Non-polynomial Interpolation Research Articles

A new type of non-polynomial based TENO scheme for hyperbolic conservation laws

For classical WENO/TENO reconstructions, the high-order polynomial interpolation is one of the main building-blocks for constructing the numerical fluxes. However, due to the inherent characteristics of polynomials, the polynomial interpolation is not suitable to represent the flow scale with sharp transitions such as high-wavenumber fluctuations or discontinuities. In this work, based on a new candidate stencil arrangement with explicit detection of non-smooth regions, a new shock-capturing framework is proposed by combining the infinitely differentiable non-polynomial RBF-based reconstruction in smooth regions with the jump-like non-polynomial interpolation for genuine discontinuities. The resulting scheme not only achieves the desired high-order accuracy for smooth flow scales but also resolves the genuine discontinuities with a sub-cell resolution. Moreover, with the observation that the built-in parameter has a significant impact on the basic shape of the Gaussian RBF kernel function, a nonlinear adaptation strategy for the shape parameter is proposed to incorporate the local features of the flow field and further minimize the numerical dissipation errors for smooth flow scales. In order to demonstrate the robust shock-capturing and high wave-resolution properties of the proposed non-polynomial based scheme, a set of challenging benchmark cases with a wide range of Mach numbers is simulated without the necessity of parameter tuning case by case.

Temporal super-resolution of echocardiography using a novel high-precision non-polynomial interpolation

Higher frame rates are very useful for improving medical diagnosis in fast-moving parts of the heart, especially in the valves. To this end, we propose a non-polynomial interpolation method for increasing the frame rate in echocardiography. Besides describing the proposed method, we present two additional contributions: (1) we obtain a closed-form solution, which is continuous and infinitely differentiable; (2) we provide an error analysis of the method. The resulting error bound indicates that the interpolation method is reliable. Finally, to show the efficiency of our proposal in temporal super-resolution, i.e., the increase in frame rate, we apply it to three types of datasets, including a 1D signal, a simulated dataset, and B-mode echocardiography images. Our experimental results show that the Mean Squared Error of the proposed method is reduced from 0.6 to 0.3, while having the same computational complexity compared to cubic B-spline. The quantitative results also indicate that, even with lower selection rates, we can reach a high performance reconstruction while the image quality is not degraded significantly.

Multistep collocation method for Fredholm integral equations of the second kind

In this paper a Multistep Collocation (MC) method for the second kind Fredholm integral equations (FIEs) is proposed and analyzed. The multistep collocation method is applied to FIE with smooth kernels under uniform mesh and weakly singular kernels |s−t|−α(0<α<1) using a graded mesh then the same convergence rate as collocation method but with a lower degree of freedom is obtained. Moreover, in order to avoid the round-off errors caused by graded mesh, a Hybrid Multistep Collocation (HMC) method by combining multistep collocation and hybrid collocation method is proposed. The HMC method converges faster with lower degrees of freedom and more efficiently captures the weakly singular properties by nonpolynomial interpolation at the first subinterval. The L∞-norm convergence results are analysed and proved. Numerical examples are presented to demonstrate the efficiency of the proposed methods.

Solving the third-kind Volterra integral equation via the boundary value technique: Lagrange polynomial versus fractional interpolation

The solution to the third-kind Volterra integral equation (VIE3) usually has unbounded derivatives near the original point t=0, which brings difficulties to numerical computation. In this paper, we analyze two kinds of modified multistep collocation methods for VIE3: collocation boundary value method with the fractional interpolation (FCBVM) and that with Lagrange interpolation (CBVMG). The former is developed based on the non-polynomial interpolation which is particularly feasible for approximating functions in the form of tη with the real number η≥0. The latter is devised by using classical polynomial interpolation. The application of the boundary value technique enables both approaches to efficiently solve long-time integration problems. Moreover, we investigate the convergence properties of these two kinds of algorithms by Grönwall’s inequality.

Application Local Polynomial and Non-polynomial Splines of the Third Order of Approximation for the Construction of the Numerical Solution of the Volterra Integral Equation of the Second Kind

The present paper is devoted to the application of local polynomial and non-polynomial interpolation splines of the third order of approximation for the numerical solution of the Volterra integral equation of the second kind. Computational schemes based on the use of the splines include the ability to calculate the integrals over the kernel multiplied by the basis function which are present in the computational methods. The application of polynomial and nonpolynomial splines to the solution of nonlinear Volterra integral equations is also discussed. The results of the numerical experiments are presented.

Non-Polynomial Interpolation of Functions with Large Gradients and Its Application

Interpolation of a function of one variable with large gradients in the boundary layer region is studied. The problem is that the use of classical polynomial interpolation formulas on a uniform mesh to functions with large gradients can lead to errors of the order of $$O(1)$$ , despite a small mesh size. An interpolation formula based on fitting to the component that defines the boundary-layer growth of the function is investigated. An error estimate, which depends on the number of interpolation nodes and is uniform over the boundary layer component and its derivatives, is obtained. It is shown how the interpolation formula derived can be used to construct formulas for numerical differentiation and integration and in the two-dimensional case. The corresponding error estimates are obtained.

Non-polynomial interpolation of functions in the presence of a boundary layer

The question of interpolation of a function of one variable with large gradients in the boundary layer region is investigated. The problem is that applying of polynomial interpolation formulas on a uniform grid to functions with large gradients can lead to unacceptable errors. We study the interpolation formulas with an arbitrarily number of interpolation nodes which are exact on the singular component. This component is responsible for the main growth of the function in the boundary layer and can be found based on asymptotic expansions. It is proved that error estimates don’t depend on the singular component and its derivatives. In the case of an exponential boundary layer these estimates don’t depend on a small parameter.

