Interpolation Nodes Research Articles

A Kriging‐based error‐reproducing and interpolating kernel method for improved mesh‐free approximations

AbstractAn error‐reproducing and interpolating kernel method (ERIKM), which is a novel and improved form of the error‐reproducing kernel method (ERKM) with the nodal interpolation property, is proposed. The ERKM is a non‐uniform rational B‐splines (NURBS)‐based mesh‐free approximation scheme recently proposed by Shaw and Roy (Comput. Mech. 2007; 40(1):127–148). The ERKM is based on an initial approximation of the target function and its derivatives by NURBS basis functions. The errors in the NURBS approximation and its derivatives are then reproduced via a family of non‐NURBS basis functions. The non‐NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation obtained in the first step. In the ERKM, the interpolating property at the boundary is achieved by repeating the knot (open knot vector). However, for most problems of practical interest, employing NURBS with open knots is not possible because of the complex geometry of the domain, and consequently ERKM shape functions turn out to be non‐interpolating. In ERIKM, the error functions are obtained through localized Kriging based on a minimization of the squared variance of the estimate with the reproduction property as a constraint. Interpolating error functions so obtained are then added to the NURBS approximant. While enriching the ERKM with the interpolation property, the ERIKM naturally possesses all the desirable features of the ERKM, such as insensitivity to the support size and ability to reproduce sharp layers. The proposed ERIKM is finally applied to obtain strong and weak solutions for a class of linear and non‐linear boundary value problems of engineering interest. These illustrations help to bring out the relative numerical advantages and accuracy of the new method to some extent. Copyright © 2007 John Wiley & Sons, Ltd.

Rational matrix functions associated with the Nevanlinna-Pick problem in the class S[a, b] and orthogonal on a compact interval

We consider the interpolation Nevanlinna-Pick problem with infinitely many interpolation nodes in the class S[a, b] and rational matrix functions associated with this problem and orthogonal on the segment [a, b]. We obtain a criterion of complete indeterminacy of the Nevanlinna-Pick problem in terms of orthogonal rational matrix functions.

A novel efficient two-phase algorithm for training interpolation radial basis function networks

Interpolation radial basis function (RBF) networks have been widely used in various applications. The output layer weights are usually determined by minimizing the sum-of-squares error or by directly solving interpolation equations. When the number of interpolation nodes is large, these methods are time consuming, difficult to control the balance between the convergence rate and the generality, and difficult to reach a high accuracy. In this paper, we propose a two-phase algorithm for training interpolation RBF networks with bell-shaped basis functions. In the first phase, the width parameters of basis functions are determined by taking into account the tradeoff between the error and the convergence rate. Then, the output layer weights are determined by finding the fixed point of a given contraction transformation. The running time of this new algorithm is relatively short and the balance between the convergence rate and the generality is easily controlled by adjusting the involved parameters, while the error is made as small as desired. Also, its running time can be further enhanced thanks to the possibility to parallelize the proposed algorithm. Finally, its efficiency is illustrated by simulations.

An interpolation‐based local differential quadrature method to solve partial differential equations using irregularly distributed nodes

AbstractTo circumvent the constraint in application of the conventional differential quadrature (DQ) method that the solution domain has to be a regular region, an interpolation‐based local differential quadrature (LDQ) method is proposed in this paper. Instead of using regular nodes placed on mesh lines in the DQ method (DQM), irregularly distributed nodes are employed in the LDQ method. That is, any spatial derivative at a nodal point is approximated by a linear weighted sum of the functional values of irregularly distributed nodes in the local physical domain. The feature of the new approach lies in the fact that the weighting coefficients are determined by the quadrature rule over the irregularly distributed local supporting nodes with the aid of nodal interpolation techniques developed in the paper. Because of this distinctive feature, the LDQ method can be consistently applied to linear and nonlinear problems and is really a mesh‐free method without the limitation in the solution domain of the conventional DQM. The effectiveness and efficiency of the method are validated by two simple numerical examples by solving boundary‐value problems of a linear and a nonlinear partial differential equation. Copyright © 2007 John Wiley & Sons, Ltd.

