We derived the formulae of central differentiation for the finding of the first and second derivatives of functions given in discrete points, with the number of points being arbitrary. The obtained formulae for the derivative calculation do not require direct construction of the interpolating polynomial. As an example of the use of the developed method we calculated the first derivative of the function having known analytical value of the derivative. The result was examined in the limiting case of infinite number of points. We studied the spectral characteristics of the weight coefficients sequence of the numerical differentiation formulae. The performed investigation enabled one to analyze the accuracy of the numerical differentiation carried out with the use of the developed technique.
This paper provides a new approach to derive various arbitrary high order finite difference formulae for the numerical differentiation of analytic functions. In this approach, various first and second order formulae for the numerical approximation of analytic functions are given with error terms explicitly expanded as Taylor series of the analytic function. These lower order approximations are successively improved by one or two (two order improvement for centered formulae) to give finite difference formulae of arbitrary high order. The new approach allows to recover the standard backward, for…
In this paper we explore orthogonal systems in $\mathrm{L}_2(\mathbb{R})$ which give rise to a skew-Hermitian, tridiagonal differentiation matrix. Surprisingly, allowing the differentiation matrix to be complex leads to a particular family of rational orthogonal functions with favourable properties: they form an orthonormal basis for $\mathrm{L}_2(\mathbb{R})$, have a simple explicit formulae as rational functions, can be manipulated easily and the expansion coefficients are equal to classical Fourier coefficients of a modified function, hence can be calculated rapidly. We show that this famil…
We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor's formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.
The $\ell_2-$ and $\ell_1-$regularized modified Lagrange interpolation formulae over $[-1,1]$ are deduced in this paper. This paper mainly analyzes the numerical characteristics of regularized barycentric interpolation formulae, which are presented in [C. An and H.-N. Wu, 2019], and regularized modified Lagrange interpolation formulae for both interpolation and extrapolation. Regularized barycentric interpolation formulae can be carried out in $\mathcal{O}(N)$ operations based on existed algorithms [H. Wang, D. Huybrechs and S. Vandewalle, Math. Comp., 2014], and regularized modified Lagrange …
For backward differentiation formulae (BDF) applied to gradient flows of semiconvex functions, quadratic stability implies the existence of a Lyapunov functional. We compute the maximum time step which can be derived from quadratic stability for the 3-step BDF method (BDF3). Applications to the asymptotic behaviour of sequences generated by the BDF3 scheme are given.
This paper considers the $ p $ ($ p=1,2,3 $) order numerical differentiation on function $ y $ in $ (0,2π) $. They are transformed into corresponding Fredholm integral equation of the first kind. Computational schemes with analytic solution formulas are designed using Galerkin method on trigonometric basis. Convergence and divergence are all analysed in Corollaries 5.1, 5.2, and a-priori error estimate is uniformly obtained in Theorem 6.1, 7.1, 7.2. Therefore, the algorithm achieves the optimal convergence rate $ O( δ^{\frac{2μ}{2μ+1}} ) \ (μ= \frac{1}{2} \ \textrm{or} \ 1)$ with periodic Sobo…
We are interested in the development of an algorithmic differentiation framework for computing approximations to tangent vectors to scalar and systems of hyperbolic partial differential equations. The main difficulty of such a numerical method is the presence of shock waves that are resolved by proposing a numerical discretization of the calculus introduced in Bressan and Marson [Rend. Sem. Mat. Univ. Padova, 94:79-94, 1995]. Numerical results are presented for the one-dimensional Burgers equation and the Euler equations. Using the essential routines of a state-of-the-art code for computationa…
The problem of numerical differentiation can be thought of as an inverse problem by considering it as solving a Volterra equation. It is well known that such inverse integral problems are ill-posed and one requires regularization methods to approximate the solution appropriately. The commonly practiced regularization methods are (external) parameter-based like Tikhonov regularization, which have certain inherent difficulties like choosing an optimal value of the regularization parameter, especially in the absence of noise information, which is a non-trivial task. An attractive alternative is i…
In applied mathematics, especially in optimization, functions are often only provided as so called "Black-Boxes" provided by software packages, or very complex algorithms, which make automatic differentation very complicated or even impossible. Hence one seeks the numerical approximation of the derivative. Unfortunately numerical differentation is a difficult task in itself, and it is well known that it is numerical instable. There are many works on this topic, including the usage of (global) Chebyshev approximations. Chebyshev approximations have the great property that they converge very fas…
Higher order numerical differentiation on the Infinity Computer
University of Calabria（卡拉布里亚大学）；Institute of High Performance Computing and Networking of the National Research Council of Italy（意大利国家研究委员会高性能计算与网络研究所）
A novel high-order numerical scheme is proposed to compute the covariant derivative, particularly for divergence and curl, on any curved surface. The proposed scheme does not require the construction of a curved axis or metric tensor, which would deteriorate the accuracy of the covariant derivative and prevent its application to complex surfaces. As an application, the Helmholtz-Hodge decomposition (HHD) is adapted in the context of the Galerkin method for displaying the irrotational, incompressible, and harmonic components of vectors on curved surfaces.
Agency for Science, Technology and Research (A*STAR)（新加坡科技研究局（A*STAR））；Nanyang Technological University（南洋理工大学）
In this study, novel physics-informed neural network (PINN) methods for coupling neighboring support points and their derivative terms which are obtained by automatic differentiation (AD), are proposed to allow efficient training with improved accuracy. The computation of differential operators required for PINNs loss evaluation at collocation points are conventionally obtained via AD. Although AD has the advantage of being able to compute the exact gradients at any point, such PINNs can only achieve high accuracies with large numbers of collocation points, otherwise they are prone to optimizi…
Automatic differentiation (AD) is an ensemble of techniques that allow to evaluate accurate numerical derivatives of a mathematical function expressed in a computer programming language. In this paper we use AD for stating and solving solid mechanics problems. Given a finite element discretization of the domain, we evaluate the free energy of the solid as the integral of its strain energy density, and we make use of AD for directly obtaining the residual force vector and the tangent stiffness matrix of the problem, as the gradient and the Hessian of the free energy respectively. The result is …
The known Complex Step Derivative (CSD) method allows easy and accurate differentiation up to machine precision of real analytic functions by evaluating them a small imaginary step next to the real number line. The current paper proposes that derivatives of holomorphic functions can be calculated in a similar fashion by taking a small step in a quaternionic direction instead. It is demonstrated that in so doing the CSD properties of high accuracy and convergence are carried over to derivatives of holomorphic functions. To demonstrate the ease of implementation, numerical experiments were perfo…
Differentiation by integration with Jacobi polynomials
INRIA Lille-Nord Europe（法国国家信息与自动化研究所里尔北欧分部）；Université de Lille 1（里尔第一大学）；Arts et Métiers ParisTech centre de Lille（巴黎国立高等工艺美术学院里尔中心）；École Centrale de Lille（里尔中央理工学院）
