Degree Of Interpolation Research Articles

Extensions of planar GC sets and syzygy matrices

The geometric characterization, introduced by Chung and Yao, identifies node sets for total degree interpolation such that the Lagrange fundamental polynomials are products of linear factors. Sets satisfying the geometric characterization are usually called GC sets. Gasca and Maeztu conjectured that planar GC sets of degree n contain n + 1 collinear points. It has been shown that the conjecture holds for degrees not greater than 5 but it is still unsolved for general degree. One promising approach consists of studying the syzygies of the ideal of polynomials vanishing at the nodes. In order to describe syzygy matrices of GC sets, we analyze the extension of a GC set of degree n to a GC set of degree n + 1, by adding a n + 2 nodes on a line.

A hybrid discontinuous Galerkin method for tokamak edge plasma simulations in global realistic geometry

Progressing toward more accurate and more efficient numerical codes for the simulation of transport and turbulence in the edge plasma of tokamaks, we propose in this work a new hybrid discontinuous Galerkin solver. Based on 2D advection–diffusion conservation equations for the ion density and the particle flux in the direction parallel to the magnetic field, the code simulates plasma transport in the poloidal section of tokamaks, including the open field lines of the Scrape-off Layer (SOL) and the closed field lines of the core region. The spatial discretization is based on a high-order hybrid DG scheme on unstructured meshes, which provides an arbitrary high-order accuracy while reducing considerably the number of coupled degrees of freedom with a local condensation process. A discontinuity sensor is employed to identify critical elements and regularize the solution with the introduction of artificial diffusion. Based on a finite-element discretization, not constrained by a flux-aligned mesh, the code is able to describe plasma facing components of any complex shape using Bohm boundary conditions and to simulate the plasma in versatile magnetic equilibria, possibly extended up to the center. Numerical tests using a manufactured solution show appropriate convergence orders when varying independently the number of elements or the degree of interpolation. Validation is performed by benchmarking the code with the well-referenced edge transport code SOLEDGE2D (Bufferand et al., 2013, 2015 [1,2]) in the WEST geometry. Final numerical experiments show the capacity of the code to deal with low-diffusion solutions.

An iterative approach to barycentric rational Hermite interpolation

In this paper we study an iterative approach to the Hermite interpolation problem, which first constructs an interpolant of the function values at $$n+1$$ nodes and then successively adds m correction terms to fit the data up to the mth derivatives. In the case of polynomial interpolation, this simply reproduces the classical Hermite interpolant, but the approach is general enough to be used in other settings. In particular, we focus on the family of rational Floater–Hormann interpolants, which are based on blending local polynomial interpolants of degree d with rational blending functions. For this family, the proposed method results in rational Hermite interpolants, which depend linearly on the data, with numerator and denominator of degree at most $$(m+1)(n+1)-1$$ and $$(m+1)(n-d)$$ , respectively. They converge at the rate of $$O(h^{(m+1)(d+1)})$$ as the mesh size h converges to 0. After deriving the barycentric form of these interpolants, we prove the convergence rate for $$m=1$$ and $$m=2$$ , and show that the approximation results compare favourably with other constructions.

A numerical study of Newton interpolation with extremely high degrees

In current textbooks the use of Chebyshev nodes with Newton interpolation is advocated as the most efficient numerical interpolation method in terms of approximation accuracy and computational effort. However, we show numerically that the approximation quality obtained by Newton interpolation with Fast Leja (FL) points is competitive to the use of Chebyshev nodes, even for extremely high degree interpolation. This is an experimental account of the analytic result that the limit distribution of FL points and Chebyshev nodes is the same when letting the number of points go to infinity. Since the FL construction is easy to perform and allows to add interpolation nodes on the fly in contrast to the use of Chebyshev nodes, our study suggests that Newton interpolation with FL points is currently the most efficient numerical technique for polynomial interpolation. Moreover, we give numerical evidence that any reasonable function can be approximated up to machine accuracy by Newton interpolation with FL points if desired, which shows the potential of this method.