Solution of Cauchy-Type Singular Integral Equations of the First Kind with Zeros of Jacobi Polynomials as Interpolation Nodes

Of concern in this paper is the numerical solution of Cauchy-type singular integral equations of the first kind at a discrete set of points. A quadrature rule based on Lagrangian interpolation, with the zeros of Jacobi polynomials as nodes, is developed to solve these equations. The problem is reduced to a system of linear algebraic equations. A theoretical convergence result for the approximation is provided. A few numerical results are given to illustrate and validate the power of the method developed. Our method is more accurate than some earlier methods developed to tackle this problem.

A Subspace‐Based Method for Solving Lagrange–Sylvester Interpolation Problems

In this paper, we study the Lagrange-Sylvester interpolation of rational matrix functions which are analytic at infinity, and propose a new interpolation algorithm based on the recent subspace-based identification methods. The proposed algorithm is numerically efficient and delivers a minimal interpolant in state-space form. The solvability condition for the subspace-based algorithm is particularly simple and depends only on the total multiplicity of the interpolation nodes. As an application, we consider subspace-based system identification with interpolation constraints, which arises, for example, in the identification of continuous-time systems with a given relative degree.

Explicit Representations for Local Lagrangian Numerical Differentiation

It is shown that in Lagrangian numerical differentiation formulas, the coefficients are explicitly expressed by means of cycle indicator polynomials of symmetric group. Moreover, asymptotic expansions of the remainders are also explicitly represented as a fixed number of interpolation nodes approaching infinitely to the point at which the derivative is evaluated. This implies that complete explicit formulas for local Lagrangian numerical differentiation can be obtained.

On the Filon and Levin methods for highly oscillatory integral [formula omitted

This paper shows that for any suitably smooth function f ( x ) and arbitrarily selected interpolation nodes c 1 , c 2 , … , c v in [ a , b ] , the Filon method and the Levin method for ∫ a b f ( x ) e i ω g ( x ) d x with the polynomial interpolation approach are identical when g ( x ) is a linear function. Based on this result, a new efficient Levin quadrature for ∫ a b f ( x ) e i ω g ( x ) d x is presented.

A comparison of interpolation grids over the triangle or the tetrahedron

A simple strategy for constructing a sequence of increasingly refined interpolation grids over the triangle or the tetrahedron is discussed with the goal of achieving uniform convergence and ensuring high interpolation accuracy. The interpolation nodes are generated based on a one-dimensional master grid comprised of the zeros of the Lobatto, Legendre, Chebyshev, and second-kind Chebyshev polynomials. Numerical computations show that the Lebesgue constant and interpolation accuracy of some proposed grids compare favorably with those of alternative grids constructed by optimization, including the Fekete set. While some sets are clearly preferable to others, no single set can claim uniformly better convergence properties as the number of nodes is raised.

3-D movement simulation techniques using FE methods: Application to eddy current non-destructive testing

In this paper, sample calculations were performed with a three-dimensional (3-D) numerical method to describe the response of an eddy current (EC) probe to defects in a conducting plate. The governing field equations are given in terms of coupled magnetic vector and electrical scalar potentials and solved using a 3-D finite element method implemented in Matlab environment. The displacement of the sensor operating in absolute or differential mode is simulated using three methods: The first and second methods known, respectively, as the nodal interpolation technique (NIT) and the Lagrange multiplier technique (LMT) consist in ensuring the connection between fixed and moved 3-D independent meshes. The third method known as the properties assignment technique (PAT) is based on the geometrical band. In this work, these methods are used to compute the real and imaginary parts of the impedance correlation at every position of the coil(s). This allows the characterisation of the presence of a defect through the EC distribution.

On Newman-type rational interpolation to |x| at the Chebyshev nodes of the second kind

Recently Brutman and Passow considered Newman-type rational interpolation to |x| induced by arbitrary set of symmetric nodes in [−1, 1] and gave the general estimation of the approximation error. By their methods one could establish the exact order of approximation for some special nodes. In the present paper we consider the special case where the interpolation nodes are the zeros of the Chebyshev polynomial of the second kind and prove that in this case the exact order of approximation is O (1/n ln n).

A Lobatto interpolation grid in the tetrahedron

A sequence of increasingly refined interpolation grids inside the tetrahedron is proposed with the goal of achieving uniform convergence and ensuring high interpolation accuracy. The number of interpolation nodes, N, corresponds to the number of terms in the complete mth-order polynomial expansion with respect to the three tetrahedral barycentric coordinates. The proposed grid is constructed by deploying Lobatto interpolation nodes over the faces of the tetrahedron, and then computing interior nodes using a simple formula that involves the zeros of the Lobatto polynomials. Numerical computations show that the Lebesgue constant and interpolation accuracy of the proposed grid compare favourably with those of alternative grids constructed by solving optimization problems. The condition number of the mass matrix is significantly lower than that of the uniform grid and comparable to that of optimal grids proposed by previous authors.

