Variational splines for the numerical solution of singular differential equations

The numerical parametrization method (PM), originally created for optimal control problems, is specificated for classical calculus of variation problems that arise in connection with singular implicit (IDEs) and differential-algebraic equations (DAEs). The PM for IDEs is based on representation of the required solution as a spline with moving knots and on minimization of the discrepancy functional with respect to the spline parameters. Such splines are named variational splines. For DAEs only finite entering functions can be represented by splines, and the functional under minimization is the discrepancy of the algebraic subsystem. The first and the second derivatives of the functionals are calculated in two ways – for DAEs with the help of adjoint variables, and for IDE directly. The PM does not use the notion of differentiation index, and it is applicable to any singular equation having a solution. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The parameterization method in optimal control problems and differential–algebraic equations

The paper is devoted to the explanation of the numerical parameterization method (PM) for optimal control (OC) problems with intermediate phase constraint and to its circumstantiation for classical calculus of variation (CV) problems that arise in connection with singular ODEs or DAEs, especially in cases of their essential degeneracy. The PM is based on a finite parameterization of control functions and on derivation of the problem functional with respect to control parameters. The first and the second derivatives are calculated with the help of adjoint vector and matrix impulses. Results of the solution to one phase constrained OC and two degenerate CV problems, connected with singular DAEs nonreducible to the normal form, are presented.     

Methods for solving singular boundary value problems using splines: a review

This paper surveys and reviews papers of spline solution of singular boundary value problems. Among a number of numerical methods used to solve two-point singular boundary value problems, spline methods provide an efficient tool. Techniques collected in this paper include cubic splines, non-polynomial splines, parametric splines, B-splines and TAGE method.     

Cubic spline for a class of singular two-point boundary value problems

In this paper we have presented a method based on cubic splines for solving a class of singular two-point boundary value problems. The original differential equation is modified at the singular point then the boundary value problem is treated by using cubic spline approximation. The tridiagonal system resulting from the spline approximation is efficiently solved by Thomas algorithm. Some model problems are solved, and the numerical results are compared with exact solution.     

L 1 C 1 polynomial spline approximation algorithms for large data sets

In this article, we address the problem of approximating data points by C ¹-smooth polynomial spline curves or surfaces using L ₁-norm. The use of this norm helps to preserve the data shape and it reduces extraneous oscillations. In our approach, we introduce a new functional which enables to control directly the distance between the data points and the resulting spline solution. The computational complexity of the minimization algorithm is nonlinear. A local minimization method using sliding windows allows to compute approximation splines within a linear complexity. This strategy seems to be more robust than a global method when applied on large data sets. When the data are noisy, we iteratively apply this method to globally smooth the solution while preserving the data shape. This method is applied to image denoising.     

On Spline Collocation for Singular Integral Equations

This paper is devoted to the approximate solution of one-dimensional singular integral equations on a closed curve by spline collocation methods. As the main result we give conditions which are sufficient and in special cases also necessary for the convergence in SOBOLEV norms. The paper is organized as follows. In chapter 1 we indicate some definitions and some facts about projection methods. In chapter 2, we generalize a technique developed in [1] and study the convergence of collocations using splines of odd degree in periodic SOBOLEV spaces. In chapter 3, we apply our method to collocations by splines of even degree and consider the case of systems of equations. And in the last chapter, 4, the results are applied to singular integral equations and compared with known facts about piecewise linear spline collocation for such equations.     

An energy-minimization framework for monotonic cubic spline interpolation

This paper describes the use of cubic splines for interpolating monotonic data sets. Interpolating cubic splines are popular for fitting data because they use low-order polynomials and have C² continuity, a property that permits them to satisfy a desirable smoothness constraint. Unfortunately, that same constraint often violates another desirable property: monotonicity. It is possible for a set of monotonically increasing (or decreasing) data points to yield a curve that is not monotonic, i.e., the spline may oscillate. In such cases, it is necessary to sacrifice some smoothness in order to preserve monotonicity.The goal of this work is to determine the smoothest possible curve that passes through its control points while simultaneously satisfying the monotonicity constraint. We first describe a set of conditions that form the basis of the monotonic cubic spline interpolation algorithm presented in this paper. The conditions are simplified and consolidated to yield a fast method for determining monotonicity. This result is applied within an energy minimization framework to yield linear and nonlinear optimization-based methods. We consider various energy measures for the optimization objective functions. Comparisons among the different techniques are given, and superior monotonic C² cubic spline interpolation results are presented. Extensions to shape preserving splines and data smoothing are described.     

