally supported basis functions for the space are constructed for order $d\ge 2$. The construction for $d=2$ is simpler than the “serendipity” quadratic finite elements that have appeared in the recent literature. The number of basis functions is similar to, but fewer than, those of the virtual element method. We use them for the numerical solution of the Poisson equation on two special types of nontriangular partitions to present a proof of concept for solving PDEs over polygonal partitions. Numerical solutions based on quadrangulations and pentagonal partitions are demonstrated to show the efficiency of these polygonal spline functions. They can lead to a more accurate solution by using fewer degrees of freedom than the traditional continuous polynomial finite element method if the solutions are smooth although assembling the mass and stiffness matrices can take more time.

Real-time, local spline interpolation schemes on bounded intervals

We develop a local polynomial spline interpolation scheme for arbitrary spline order on bounded intervals. Our method’s local formulation, eective boundary considerations and optimal interpolation error rate make it particularly useful for real-time implementation in real-world applications. Mathematics Subject Classification: 41A05, 41A15

α-Spline design of finite impulse response digital filters

The use of transition functions to connect the passband and the stopband in the ideal amplitude frequency response of an FIR filter reduces the oscillations near the band edges caused by the Gibbs phenomenon in filters designed by windowing. To this end, several methods based on β-splines have been proposed, in which an integer order of the spline is used to properly shape the transition band of the filter. However, it has been reported in the literature that the optimal value of the spline order that minimizes the approximation׳s mean square error may take non-integer values. Therefore, there is a lack of a precise formulation that allows the use of real positive numbers to properly design spline-based digital filters. The aim of this paper is to solve this problem, presenting a new formulation based on α-splines. By means of the proposed approach, non-integer values for the spline order can be employed in an appropriate mathematical framework, allowing its use as a real variable in any design procedure. Design examples demonstrate that smaller approximation errors than other approaches based on conventional windows can be obtained.

A mixed decomposition-spline approach for the numerical solution of a class of singular boundary value problems

In this paper, we formulate a patching scheme to solve numerically a class of boundary value problems. The strategy is being founded on a merger of two methods, a modified decomposition technique applied on a small interval near the singularity and a fourth order spline collocation technique which is used on the rest of the domain of the problem. The convergence of the proposed method is analyzed and a fourth-order rate of convergence is proved. Some examples are given to test the efficiency of the scheme and to verify the order of the rate of convergence. The numerical results are compared with exact solutions and the outcomes of other existing numerical methods.

Free vibration of anti-symmetric angle-ply plates with variable thickness

Free vibration of anti-symmetric angle-ply plates with variable thickness is analysed using spline function approximation including the effect of shear deformation. The equations of motion for the plate are derived using the theory of Yang, Norris and Stavsky. Assuming the solution in a separable form, a system of coupled differential equations in displacement and rotational functions are obtained and these functions are approximated by Bickley-type splines of order three. A generalised eigenvalue problem is obtained and solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. Two and four layered plates consisting of two different materials and plies comprising of same as well as different materials for two different boundary conditions are analysed. The effect of material properties, ply orientation, number of lay ups, aspect ratio and coefficients of thickness variations on the frequency parameter are presented. The accuracy of the result is ascertained by convergence and comparative study.

Isogeometric analysis of the Cahn–Hilliard equation – a convergence study

Herein, we present a numerical convergence study of the Cahn-Hilliard phase-field model within an isogeometric finite element analysis framework. Using a manufactured solution, a mixed formulation of the Cahn-Hilliard equation and the direct discretisation of the weak form, which requires a C 1 -continuous approximation, are compared in terms of convergence rates. For approximations that are higher than second-order in space, the direct discretisation is found to be superior. Suboptimal convergence rates occur when splines of order p = 2 are used. This is validated with a priori error estimates for linear problems. The convergence analysis is completed with an investigation of the temporal discretisation. Second-order accuracy is found for the generalised-α method. This ensures the functionality of an adaptive time stepping scheme which is required for the efficient numerical solution of the Cahn-Hilliard equation. The isogeometric finite element framework is eventually validated by two numerical examples of spinodal decomposition.