A Lobatto interpolation grid over the triangle

A sequence of increasingly refined interpolation grids over the triangle is proposed, with the goal of achieving uniform convergence and ensuring high interpolation accuracy. The number of interpolation nodes, N , corresponds to a complete mth-order polynomial expansion with respect to the triangle barycentric coordinates, which arises by the horizontal truncation of the Pascal triangle. The proposed grid is generated by deploying Lobatto interpolation nodes along the three edges of the triangle, and then computing interior nodes by averaged intersections to achieve three-fold rotational symmetry. Numerical computations show that the Lebesgue constant and interpolation accuracy of the proposed grid compares favorably with those of the best-known grids consisting of the Fekete points. Integration weights corresponding to the set of Lobatto triangle base points are tabulated.

Stable Lagrangian numerical differentiation with the highest order of approximation

Some asymptotic representations for the truncation error for the Lagrangian numerical differentiation are presented, when the ratio of the distance between each interpolation node and the differentiated point to step-parameter h is known. Furthermore, if the sampled values of the function at these interpolation nodes have perturbations which are bounded by e, a method for determining step-parameter h by means of perturbation bound e and order n of interpolation is provided to saturate the order of approximation. And all the investigations in this paper can be generalized to the set of quasi-uniform nodes.

A Criterion for the Uniform Convergence of the Lagrange-Jacobi Interpolation Process

We obtain a uniform and sufficient condition for the convergence of the Lagrange interpolation process with Jacobi nodes on a closed interval [a, b] ⊂ (−1, 1). The condition is stated in terms of the second differences of the interpolated function and uses its values only at the interpolation nodes. Some well-known criteria for uniform convergence are obtained as a consequence of our result.

Matrix-free Interpolation on the Sphere

We study a subspace of bivariate trigonometric polynomials for interpolating functions on the sphere. We give an explicit construction for a system of interpolation nodes, and the corresponding basis for this space, that allows a (discrete) fast Fourier transform-type formula for the interpolant. We prove that the uniform norm of our interpolation operator is of the order log M2, where M is the number of interpolation points. We also construct a minimal quadrature rule for our space (with a number of points equal to the dimension of the space), and describe an associated interpolation operator.

Some researches on trivariate Lagrange interpolation

In this paper, in order to go a step further research on the problem of trivariate Lagrange interpolation, we pose the concepts of sufficient intersection of algebraic surfaces and Lagrange interpolation along a space algebraic curve, and extend Cayley–Bacharach theorem in algebraic geometry from R 2 to R 3 . By using the conclusion of the extended theorem, we deduce a general method of constructing properly posed set of nodes for Lagrange interpolation along a space algebraic curve, and give a series of corollaries for the practical applications. Moreover, we give a new method of constructing properly posed set of nodes for Lagrange interpolation along an algebraic surface, and as a result we make clear the geometrical structure of it.

Bidimensional interpolation operators of finite element type and degree of exactness two

For a given arbitrary triangulation of \(\mathbb R^2\), we construct an interpolating operator which is exact for the polynomials in two variables of total degree \(\leq 2\). This operator is local, in the sense that the information around an interpolation node is taken from a small region around this point. We study the remainder of the interpolation formula.

Flexible piecewise approximations based on partition of unity

In this paper, we study a flexible piecewise approximation technique based on the use of the idea of the partition of unity. The approximations are piecewisely defined, globally smooth up to any order, enjoy polynomial reproducing conditions, and satisfy nodal interpolation conditions for function values and derivatives of any order. We present various properties of the approximations, that are desirable properties for optimal order convergence in solving boundary value problems.

Convergence Conditions for Interpolation Fractions at the Nodes Distinct from the Singular Points of a Function

We consider an interpolation process for the class of functions with finitely many singular points by means of the rational functions whose poles coincide with the singular points of the function under interpolation. The interpolation nodes constitute a triangular matrix and are distinct from the singular points of the function. We find a necessary and sufficient condition for uniform convergence of sequences of interpolation fractions to the function under interpolation on every compact set disjoint from the singular points of the function and other conditions for convergence.