Curvature variation minimizing cubic Hermite interpolants

In this paper, planar parametric Hermite cubic interpolants with small curvature variation are studied. By minimization of an appropriate approximate functional, it is shown that a unique solution of the interpolation problem exists, and has a nice geometric interpretation. The best solution of such a problem is a quadratic geometric interpolant. The optimal approximation order 4 of the solution is confirmed. The approach is combined with strain energy minimization in order to obtain G¹ cubic interpolatory spline.     

Trivariate near-best blending spline quasi-interpolation operators

A method to define trivariate spline quasi-interpolation operators (QIOs) is developed by blending univariate and bivariate operators whose linear functionals allow oversampling. In this paper, we construct new operators based on univariate B-splines and bivariate box splines, exact on appropriate spaces of polynomials and having small infinity norms. An upper bound of the infinity norm for a general blending trivariate spline QIO is derived from the Bernstein-Bézier coefficients of the fundamental functions associated with the operators involved in the construction. The minimization of the resulting upper bound is then proposed and the existence of a solution is proved. The quadratic and quartic cases are completely worked out and their exact solutions are explicitly calculated.     

Pseudo-polyharmonic vectorial approximation for div-curl and elastic semi-norms

Vector field reconstruction is a problem arising in many scientific applications. In this paper, we study a div-curl approximation of vector fields by pseudo-polyharmonic splines. This leads to the variational smoothing and interpolating spline problems with minimization of an energy involving the curl and the divergence of the vector field. The relationship between the div-curl energy and elastic energy is established. Some examples are given to illustrate the effectiveness of our approach for a vector field reconstruction.     

Konstruktion und algebraische Eigenschaften vonM-Spline-Interpolierenden

Summary This paper generates interpolatingM-splines in the sense of Lucas [J. of Approx. Th. 5, 1–14 (1972)] by a simple algebraic construction. The method yieldsM-spline interpolants for every finite family of functionals commuting with the remainder term of a generalized Taylor formula. These assumptions are fulfilled for a large class of spline interpolation problems (e.g. splines generated by certain singular differential operators and splines of several variables) without any further requirements about the geometrical distribution or denseness of the interpolation points. A generalization ofB-splines is used to improve the numerical behaviour of the interpolation process.     

Numerical solution of the inverse electrocardiography problem with the use of the Tikhonov regularization method

The inverse electrocardiography problem related to medical diagnostics is considered in terms of potentials. Within the framework of the quasi-stationary model of the electric field of the heart, the solution of the problem is reduced to the solution of the Cauchy problem for the Laplace equation in R ³. A numerical algorithm based on the Tikhonov regularization method is proposed for the solution of this problem. The Cauchy problem for the Laplace equation is reduced to an operator equation of the first kind, which is solved via minimization of the Tikhonov functional with the regularization parameter chosen according to the discrepancy principle. In addition, an algorithm based on numerical solution of the corresponding Euler equation is proposed for minimization of the Tikhonov functional. The Euler equation is solved using an iteration method that involves solution of mixed boundary value problems for the Laplace equation. An individual mixed problem is solved by means of the method of boundary integral equations of the potential theory. In the study, the inverse electrocardiography problem is solved in region Ω close to the real geometry of the torso and heart.     

Integral representation of slowly growing equidistant splines

In this paper we consider polynomial splines S(x) with equidistant nodes which may grow as O (|x|^s). We present an integral representation of such splines with a distribution kernel. This representation is related to the Fourier integral of slowly growing functions. The part of the Fourier exponentials herewith play the so called exponential splines by Schoenberg. The integral representation provides a flexible tool for dealing with the growing equidistant splines. First, it allows us to construct a rich library of splines possessing the property that translations of any such spline form a basis of corresponding spline space. It is shown that any such spline is associated with a dual spline whose translations form a biorthogonal basis. As examples we present solutions of the problems of projection of a growing function onto spline spaces and of spline interpolation of a growing function. We derive formulas for approximate evaluation of splines projecting a function onto the spline space and establish therewith exact estimations of the approximation errors.     

