Order Spline Research Articles

Limit theorems of the theory of discrete periodic splines

It is known from the discrete harmonic analysis that the interpolation problem with equidistant interpolation points has a unique solution. If the right-hand sides in the interpolation problem are fixed, the spline depends on two parameters: the spline order and the number of points located between neighboring interpolation points. We find explicit expressions for the limits of interpolation spllines with respect to each parameter separately and show that both repeated limits exist. We also prove that these repeated limits are equal and their value is an interpolation trigonometric polynomial. Bibliography: 10 titles. Illustrations: 2 figures.

Privatization of Businesses and Flexible Investment: A Real Option Approach

In this article, we study the situation, where by partial privatization a private company is given the opportunity to invest into a government owned business. After payment of an initial instalment cost, the private company's investments are flexible within a range $[0,k]$ until the business is completed. We model the problem in a real option framework, using geometric mean reversion to describe the dynamics of the business. We determine the optimal time for the private company to enter and pay the initial instalment cost as well as the optimal dynamic investment strategy that it follows afterward. For the latter, analytic solution cannot be obtained. We use quadratic splines in order to solve the corresponding dynamic programming problem. Finally we determine the optimal degree of privatization in our model from the government's perspective.

Vector partition functions and index of transversally elliptic operators

Let G be a torus acting linearly on a complex vector space M and let X be the list of weights of G in M. We determine the topological equivariant K-theory of the open subset M f of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the character lattice \( \widehat{G} \), satisfying the cocircuit difference equations associated to X, introduced by Dahmen and Micchelli in the context of the theory of splines in order to study vector partition functions (cf. [7]).This allows us to determine the range of the index map from G-transversally elliptic operators on M to generalized functions on G and to prove that the index map is an isomorphism on the image. This is a setting studied by Atiyah and Singer [1] which is in a sense universal for index computations.

Minimizing Sequences for a Family of Functional Optimal Estimation Problems

Rates of convergence are derived for approximate solutions to optimization problems associated with the design of state estimators for nonlinear dynamic systems. Such problems consist in minimizing the functional given by the worst-case ratio between the ℒ p -norm of the estimation error and the sum of the ℒ p -norms of the disturbances acting on the dynamic system. The state estimator depends on an innovation function, which is searched for as a minimizer of the functional over a subset of a suitably-defined functional space. In general, no closed-form solutions are available for these optimization problems. Following the approach proposed in (Optim. Theory Appl. 134:445–466, 2007), suboptimal solutions are searched for over linear combinations of basis functions containing some parameters to be optimized. The accuracies of such suboptimal solutions are estimated in terms of the number of basis functions. The estimates hold for families of approximators used in applications, such as splines of suitable orders.

On the exact values of the best approximations of classes of differentiable periodic functions by splines

We obtain the exact values of the best L1-approximations of classes WrF, r ∈ ℕ, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F, as well as of classes WrHω of periodic functions whose rth derivative has a given convex (upward) majorant ω(t) of the modulus of continuity, by subspaces of polynomial splines of order m ≥ r + 1 and of deficiency 1 with nodes at the points 2kπ/n and 2kπ/n + h, n ∈ ℕ, k ∈ ℤ, h ∈ (0, 2π/n). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.

On one version of wavelet decompositions of spaces of polynomial splines

We study wavelet decompositions of the spaces of polynomial splines of order m on nonuniform grids, constructed via projections of Lagrange type. Bibliography: 3 titles.

Selection of locations of knots for linear splines in random regression test‐day models

Using spline functions (segmented polynomials) in regression models requires the knowledge of the location of the knots. Knots are the points at which independent linear segments are connected. Optimal positions of knots for linear splines of different orders were determined in this study for different scenarios, using existing estimates of covariance functions and an optimization algorithm. The traits considered were test-day milk, fat and protein yields, and somatic cell score (SCS) in the first three lactations of Canadian Holsteins. Two ranges of days in milk (from 5 to 305 and from 5 to 365) were taken into account. In addition, four different populations of Holstein cows, from Australia, Canada, Italy and New Zealand, were examined with respect to first lactation (305 days) milk only. The estimates of genetic and permanent environmental covariance functions were based on single- and multiple-trait test-day models, with Legendre polynomials of order 4 as random regressions. A differential evolution algorithm was applied to find the best location of knots for splines of orders 4 to 7 and the criterion for optimization was the goodness-of-fit of the spline covariance function. Results indicated that the optimal position of knots for linear splines differed between genetic and permanent environmental effects, as well as between traits and lactations. Different populations also exhibited different patterns of optimal knot locations. With linear splines, different positions of knots should therefore be used for different effects and traits in random regression test-day models when analysing milk production traits.

