Interpolation Nodes Research Articles

Numerical investigation of a high order hybridizable discontinuous Galerkin method for 2d time‐harmonic Maxwell's equations

PurposeThis work is concerned with the development and the numerical investigation of a hybridizable discontinuous Galerkin (HDG) method for the simulation of two‐dimensional time‐harmonic electromagnetic wave propagation problems.Design/methodology/approachThe proposed HDG method for the discretization of the two‐dimensional transverse magnetic Maxwell equations relies on an arbitrary high order nodal interpolation of the electromagnetic field components and is formulated on triangular meshes. In the HDG method, an additional hybrid variable is introduced on the faces of the elements, with which the element‐wise (local) solutions can be defined. A so‐called conservativity condition is imposed on the numerical flux, which can be defined in terms of the hybrid variable, at the interface between neighbouring elements. The linear system of equations for the unknowns associated with the hybrid variable is solved here using a multifrontal sparse LU method. The formulation is given, and the relationship between the considered HDG method and a standard upwind flux‐based DG method is also examined.FindingsThe approximate solutions for both electric and magnetic fields converge with the optimal order of p+1 in L2 norm, when the interpolation order on every element and every interface is p and the sought solution is sufficiently regular. The presented numerical results show the effectiveness of the proposed HDG method, especially when compared with a classical upwind flux‐based DG method.Originality/valueThe work described here is a demonstration of the viability of a HDG formulation for solving the time‐harmonic Maxwell equations through a detailed numerical assessment of accuracy properties and computational performances.

A Flattest Constrained Envelope Approach for Empirical Mode Decomposition

Empirical mode decomposition (EMD) is an adaptive method for nonlinear, non-stationary signal analysis. However, the upper and lower envelopes fitted by cubic spline interpolation (CSI) may often occur overshoots. In this paper, a new envelope fitting method based on the flattest constrained interpolation is proposed. The proposed method effectively integrates the difference between extremes into the cost function, and applies a chaos particle swarm optimization method to optimize the derivatives of the interpolation nodes. The proposed method was tested on three different types of data: ascertain signal, random signals and real electrocardiogram signals. The experimental results show that: (1) The proposed flattest envelope effectively solves the overshoots caused by CSI method and the artificial bends caused by piecewise parabola interpolation (PPI) method. (2) The index of orthogonality of the intrinsic mode functions (IMFs) based on the proposed method is 0.04054, 0.02222±0.01468 and 0.04013±0.03953 for the ascertain signal, random signals and electrocardiogram signals, respectively, which is lower than the CSI method and the PPI method, and means the IMFs are more orthogonal. (3) The index of energy conversation of the IMFs based on the proposed method is 0.96193, 0.93501±0.03290 and 0.93041±0.00429 for the ascertain signal, random signals and electrocardiogram signals, respectively, which is closer to 1 than the other two methods and indicates the total energy deviation amongst the components is smaller. (4) The comparisons of the Hilbert spectrums show that the proposed method overcomes the mode mixing problems very well, and make the instantaneous frequency more physically meaningful.

Convex combinations for diffusion schemes

An interpolation method for diffusion in anisotropic discontinuous media based on convex combinations and physical relationships is presented. The method is exact for any piecewise linear solution, even if the interpolation nodes lie on the opposite sides of a material discontinuity. Values in points that do not lie within the convex hull of interpolation nodes are computed using flux boundary conditions. The method permits interpolation in every point within most domains, while preserving the maximum and minimum principles. We propose to replace the interpolation techniques in several non-linear finite volume schemes with the present method. Additionally, it is demonstrated that the construction of a convex combination by a simple search strategy may not be economical or even feasible if the grid is distorted. An alternative search structure that behaves well in such cases is proposed.