Hilbertian Approach for Univariate Spline with Tension

In this work, a new approach is proposed for constructing splines with tension. The basic idea is in the use of distributions theory, which allows us to define suitable Hilbert spaces in which the tension spline minimizes some energy functional. Classical orthogonal conditions and characterizations of the spline in terms of a fundamental solution of a differential operator are provided. An explicit representation of the tension spline is given. The tension spline can be computed by solving a linear system. Some numerical examples are given to illustrate this approach.     

Hilbertian Approach for Univariate Spline with Tension

In this work, a new approach is proposed for constructing splines with tension. The basic idea is in the use of distributions theory, which allows us to define suitable Hilbert spaces in which the tension spline minimizes some energy functional. Classical orthogonal conditions and characterizations of the spline in terms of a fundamental solution of a differential operator are provided. An explicit representation of the tension spline is given. The tension spline can be computed by solving a linear system. Some numerical examples are given to illustrate this approach.     

G2 composite cubic Bézier curves

We present an algorithm for creating planar G² spline curves using rational Bézier cubic segments. The splines interpolate a sequence of points, tangents and curvatures. In addition each segment has two more geometric shape handles. These are obtained from an analysis of the singular point of the curve. The individual segments are convex, but zero curvature can be assigned at a junction point, hence inflection points can be placed where desired but cannot occur otherwise.     

Minimal splines and wavelets

This paper is dedicated to the memory of the prominent mathematician S.G. Mikhlin. Here, Mikhlin’s idea of approximation relations is used for construction of wavelet resolution in the case of spline spaces of zero height. These approximation relations allow one to establish the embedding of the spline spaces corresponding to nested grids. Systems of functionals which are biorthogonal to the basic splines are constructed using the relations; then the systems obtained are used for wavelet decomposition. It is established that, for a fixed pair of grids of which one is embedded into the other and for an arbitrary fixed (on the coarse grid) spline space, there exists a continuum of spline spaces (on the fine grid) which contain the aforementioned spline space on the coarse grid. The wavelet decomposition of such embedding is given and the corresponding formulas of decomposition and formulas of reconstruction are deduced. The space of ( , φ)-splines is introduced with three objects: the full chain of vectors, prescribed infinite grid on real axis and the preassigned vector-function φ with m + 1 components (m is called the order of the splines). Under certain assumptions, the splines belong to the class C ^{m ? 1}. The gauge relations between the basic splines on the coarse grid and the basic splines on the fine grid are deduced. A general method for construction of a biorthogonal system of functionals (to basic spline system) is suggested. In this way, a chain of nested spline spaces is obtained, and the wavelet decomposition of the chain is discussed. The spaces and chains of spaces are completely classified in the terms of manifolds. The manifold of spaces considered is identified with the manifold of complete sequences of points of the direct product of an interval on the real axis and the projective space ? ^m ; the manifold of nested spaces is identified with the manifold of nested sequences of points of the direct product mentioned above.     

On the problems of smoothing and near-interpolation

In the first part of this paper we apply a saddle point theorem from convex analysis to show that various constrained minimization problems are equivalent to the problem of smoothing by spline functions. In particular, we show that near-interpolants are smoothing splines with weights that arise as Lagrange multipliers corresponding to the constraints in the problem of near-interpolation. In the second part of this paper we apply certain fixed point iterations to compute these weights. A similar iteration is applied to the computation of the smoothing parameter in the problem of smoothing.

     

Stability Issues in Regular and Noncritical Singular DAEs

This paper addresses equilibrium stability issues in both regular and singular differential-algebraic equations (DAEs). We present a survey of available results and discuss some commonly-used methods in the qualitative analysis of low-index autonomous systems. Additionally, we extend the use of matrix pencil theory to the stability study of singular problems, pointing out some interesting relations between regular and singular DAEs. This framework is applied to the qualitative analysis of singular equations arising in the context of the Singularity Induced Bifurcation theorem, and also to the stability study of stationary equilibria in singular DAEs.     

Least-squares Spline Regression with Block-diagonal Variance Matrices

A numerical method of solution is presented for the least squaresfitting of experimental data by spline functions in the casewhere the data errors are correlated and for which the variancematrix is specified. The method is general in that it permits(a) splines of any order, (b) the knots of the spline to bearbitrary in number and position, and (c) variance matricesthat are block diagonal in form. Since limiting forms of (c)are diagonal and full variance matrices, the method can handle,as special cases, both conventional spline regression problemsand spline regression problems with general, unstructured variancematrices. An application to gamma spectrometry, in which theblocks of the variance matrix have special structure, is fullytreated.