Analysis of spatial interpolation in the material-point method

This paper analyses different types of spatial interpolation for the material-point method. The interpolations include quadratic elements and cubic splines in addition to the standard linear shape functions usually applied. For the small-strain problem of a vibrating bar, the best results are obtained using quadratic elements. It is shown that for more complex problems, the use of partially negative shape functions is inconsistent with the material-point method in its current form, necessitating other types of interpolation such as cubic splines in order to obtain smoother representations of field quantities. The properties of different interpolation functions are analysed using numerical examples, including the classical cantilevered beam problem.

Free Vibration of Layered Circular Cylindrical Shells of Variable Thickness Using Spline Function Approximation

Free vibration of layered circular cylindrical shells of variable thickness is studied using spline function approximation by applying a point collocation method. The shell is made up of uniform layers of isotropic or specially orthotropic materials. The equations of motions in longitudinal, circumferential and transverse displacement components, are derived using extension of Love's first approximation theory. The coupled differential equations are solved using Bickley-type splines of suitable order, which are cubic and quintic, by applying the point collocation method. This results in the generalized eigenvalue problem by combining the suitable boundary conditions. The effect of frequency parameters and the corresponding mode shapes of vibration are studied with different thickness variation coefficients, and other parameters. The thickness variations are assumed to be linear, exponential, and sinusoidal along the axial direction. The results are given graphically and comparisons are made with those results obtained using finite element method.

A note on: Spline technique for modeling roadway profile to minimize earthwork cost

The gap constraint used in A. A. Moreb, Spline technique for modeling roadway profile to minimize earthwork cost, J. Ind. Man. & Opt., 5 (2) (2009), 275-283 introduces unnecessary errors, while the slope constraint may be violated for second- and higher-order splines. In this note we amend the gap constraint, while maintaining the linearity of the model. We also present an improved slope constraint for linear and quadratic splines, and show that it becomes nonlinear for cubic and higher order splines. The improvements also apply to A. Moreb, M. Aljohani, Quadratic representation for roadway profile that minimizes earthwork cost, J. Sys. Sci. & Sys. Eng., 13 (2) (2004), 245-252.

Calibration relations for nonpolynomial splines

Nonpolynomial (X, A, ϕ)-splines of the third order and the special case of Bϕ-splines of class C2 are studied. For such splines calibration relations are obtained, owing to which the coordinate splines on the original grid is represented in terms of the coordinate splines on a refined grid. A nonlinear mapping (ℝ4)9 ↦ ℝ4 and locally orthogonal chains of vectors are used for this purpose. Bibliography: 22 titles.

An approach for optimization of loading cycles of hydraulic mining excavators

AbstractThe present paper describes an optimization multi‐stage approach for computing the optimal loading cycle of a heavy weight excavator along a predefined trajectory under dynamic constraints such as maximal ensuing forces, accelerations, and maximal allowed hydraulic power. The approach involves the use of splines of 3rd and 5th order for smooth interpolation of the spatial trajectory up to the level of linear and angular accelerations, as well as an SQP optimization routine to find the optimal end‐effector motion law along the trajectory, which finally leads to strokes and pressures at the hydraulic actuators as a function of time. The approach can also be extended to other multi‐body systems following a predetermined trajectory. The solution scheme can be included in a more general optimization routine aiming to optimize the trajectory itself. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Implementation of high order spline interpolations for tracking test particles in discretized fields

A systematic approach for constructing high order spline interpolation methods is proposed for fields known on regular, rectangular grids. These interpolation methods are tested in tracking trajectories of particles submitted to a force that derives from a potential known on a grid. The interplay between the time advancement scheme and the spatial interpolation is studied in detail and it is shown how the order of the trajectory solver is directly affected by the order of the spline interpolation. It is also shown how an interpolation method that preserves topological properties of physical fields can be better exploited with these higher order spline approximations.