An Extension of the Floater–Hormann Family of Barycentric Rational Interpolants

The barycentric rational interpolants introduced by Floater and Hormann in 2007 are “blends” of polynomial interpolants of fixed degree d. In some cases these rational functions achieve approximation of much higher quality than the classical polynomial interpolants, which, e.g., are ill-conditioned and lead to Runge’s phenomenon if the interpolation nodes are equispaced. For such nodes, however, the condition of Floater–Hormann interpolation deteriorates exponentially with increasing d. In this paper, an extension of the Floater–Hormann family with improved condition at equispaced nodes is presented and investigated. The efficiency of its applications such as the approximation of derivatives, integrals and primitives of functions is compared to the corresponding results recently obtained with the original family of rational interpolants. Math Subject Classification: 65D05, 65L12, 65D32, 41A05, 41A20, 41A25

Study of the approximate approaches to the POD based spectral representation method

The spectral representation method (SRM) is most widely used in simulating the stochastic field. The proper orthogonal decomposition (POD) based SRM is an important form. This paper investigates the approximate approaches to the POD-based SRM in simulating two typical problems, i.e., the seismic ground motion and wind velocity fields simulations. Then, the accuracy resulting from the power spectral density matrix-based POD method (PSRM) is compared to that of the coherency matrix-based POD method (CPSRM). It is concluded that the CPSRM maintains a much higher accuracy than the PSRM. In the CPSRM, the linear interpolation of eigenvectors and third-order polynomial interpolation of eigenvalues can be accepted to attain high accuracy; the linearly distributed interpolation nodes are effective in the ground motions simulation; however, the exponentially distributed interpolation nodes are effective in the wind velocity simulation.

Interpolation and superpositions of multivariate continuous functions

A new approach to proving the unrepresentability of multivariate continuous functions by superpositions of continuous functions of fewer variables is suggested. The approach is based on the idea of interpolating functions with the use of sufficiently sparse interpolation nodes but with relatively high accuracy. The approach is illustrated by Vitushkin’s well-known results on superpositions of functions that depend on a given number of variables and have derivatives of a given order.

Bounding the Lebesgue constant for Berrut’s rational interpolant at general nodes

It has recently been shown that the Lebesgue constant for Berrut’s rational interpolant at equidistant nodes grows logarithmically in the number of interpolation nodes. In this paper we show that the same holds for a very general class of well-spaced nodes and essentially any distribution of nodes that satisfies a certain regularity condition, including Chebyshev–Gauss–Lobatto nodes as well as extended Chebyshev nodes.

A Novel Thinning Algorithm of 3D Image Model Based on Spatial Wavelet Interpolation

In dynamic modeling process of 3D image, too much few characteristics extracted in the binocular stereo vision modeling process cause the cloud characteristics of the model node could not be recovered and low fidelity of the model. In order to solve this problem, this paper presents a thinning algorithm of 3D image model based on improved spatial wavelet interpolation. Firstly, several coordinates of dynamic characteristic nodes are collected by the binocular stereo vision method to compose a characteristic' assemblage which is used to build an interpolation region. Then the characteristic nodes of interpolation are built in the region by the dynamic wavelet recovering method, and modified in direction. Whether the characteristic is reasonable is judged by the modified result. Finally the new characteristic is taken as the vertex to construct a new region and complete the iterative interpolation. Simulation results show that the accuracy error of the interpolation is less than 15 pixels when image interpolation is carried out based on a few characteristics which basically meet the requirements of stability, accuracy and anti-interference ability of 3D interpolation.

Approximate Solutions of Fisher's Type Equations with Variable Coefficients

The spectral collocation approximations based on Legendre polynomials are used to compute the numerical solution of time-dependent Fisher’s type problems. The spatial derivatives are collocated at a Legendre-Gauss-Lobatto interpolation nodes. The proposed method has the advantage of reducing the problem to a system of ordinary differential equations in time. The four-stage A-stable implicit Runge-Kutta scheme is applied to solve the resulted system of first order in time. Numerical results show that the Legendre-Gauss-Lobatto collocation method is of high accuracy and is efficient for solving the Fisher’s type equations. Also the results demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations.