A path-following technique implemented in a Lagrangian formulation to model quasi-brittle fracture

A path-following strategy based on an energy release criterion is proposed and implemented in an augmented Lagrangian formulation to model localized cohesive cracks. The model is formulated in a rigourous variational framework of continuous spaces. A procedure for the solution of the proposed formulation in a continuous spaces context is also presented. The discrete form of the problem for the implementation within the finite element method is also exposed. The ability and robustness of the proposed method to track equilibrium paths including snap-back phenomenon are analyzed and discussed by means of several numerical tests. The restriction based on an energy release criterion is introduced in the formulation by using a augmented Lagrangian method. The effect of the interpolation order for the Lagrange multiplier (so-called Lagrange field), its influence on the proposed constraint equation and the mesh density dependency are studied and illustrated in the numerical simulations. Different crack patterns are analyzed and some recommendations concerning the interpolation degree that should be used for the Lagrangian field to avoid spurious energy release are also included.

Gaussian process regression for geometry optimization

We implemented a geometry optimizer based on Gaussian process regression (GPR) to find minimum structures on potential energy surfaces. We tested both a two times differentiable form of the Matérn kernel and the squared exponential kernel. The Matérn kernel performs much better. We give a detailed description of the optimization procedures. These include overshooting the step resulting from GPR in order to obtain a higher degree of interpolation vs. extrapolation. In a benchmark against the Limited-memory Broyden–Fletcher–Goldfarb–Shanno optimizer of the DL-FIND library on 26 test systems, we found the new optimizer to generally reduce the number of required optimization steps.

A new incompatible mode element with selective mass scaling for saturated soil dynamics

It is a well-known fact that addressing hydromechanical problems in saturated soils with the finite element method and equal-order interpolation formulations in displacements and pore pressures produces unstable results. Classically, stabilization has been achieved by increasing the interpolation degree of displacement with respect to pore pressure, hence fulfilling the Babuska–Brezzi condition. However, the use of quadratic elements involves high computational costs. From that point of view, the use of stabilized low-order elements is a more desirable option. Much research has been carried out in different directions in the stabilization of low-order formulations for saturated soils in quasistatic conditions, among others with the technique based on strain field enhancement through internal degrees of freedom. This article presents an alternative displacement–pore pressure formulation for saturated soil dynamics based on the enhancement of the displacement field through incompatible modes.

Spectral element method for band-structure calculations of 3D phononic crystals

The spectral element method (SEM) is a special kind of high-order finite element method (FEM) which combines the flexibility of a finite element method with the accuracy of a spectral method. In contrast to the traditional FEM, the SEM exhibits advantages in the high-order accuracy as the error decreases exponentially with the increase of interpolation degree by employing the Gauss–Lobatto–Legendre (GLL) polynomials as basis functions. In this study, the spectral element method is developed for the first time for the determination of band structures of 3D isotropic/anisotropic phononic crystals (PCs). Based on the Bloch theorem, we present a novel, intuitive discretization formulation for Navier equation in the SEM scheme for periodic media. By virtue of using the orthogonal Legendre polynomials, the generalized eigenvalue problem is converted to a regular one in our SEM implementation to improve the efficiency. Besides, according to the specific geometry structure, 8-node and 27-node hexahedral elements as well as an analytic mesh have been used to accurately capture curved PC models in our SEM scheme. To verify its accuracy and efficiency, this study analyses the phononic-crystal plates with square and triangular lattice arrangements, and the 3D cubic phononic crystals consisting of simple cubic (SC), bulk central cubic (BCC) and faced central cubic (FCC) lattices with isotropic or anisotropic scatters. All the numerical results considered demonstrate that SEM is superior to the conventional FEM and can be an efficient alternative method for accurate determination of band structures of 3D phononic crystals.

Convergence rates of derivatives of Floater–Hormann interpolants for well-spaced nodes

Floater–Hormann interpolants constitute a family of barycentric rational interpolants which are based on blending local polynomial interpolants of degree d. Recent results suggest that the k-th derivatives of these interpolants converge at the rate of O(hd+1−k) for k≤d as the mesh size h converges to zero. So far, this convergence rate has been proven for k=1,2 and for k≥3 under the assumption of equidistant or quasi-equidistant interpolation nodes. In this paper we extend these results and prove that Floater–Hormann interpolants and their derivatives converge at the rate of O(hjd+1−k), where hj is the local mesh size, for any k≥0 and any set of well-spaced nodes.