Fast non-polynomial interpolation and integration for functions with logarithmic singularities

A fast non-polynomial interpolation is proposed in this paper for functions with logarithmic singularities. It can be executed fast with the discrete cosine transform. Based on this interpolation, a new quadrature is proposed for a kind of logarithmically singular integrals. The interpolation and integration errors are also analyzed. Numerical examples of the interpolation and integration are shown to validate the efficiency of the proposed new interpolation and the new quadrature.

A RBF-WENO finite volume method for hyperbolic conservation laws with the monotone polynomial interpolation method

Essentially non-oscillatory (ENO) and weighted ENO (WENO) methods are efficient high order numerical methods for solving hyperbolic conservation laws designed to reduce the Gibbs oscillations. The original ENO and WENO methods are based on the polynomial interpolation and the overall convergence rate is uniquely determined by the total number of interpolation points involved for the approximation. In this paper, we propose non-polynomial ENO and WENO finite volume methods in order to enhance the local accuracy and convergence. The infinitely smooth radial basis functions (RBFs) are adopted as a non-polynomial interpolation basis. Particularly we use the multi-quadratic and Gaussian RBFs. The non-polynomial interpolation such as the RBF interpolation offers the flexibility to control the local error by optimizing the free parameter. Then we show that the non-polynomial interpolation can be represented as a perturbation of the polynomial interpolation. To guarantee the essentially non-oscillatory property, the monotone polynomial interpolation method is introduced as a switching method to the polynomial reconstruction adaptively near the non-smooth area. The numerical results show that the developed non-polynomial ENO and WENO methods with the monotone polynomial interpolation method enhance the local accuracy and give sharper solution profile than the ENO/WENO methods based on the polynomial interpolation.

Some studies on nonpolynomial interpolation and error analysis

In this paper, we propose nonpolynomial and Hermite nonpolynomial interpolation with multiple parameters and present method to determine optimal value of parameters which generate minimum error in approximation. The generalized error analysis results for nonpolynomial and Hermite nonpolynomial interpolations are derived. We establish theoretical relationship among nonpolynomial, polynomial interpolation and the Fourier series, and propose solution to Runge’s phenomenon through nonpolynomial interpolation. The Hermite nonpolynomial cubic spline, nonpolynomial cubic spline interpolation methods and their error analysis are presented. Numerical simulations are carried out for the analysis of error in cubic spline interpolations. Proposed method is applied to the analysis of various time series to show comparison in errors between polynomial and nonpolynomial spline interpolations, and to Empirical Mode Decomposition (EMD) to illustrate practical usefulness of the results.

An analogue of the four-point Newton-Cotes formula for a function with a boundary-layer component

The construction of the Newton-Cotes formulas is based on approximating the integrand by a Lagrange polynomial. The error of such quadrature formulas can be great for a function with a boundary-layer component. In this paper, an analog of the four-point Newton-Cotes rule is constructed. The construction is based on using a nonpolynomial interpolation that is exact for the boundary layer component. Error estimates of the quadrature rule independent of the boundary layer component gradients are obtained. Numerical experiments are performed.

How good can polynomial interpolation on the sphere be

This paper explores the quality of polynomial interpolation approximations over the sphere S r−1⊂R r in the uniform norm, principally for r=3. Reimer [17] has shown there exist fundamental systems for which the norm ‖Λ n ‖ of the interpolation operator Λ n , considered as a map from C(S r−1) to C(S r−1), is bounded by d n , where d n is the dimension of the space of all spherical polynomials of degree at most n. Another bound is d n 1/2(λavg/λmin )1/2, where λavg and λmin are the average and minimum eigenvalues of a matrix G determined by the fundamental system of interpolation points. For r=3 these bounds are (n+1)2 and (n+1)(λavg/λmin )1/2, respectively. In a different direction, recent work by Sloan and Womersley [24] has shown that for r=3 and under a mild regularity assumption, the norm of the hyperinterpolation operator (which needs more point values than interpolation) is bounded by O(n 1/2), which is optimal among all linear projections. How much can the gap between interpolation and hyperinterpolation be closed? For interpolation the quality of the polynomial interpolant is critically dependent on the choice of interpolation points. Empirical evidence in this paper suggests that for points obtained by maximizing λmin , the growth in ‖Λ n ‖ is approximately n+1 for n<30. This choice of points also has the effect of reducing the condition number of the linear system to be solved for the interpolation weights. Choosing the points to minimize the norm directly produces fundamental systems for which the norm grows approximately as 0.7n+1.8 for n<30. On the other hand, ‘minimum energy points’, obtained by minimizing the potential energy of a set of (n+1)2 points on S 2, turn out empirically to be very bad as interpolation points. This paper also presents numerical results on uniform errors for approximating a set of test functions, by both interpolation and hyperinterpolation, as well as by non-polynomial interpolation with certain global basis functions.

A torsion bending element for thin-walled beams with open and closed cross sections

A finite element is formulated for the torsion problems of thin-walled beams. The element is based on Benscoter's beam theory, which is valid for open and also closed cross-sections. The non-polynomial interpolation presented in this paper allows the exact static solution to be obtained with only one element. Numerical results are presented for three thin-walled cantilever beams, one with a channel cross-section and the two others with rectangular cross-sections. The influence of the transverse shear strain is investigated and the different models of torsion are compared. For one example, the results obtained with one-dimensional torsion elements are compared with those obtained using shell elements.