Geometrically designed, variable knot regression splines

A new method of Geometrically Designed least squares (LS) splines with variable knots, named GeDS, is proposed. It is based on the property that the spline regression function, viewed as a parametric curve, has a control polygon and, due to the shape preserving and convex hull properties, it closely follows the shape of this control polygon. The latter has vertices whose x-coordinates are certain knot averages and whose y-coordinates are the regression coefficients. Thus, manipulation of the position of the control polygon may be interpreted as estimation of the spline curve knots and coefficients. These geometric ideas are implemented in the two stages of the GeDS estimation method. In stage A, a linear LS spline fit to the data is constructed, and viewed as the initial position of the control polygon of a higher order ($$n>2$$n>2) smooth spline curve. In stage B, the optimal set of knots of this higher order spline curve is found, so that its control polygon is as close to the initial polygon of stage A as possible and finally, the LS estimates of the regression coefficients of this curve are found. The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline coefficients. Numerical examples are provided and further supplemental materials are available online.

A simple and general approach to fitting the discount curve under no-arbitrage constraints

We suggest a simple and general approach to fitting the discount curve under no-arbitrage constraints based on a penalized shape-constrained B-spline. The approach accommodates B-splines of any order and fitting both under the L1 and the L2 loss functions. An application to US STRIPS data from 2001–2015 suggests that polynomial splines of order three and four are mandatory to obtain reasonable fits. The choice of the loss function appears to be less relevant.

Continuous-Time Estimation of Attitude Using B-Splines on Lie Groups

Filtering algorithms are the workhorse of spacecraft attitude estimation, but recent research has shown that the use of batch estimation techniques can result in higher accuracy per unit of computational cost. This paper presents an approach for singularity-free batch estimation of attitude in continuous time using B-Spline curves on unit-length quaternions. It extends existing theory of unit-length quaternion B-splines to general Lie groups and arbitrary B-spline order. It is shown how to use these curves for continuous-time batch estimation using Gauss–Newton or Levenberg–Marquardt, including efficient curve initialization, a parameter update step that preserves the Lie group constraint within an unconstrained optimization framework, and the derivation of Jacobians of the B-spline’s value and its time derivatives with respect to an update of its parameters. For unit-length quaternion splines, the equations for angular velocity and angular acceleration are derived. An implementation of this algorithm is evaluated on two problems: spacecraft attitude estimation using a three-axis magnetometer, a sun sensor, and 1) a three-axis gyroscope, and 2) a continuous-time vehicle dynamics model based on Euler’s equation. Its performance is compared against a standard multiplicative extended Kalman filter and a recently published batch attitude estimation algorithm. The results show that B-splines have equal or superior performance over all test cases and provide two key tuning parameters, the number of knots and the spline order, that an engineer can use to trade off accuracy and computational efficiency when choosing a spline representation for a given estimation problem.

On topology optimization with embedded boundary resolution and smoothing

Honeycomb tessellation, by virtue of the edge connectivity it offers, helps eliminate various geometric singularities and thus shows promise in topology optimization. Numerous V-notches, which are sites of stress concentration and also pose difficulty in manufacturing, get retained on the continuum boundaries. Explicit boundary resolution and smoothing scheme is embedded within each gradient step when performing topology optimization with negative circular masks to assuage these serrations. This procedure, in particular, is deemed necessary when synthesizing contact aided compliant mechanisms. Continuum boundaries of each intermediate design are identified, and multiple smoothing steps are performed on the boundary nodes by shifting them systematically. Consequently, many regular hexagonal cells get morphed into concave ones. Shape functions based on Mean Value Coordinates (MVC) are implemented to cater to hexagonal finite elements of generic shapes. A continuation approach is used on the logistic material assignment model to tend to Heaviside function and thus yield close to black and white solutions in addition to avoiding early convergence and numerical instability. Number of boundary smoothing steps is increased with iterations as well. Deletion/addition of masks is performed to eliminate those not contributing in topology optimization and to introduce others in cases when singularities like local islands or layers prevail. To enhance continuity and to facilitate manufacturing, boundary nodes are interpolated using order m splines. For the examples presented, it is noted that splines of order 4 are adequate in representing the continuum boundaries accurately.