Intact soluble P-glycoprotein is secreted by sinonasal epithelial cells.

P-glycoprotein (P-gp) is a 170 kDa transmembrane efflux pump, which is upregulated in chronic rhinosinusitis. Studies of leukemia demonstrated that P-gp may also be secreted in an intact soluble form. The purpose of this study was to explore whether sinonasal epithelial cells were capable of secreting soluble P-gp and whether P-gp has any functional role. Soluble and cytoplasmic P-gp were quantified in vehicle and lipopolysaccharide exposed cultures by enzyme-linked immunosorbent assay. The molecular weight of the soluble P-gp was determined by Western blot. Naive cultures were exposed to recombinant human P-gp at 0-2000 ng/mL. The degree of membranous interpolation was determined by quantitative fluorescent immunocytochemistry and function was determined by a calcein acetoxymethyl ester assay. Soluble P-gp was secreted intact at 170 kDa. Mean (standard deviation) secretion was detected within vehicle wells at 55.43 ± 26.26 ng/mL, which significantly increased to 333.27 ± 305.98 ng/mL (p < 0.001) after lipopolysaccharide stimulation. Soluble P-gp strongly and significantly correlated with cytoplasmic P-gp (r = 0.57, p = 0.000001). Exposure to 2000 ng/mL of recombinant P-gp significantly increased corrected total cell fluorescence (1.34 ± 1.85) relative to vehicle control 0.29 ± 0.26 (p = 0.01) and significantly reduced calcein acetoxymethyl ester fluorescence (82.03 ± 43.69) relative to 100 ng/mL recombinant P-gp exposed cells (123.11 ± 42.16, p = 0.001). Cultured sinonasal epithelial cells were able to both secrete intact P-gp and could functionally interpolate soluble P-gp into their cell membrane. These in vitro findings indicated that soluble P-gp may be present in nasal mucus as a biomarker and could participate in the maintenance of P-gp overexpression in chronic rhinosinusitis and associated inflammation.

The Leja Method Revisited: Backward Error Analysis for the Matrix Exponential

The Leja method is a polynomial interpolation procedure that can be used to compute matrix functions. In particular, computing the action of the matrix exponential on a given vector is a typical application. This quantity is required, e.g., in exponential integrators. The Leja method essentially depends on three parameters: the scaling parameter, the location of the interpolation points, and the degree of interpolation. We present here a backward error analysis that allows us to determine these three parameters as a function of the prescribed accuracy. Additional aspects that are required for an efficient and reliable implementation are discussed. Numerical examples illustrating the performance of our MATLAB code are included.

On the unique minimal monomial basis of Birkhoff interpolation problem

This paper studies the minimal monomial basis of the n-variable Birkhoff interpolation problem. First, the authors give a fast B-Lex algorithm which has an explicit geometric interpretation to compute the minimal monomial interpolation basis under lexicographic order and the algorithm is in fact a generalization of lex game algorithm. In practice, people usually desire the lowest degree interpolation polynomial, so the interpolation problems need to be solved under, for example, graded monomial order instead of lexicographic order. However, there barely exist fast algorithms for the nonlexicographic order problem. Hence, the authors in addition provide a criterion to determine whether an n-variable Birkhoff interpolation problem has unique minimal monomial basis, which means it owns the same minimal monomial basis w.r.t. arbitrary monomial order. Thus, for problems in this case, the authors can easily get the minimal monomial basis with little computation cost w.r.t. arbitrary monomial order by using our fast B-Lex algorithm.