Polynomial interpolation and asymptotic representations for zeta functions

We develop various asymptotic relations between the Riemann zeta function $\zeta(s)$ and the interpolation errors of Lagrange and Hermite interpolation to functions like $|y|^s$ and $y^{2m}\log |y|$. We show that the interpolation nodes of these int

Constructing Nested Nodal Sets for Multivariate Polynomial Interpolation

We present a robust method for choosing multivariate polynomial interpolation nodes. Our algorithm is an optimization method to greedily minimize a measure of interpolant sensitivity, a variant of a weighted Lebesgue function. Nodes are therefore chosen that tend to control oscillations in the resulting interpolant. This method can produce an arbitrary number of nodes and is not constrained by the dimension of a complete polynomial space. Our method is therefore flexible: nested nodal sets are produced in spaces of arbitrary dimensions, and the number of nodes added at each stage can be arbitrary. The algorithm produces a nodal set given a probability measure on the input space, thus parameterizing interpolants with respect to finite measures. We present examples to show that the method yields nodal sets that behave well with respect to standard interpolation diagnostics: the Lebesgue constant, the Vandermonde determinant, and the Vandermonde condition number. We also show that a nongreedy version of the nodal array has a strong connection with equilibrium measures from weighted pluripotential theory.

Fast Reduction of Generalized Companion Matrix Pairs for Barycentric Lagrange Interpolants

For a barycentric Lagrange interpolant $p(z)$, the roots of $p(z)$ are exactly the eigenvalues of a generalized companion matrix pair $(\mathbf{A},\mathbf{B})$. For real interpolation nodes, the matrix pair $(\mathbf{A},\mathbf{B})$ can be reduced to a pair $(\mathbf{H},\mathbf{B})$, where $\mathbf{H}$ has tridiagonal plus rank-one structure. In this paper we propose two fast algorithms for reducing the pair $(\mathbf{A},\mathbf{B})$ to Hessenberg-triangular form. The matrix pair $(\mathbf{A},\mathbf{B})$ has two spurious infinite eigenvalues, and if the leading coefficients of the interpolant are zero, there will also be other infinite eigenvalues. We propose tools for detecting when the leading coefficients of $p(z)$ are zero, and describe a procedure to deflate all of the infinite eigenvalues from the reduced matrix pair $(\mathbf{H},\mathbf{B})$, while still maintaining the tridiagonal plus rank-one structure of the resulting standard eigenvalue problem. Since fast $QR$ algorithms exist for such struc...

On the Lebesgue constant of subperiodic trigonometric interpolation

We solve a recent conjecture, proving that the Lebesgue constant of Chebyshev-like angular nodes for trigonometric interpolation on a subinterval [−ω,ω] of the full period [−π,π] is attained at ±ω, and its value is independent of ω and coincides with the Lebesgue constant of algebraic interpolation at the classical Chebyshev nodes in (−1,1).

A note on the W1,p‐stability of piecewise linear interpolation

AbstractIt is known that piecewise linear interpolation of functions of one variable is uniformly bounded with an H1‐stability constant of one. In [1], we considered the nodal interpolation operator acting between spaces of piecewise linear functions and presented an elementary proof by minimizing a functional representing the H1‐semi‐norm. In this note, a standard approximation argument is applied generalizing the result to piecewise linear interpolation of all functions in W1,p on a real interval, 1 ≤ p ≤ ∞. We also comment on alternative proofs and finally give a counterexample for piecewise linear interpolation in 2D. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Reduction of distortion and improvement of efficiency for gridding of scattered gravity and magnetic data

This paper presents a reasonable gridding-parameters extraction method for setting the optimal interpolation nodes in the gridding of scattered observed data. The method can extract optimized gridding parameters based on the distribution of features in raw data. Modeling analysis proves that distortion caused by gridding can be greatly reduced when using such parameters. We also present some improved technical measures that use human-machine interaction and multi-thread parallel technology to solve inadequacies in traditional gridding software. On the basis of these methods, we have developed software that can be used to grid scattered data using a graphic interface. Finally, a comparison of different gridding parameters on field magnetic data from Ji Lin Province, North China demonstrates the superiority of the proposed method in eliminating the distortions and enhancing gridding efficiency.