Free vibration of symmetric angle-ply laminated annular circular plate of variable thickness under shear deformation theory

Free vibrational study of symmetric angle-ply laminated annular circular plate of variable thickness including first order shear deformation theory using spline function approximation is studied. The equations of motion for the plates are derived using first order shear deformation theory. The solutions of displacement functions are assumed in a separable form to obtain a system of coupled differential equations in terms displacement and rotational functions and these functions are approximated by Bickley-type splines of order three. The vibration of 3- and 5-layered plates, made up of two types of materials and two types of boundary conditions are considered. A generalized eigenvalue problem is obtained and solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. Parametric studies are made of the variation of frequency parameter with respect to the radii ratio, circumferential node number, different sequence, number of lay-ups and other parameters characterizing the nature of variation of thickness for 3- and 5-layers.

Isogeometric analysis of free vibration of simple shaped elastic samples.

The paper is devoted to numerical solution of free vibration problems for elastic bodies of canonical shapes by means of a spline based finite element method (FEM), called Isogeometric Analysis (IGA). It has an advantage that the geometry is described exactly and the approximation of unknown quantities is smooth due to higher-order continuous shape functions. IGA exhibits very convenient convergence rates and small frequency errors for higher frequency spectrum. In this paper, the IGA strategy is used in computation of eigen-frequencies of a block and cylinder as benchmark tests. Results are compared with the standard FEM, the Rayleigh-Ritz method, and available experimental data. The main attention is paid to the comparison of convergence rate, accuracy, and time-consumption of IGA against FEM and also to show a spline order and parameterization effects. In addition, the potential of IGA in Resonant Ultrasound Spectroscopy measurements of elastic properties of general anisotropy solids is discussed.

Modeling of Auditory Neuron Response Thresholds with Cochlear Implants.

The quality of the prosthetic-neural interface is a critical point for cochlear implant efficiency. It depends not only on technical and anatomical factors such as electrode position into the cochlea (depth and scalar placement), electrode impedance, and distance between the electrode and the stimulated auditory neurons, but also on the number of functional auditory neurons. The efficiency of electrical stimulation can be assessed by the measurement of e-CAP in cochlear implant users. In the present study, we modeled the activation of auditory neurons in cochlear implant recipients (nucleus device). The electrical response, measured using auto-NRT (neural responses telemetry) algorithm, has been analyzed using multivariate regression with cubic splines in order to take into account the variations of insertion depth of electrodes amongst subjects as well as the other technical and anatomical factors listed above. NRT thresholds depend on the electrode squared impedance (β = −0.11 ± 0.02, P < 0.01), the scalar placement of the electrodes (β = −8.50 ± 1.97, P < 0.01), and the depth of insertion calculated as the characteristic frequency of auditory neurons (CNF). Distribution of NRT residues according to CNF could provide a proxy of auditory neurons functioning in implanted cochleas.

Spline Galerkin Methods for the Sherman--Lauricella Equation on Contours with Corners

Spline Galerkin approximation methods for the Sherman-Lauricella integral equation on simple closed piecewise smooth contours are studied, and necessary and sufficient conditions for their stability are obtained. It is shown that the method under consideration is stable if and only if certain operators associated with the corner points of the contour are invertible. Numerical experiments demonstrate a good convergence of the spline Galerkin methods and validate theoretical results. Moreover, it is shown that if all corners of the contour have opening angles located in interval $(0.1\pi, 1.9\pi)$, then the corresponding Galerkin method based on splines of order $0$, $1$ and $2$ is always stable. These results are in strong contrast with the behaviour of the Nystr\"om method, which has a number of instability angles in the interval mentioned.