High-order approximation to Caputo derivatives and Caputo-type advection–diffusion equations (III)

In this paper, a series of new high-order numerical approximations to αth (0<α<1) order Caputo derivative is constructed by using rth degree interpolation approximation for the integral function, where r≥4 is a positive integer. As a result, the new formulas can be viewed as the extensions of the existing jobs (Cao et al., 2015; Li et al., 2014), the convergence orders are O(τr+1−α), where τ is the time stepsize. Two test examples are given to demonstrate the efficiency of these schemes. Then we adopt the derived schemes to solve the Caputo type advection–diffusion equation with Dirichlet boundary conditions. The local truncation error of the derived difference scheme is O(τr+1−α+h2), where τ is the time stepsize, and h the space one. The stability and convergence of the proposed schemes for r=4 are also considered. Without loss of generality, we only display the numerical examples for r=4,5, which support the numerical algorithms.

On the adaptive finite element analysis of the Kohn–Sham equations: methods, algorithms, and implementation

SummaryIn this paper, details of an implementation of a numerical code for computing the Kohn–Sham equations are presented and discussed. A fully self‐consistent method of solving the quantum many‐body problem within the context of density functional theory using a real‐space method based on finite element discretisation of realspace is considered. Various numerical issues are explored such as (i) initial mesh motion aimed at co‐aligning ions and vertices; (ii) a priori and a posteriori optimization of the mesh based on Kelly's error estimate; (iii) the influence of the quadrature rule and variation of the polynomial degree of interpolation in the finite element discretisation on the resulting total energy. Additionally, (iv) explicit, implicit and Gaussian approaches to treat the ionic potential are compared. A quadrupole expansion is employed to provide boundary conditions for the Poisson problem. To exemplify the soundness of our method, accurate computations are performed for hydrogen, helium, lithium, carbon, oxygen, neon, the hydrogen molecule ion and the carbon‐monoxide molecule. Our methods, algorithms and implementation are shown to be stable with respect to convergence of the total energy in a parallel computational environment. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Piecewise trigonometric Hermite interpolation

Based on the symmetric, nonnegative and normalized basis of the trigonometric polynomial space, the piecewise trigonometric Hermite interpolation methods are presented. The Cn−1 and Cn continuous piecewise trigonometric Hermite interpolants of degree n are constructed and the interpolation methods are local. The integral and the differential representations of the errors of the trigonometric Hermite interpolants are given. Several examples are supplied to support the practical value of the given interpolation methods.

3D SEM Approach to Evaluate the Stability of Large-Scale Landslides in Nepal Himalaya

This paper deals with the stability of large-scale landslides with the aim of exploring reliable safety factor in 3-dimensions (3D) for the effective implementation of landslide stability enhancement measures (LSEM) in the mountainous country like Nepal. The paper uses ‘Specfem3D_Slope’, an open-source spectral element method (SEM) based program that can effectively handle the major challenges of 3D modeling to evaluate the stability of large-scale landslides. The SEM prefers p-refinement techniques (increasing the degree of interpolation or polynomial degree or spectral degree) instead of h-refinement (refining the mesh related to elemental budgets) unlike to FEM. Then the safety factors, thus obtained seem to be reliable values as per the material properties and slope model supplied, and that can encouragingly be used in effective design and implementation of various LSEM projects. Moreover, a complete draw down effect after the LSEM gives a complete sense of the effectiveness of the measures itself. In addition, the paper emphasizes some experimental tests to know the displacement behavior of landslide’s soils (e.g., Krishnabhir landslide). A theoretical model is chosen to do some deeper exercises on 3D for the reliability of the results and later implements such criteria for ‘Laprak landslide’ of Nepal Himalaya. With this modeling method (elasto-plastic) the paper incorporates all possible instability conditions like pseudo-static earthquake loading, structural loading over the sliding mass, partially saturated ground water profile, etc., the same conditions as implemented by previous researchers for Laprak landslide. The study of two major landslides of Nepal ‘Krishnabhir landslides’ (relates to major highway) and ‘Laprak landslides’ (relates to a large number of settlements over the moving mass) reflects the major issues of large-scale landslide stability modeling of Nepal Himalaya.