Form-preserving exponential approximation

In this paper we continue our research in constructing linear methods for approximating functions by parabolic splines with equidistant knots, inheriting such properties of approximated functions as monotony and convexity. The obtained results are generalized to the case of nonequidistant knots and to some kinds of exponential splines of the third order.

Optimal a priori estimates for higher order finite elements for elliptic interface problems

We analyze higher order finite elements applied to second order elliptic interface problems. Our a priori error estimates in the L 2 - and H 1 -norm are expressed in terms of the approximation order p and a parameter δ that quantifies how well the interface is resolved by the finite element mesh. The optimal p-th order convergence in the H 1 ( Ω ) -norm is only achieved under stringent assumptions on δ, namely, δ = O ( h 2 p ) . Under weaker conditions on δ, optimal a priori estimates can be established in the L 2 - and in the H 1 ( Ω δ ) -norm, where Ω δ is a subdomain that excludes a tubular neighborhood of the interface of width O ( δ ) . In particular, if the interface is approximated by an interpolation spline of order p and if full regularity is assumed, then optimal convergence orders p + 1 and p for the approximation in the L 2 ( Ω ) - and the H 1 ( Ω δ ) -norm can be expected but not order p for the approximation in the H 1 ( Ω ) -norm. Numerical examples in 2D and 3D illustrate and confirm our theoretical results.

Inequalities of the Bernstein type for splines of defect 2

We obtain new exact inequalities of the Bernstein type for periodic polynomial splines of order r and defect 2.

A multiresolution method for fitting scattered data on the sphere

In this work, we propose an efficient multiresolution method for fitting scattered data functions on a sphere S, using a tensor product method of periodic algebraic trigonometric splines of order 3 and quadratic polynomial splines defined on a rectangular map of S. We describe the decomposition and reconstruction algorithms corresponding to the polynomial and periodic algebraic trigonometric wavelets. As application of this method, we give an algorithm which allows to compress scattered data on spherelike surfaces. In order to illustrate our results, some numerical examples are presented.

Exact values of best approximations for classes of periodic functions by splines of deficiency 2

We obtain exact values of best L1-approximations for the classes WrF, r ∈ ℕ, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F as well as for the classes WrHω of periodic functions whose rth derivative has a given convex (up) majorant ω(t) of the modulus of continuity by subspaces of polynomial splines of order m ≥ r + 1 of deficiency 2 with nodes at the points 2kπ/n, n ∈ ℕ, k ∈ ℤ. It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding function classes.

Meshless methods for physics-based modeling and simulation of deformable models

As 3D digital photographic and scanning devices produce higher resolution images, acquired geometric data sets grow more complex in terms of the modeled objects’ size, geometry, and topology. As a consequence, point-sampled geometry is becoming ubiquitous in graphics and geometric information processing, and poses new challenges which have not been fully resolved by the state-of-art graphical techniques. In this paper, we address the challenges by proposing a meshless computational framework for dynamic modeling and simulation of solids and thin-shells represented as point samples. Our meshless framework can directly compute the elastic deformation and fracture propagation for any scanned point geometry, without the need of converting them to polygonal meshes or higher order spline representations. We address the necessary computational techniques, such as Moving Least Squares, Hierarchical Discretization, and Modal Warping, to effectively and efficiently compute the physical simulation in real-time. This meshless computational framework aims to bridge the gap between the point-sampled geometry with physics-based modeling and simulation governed by partial differential equations.

Solitary waves for the MRLW equation

This work presents a computational comparison study of quadratic, cubic, quartic and quintic splines for solving the modified regularized long wave (MRLW) equation. Collocation schemes with quadratic and cubic splines are found to be unconditionally stable. The fourth-order Runge–Kutta method has been used to solve the collocation schemes when quartic and quintic B-splines are used. The three invariants of motion have been evaluated to determine the conservation properties of the suggested algorithms. Comparisons of results due to different schemes with the exact values shows the accuracy and efficiency of the proposed schemes. Results corresponding to higher order splines are more accurate than those corresponding to lower order splines.