Approximations by the Cauchy-type integrals with piecewise linear densities

The paper is a contribution to the complex variable boundary element method, shortly CVBEM. It is focused on Jordan regions having piecewise regular boundaries without cusps. Dini continuous densities whose modulus of continuity ω(·) satisfies $$\mathop {\lim \sup }\limits_{s \downarrow 0} \omega (s)\ln {1 \over s} = 0$$ are considered on these boundaries. Functions satisfying the Hölder condition of order α, 0 < α ⩾ 1, belong to them. The statement that any Cauchy-type integral with such a density can be uniformly approximated by a Cauchy-type integral whose density is a piecewise linear interpolant of the original one is proved under the assumption that the mesh of the interpolation nodes is sufficiently fine and uniform. This result ensures the existence of approximate CVBEM solutions of some planar boundary value problems, especially of the Dirichlet ones.

Convergence and superconvergence of staggered discontinuous Galerkin methods for the three-dimensional Maxwell’s equations on Cartesian grids

In this paper, a new type of staggered discontinuous Galerkin methods for the three dimensional Maxwell’s equations is developed and analyzed. The spatial discretization is based on staggered Cartesian grids so that many good properties are obtained. First of all, our method has the advantages that the numerical solution preserves the electromagnetic energy and automatically fulfills a discrete version of the Gauss law. Moreover, the mass matrices are diagonal, thus time marching is explicit and is very efficient. Our method is high order accurate and the optimal order of convergence is rigorously proved. It is also very easy to implement due to its Cartesian structure and can be regarded as a generalization of the classical Yee’s scheme as well as the quadrilateral edge finite elements. Furthermore, a superconvergence result, that is the convergence rate is one order higher at interpolation nodes, is proved. Numerical results are shown to confirm our theoretical statements, and applications to problems in unbounded domains with the use of PML are presented. A comparison of our staggered method and non-staggered method is carried out and shows that our method has better accuracy and efficiency.

Extremal holomorphic maps and the symmetrized bidisc

We introduce the class of $n$-extremal holomorphic maps, a class that generalises both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrised bidisc $\Gamma$. We show that a well-known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence $\mathcal{C}_\nu, \nu \geq 0,$ of necessary conditions for solvability, prove that they are of strictly increasing strength and show that $\mathcal{C}_{n-3}$ is insufficient for the solvability of an $n$-point problem for $n\geq 3$. We propose the conjecture that condition $\mathcal{C}_{n-2}$ is necessary and sufficient for the solvability of an $n$-point interpolation problem for $\Gamma$ and we explore the implications of this conjecture. We introduce a classification of rational $\Gamma$-inner functions, that is, analytic functions from the disc into $\Gamma$ whose radial limits at almost all points on the unit circle lie in the distinguished boundary of $\Gamma$. The classes are related to $n$-extremality and the conditions $\mathcal{C}_\nu$; we prove numerous strict inclusions between the classes.

Interpolation of analytic functions with finitely many singularities

We consider an interpolation process for the class of functions with finitely many singular points by means of rational functions whose poles coincide with the singular points of the function under interpolation. The interpolation nodes form a triangular matrix. We find necessary and sufficient conditions for the 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. We generalize and improve the familiar results on the interpolation of functions with finitely many singular points by rational fractions and of entire functions by polynomials.

A discrete adapted hierarchical basis solver for radial basis function interpolation

In this paper we develop a discrete Hierarchical Basis (HB) to efficiently solve the Radial Basis Function (RBF) interpolation problem with variable polynomial degree. The HB forms an orthogonal set and is adapted to the kernel seed function and the placement of the interpolation nodes. Moreover, this basis is orthogonal to a set of polynomials up to a given degree defined on the interpolating nodes. We are thus able to decouple the RBF interpolation problem for any degree of the polynomial interpolation and solve it in two steps: (1) The polynomial orthogonal RBF interpolation problem is efficiently solved in the transformed HB basis with a GMRES iteration and a diagonal (or block SSOR) preconditioner. (2) The residual is then projected onto an orthonormal polynomial basis. We apply our approach on several test cases to study its effectiveness.