Complex Function Approximation Using Two-Dimensional Interpolation

This paper presents a new scheme for evaluating complex reciprocal and exponential functions in hardware. The proposed method utilizes a two-dimensional convolution algorithm to interpolate bivariate functions from tabulated function values in the complex domain. To reduce the memory requirements for lookup tables, the interpolation is decomposed into independent row and column computations, such that the same coefficient table can be shared. Three different interpolation kernels from degree-1 (linear) to degree-2 (quadratic Lagrange) and degree-3 (cubic Lagrange) are explored to find the optimal design parameters and the most acceptable trade-offs between performance and hardware resources. Moreover, a generic hardware architecture is designed to provide scalable implementation capabilities for computation precision and interpolation degree. To verify the proposed architecture, eight complex reciprocal and eight complex exponential design instances are implemented. The ASIC- and FPGA-based experimental results show that the proposed scheme can efficiently approximate the complex reciprocal and exponential functions with up to 16-bit precision, as well as achieve a considerable reduction of memory requirements compared with traditional bipartite and multipartite schemes. The proposed method is also applicable to other complex functions.

Statistical and systematic uncertainties in pixel-based source reconstruction algorithms for gravitational lensing

Gravitational lens modeling of spatially resolved sources is a challenging inverse problem with many observational constraints and model parameters. We examine established pixel-based source reconstruction algorithms for de-lensing the source and constraining lens model parameters. Using test data for four canonical lens configurations, we explore statistical and systematic uncertainties associated with gridding, source regularisation, interpolation errors, noise, and telescope pointing. Specifically, we compare two gridding schemes in the source plane: a fully adaptive grid that follows the lens mapping but is irregular, and an adaptive Cartesian grid. We also consider regularisation schemes that minimise derivatives of the source (using two finite difference methods) and introduce a scheme that minimises deviations from an analytic source profile. Careful choice of gridding and regularisation can reduce "discreteness noise" in the $\chi^2$ surface that is inherent in the pixel-based methodology. With a gridded source, some degree of interpolation is unavoidable, and errors due to interpolation need to be taken into account (especially for high signal-to-noise data). Different realisations of the noise and telescope pointing lead to slightly different values for lens model parameters, and the scatter between different "observations" can be comparable to or larger than the model uncertainties themselves. The same effects create scatter in the lensing magnification at the level of a few percent for a peak signal-to-noise ratio of 10, which decreases as the data quality improves.

Quartic Spline Interpolation

1. INTRODUCTIONlinear and higher degree interpolation are widely used schemes for Piecewise Polynomial approximation. But at joint of two linear pieces, piecewise linear functions have corners and therefore to achieve a prescribed accuracy usually more data are required then higher order method therefore higher order method are beneficial for best approximation. In the direction of higher order method, Kopotun [4] has obtained univariate splines equivalence of moduli of smoothness and application. Marker and Remier [5] have investigate an unconditionally convergent method for computing zero's of splines and polynomials.(also we refer to Howell and Varma [ 6], Dikshit and Rana [2 ] ,Rana [ 7.8], Agrwal and Wong [9] and Gmeling –Mayling [10] ) 1 ....

Effect of interpolation on parameters extracted from seating interface pressure arrays.

Interpolation is a common data processing step in the study of interface pressure data collected at the wheelchair seating interface. However, there has been no focused study on the effect of interpolation on features extracted from these pressure maps, nor on whether these parameters are sensitive to the manner in which the interpolation is implemented. Here, two different interpolation paradigms, bilinear versus bicubic spline, are tested for their influence on parameters extracted from pressure array data and compared against a conventional low-pass filtering operation. Additionally, analysis of the effect of tandem filtering and interpolation, as well as the interpolation degree (interpolating to 2, 4, and 8 times sampling density), was undertaken. The following recommendations are made regarding approaches that minimized distortion of features extracted from the pressure maps: (1) filter prior to interpolate (strong effect); (2) use of cubic interpolation versus linear (slight effect); and (3) nominal difference between interpolation orders of 2, 4, and 8 times (negligible effect). We invite other investigators to perform similar benchmark analyses on their own data in the interest of establishing a community consensus of best practices in pressure array data processing.