Generalized Vandermonde's system and Lagrange's interpolation
Research center of the French air force (CReA)（法国空军研究中心）；Morpho-Analysis in Signal processing lab., Research center of the French air force (CReA)（法国空军研究中心形态信号处理实验室）
We give explicit formulas as well as a quadratic time algorithm to solve (so called) generalized Vandermonde's systems of p linear equations and n variables. It allows in particular to find all (so called Lagrange's) interpolation polynoms with degree n-1 taking given values in p distinct points.
Inverse Vandermonde matrix calculation is a long-standing problem to solve nonsingular linear system $Vc=b$ where the rows of a square matrix $V$ are constructed by progression of the power polynomials. It has many applications in scientific computing including interpolation, super-resolution, and construction of special matrices applied in cryptography. Despite its numerous applications, the matrix is highly ill-conditioned where specialized treatments are considered for approximation such as conversion to Cauchy matrix, spectral decomposition, and algorithmic tailoring of the numerical solut…
In this paper we investigate polynomial interpolation using orthogonal polynomials. We use weight functions associated with orthogonal polynomials to define a weighted form of Lagrange interpolation. We introduce an upper bound of error estimation for such kinds of approximations. Later, we introduce the sufficient condition of Stenger's conjecture for orthogonal polynomials and numerical verification for such conjecture.
This paper describes the analysis of Lagrange interpolation errors on tetrahedrons. In many textbooks, the error analysis of Lagrange interpolation is conducted under geometric assumptions such as shape regularity or the (generalized) maximum angle condition. In this paper, we present a new estimation in which the error is bounded in terms of the diameter and projected circumradius of the tetrahedron. Because we do not impose any geometric restrictions on the tetrahedron itself, our error estimation may be applied to any tetrahedralizations of domains including very thin tetrahedrons.
Lectures on error analysis of interpolation on simplicial triangulations without the shape-regularity assumption, Part 2: Lagrange interpolation on tetrahedrons
In this work, we introduce the new class of functions which can use to solve the nonlinear/linear multi-dimensional differential equations. Based on these functions, a numerical method is provided which is called the Developed Lagrange Interpolation (DLI). For this, firstly, we define the new class of the functions, called the Developed Lagrange Functions (DLFs), which satisfy in the Kronecker Delta at the collocation points. Then, for the DLFs, the first-order derivative operational matrix of $\textbf{D}^{(1)}$ is obtained, and a recurrence relation is provided to compute the high-order deriv…
For the Lagrange interpolation over a triangular domain, we propose an efficient algorithm to rigorously evaluate the interpolation error constant under the maximum norm by using the finite element method (FEM). In solving the optimization problem corresponding to the interpolation error constant, the maximum norm in the constraint condition is the most difficult part to process. To handle this difficulty, a novel method is proposed by combining the orthogonality of the interpolation associated to the Fujino--Morley FEM space and the convex-hull property of the Bernstein representation of func…
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 …
Explicit Bound for Quadratic Lagrange Interpolation Constant on Triangular Finite Elements
Graduate School of Science and Technology, Niigata University（新潟大学科学技术研究生院）；LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院 LSEC、ICMSEC）
An upper bound for the Lebesgue constant (the supremum norm) of the operator of interpolation of a function in equally spaced points of a triangle by a polynomial of total degree less than or equal to n is obtained. Earlier, the rate of increase of the Lebesgue constants with respect to n for an arbitrary d-dimensional simplex was established by the author. The explicit upper bound proved in this article refines this result for d=2.
A kernel-independent treecode (KITC) is presented for fast summation of particle interactions. The method employs barycentric Lagrange interpolation at Chebyshev points to approximate well-separated particle-cluster interactions. The KITC requires only kernel evaluations, is suitable for non-oscillatory kernels, and it utilizes a scale-invariance property of barycentric Lagrange interpolation. For a given level of accuracy, the treecode reduces the operation count for pairwise interactions from $O(N^2)$ to $O(N\log N)$, where $N$ is the number of particles in the system. The algorithm is demon…
Lagrange interpolation at real projections of Leja sequences for the unit disk
Université Paul Sabatier（图卢兹第三大学）；Institut de Mathématiques, Université de Toulouse III and CNRS (UMR 5219)（图卢兹第三大学数学研究所及法国国家科学研究中心）；Hanoi University of Education（河内师范大学）
Lagrange interpolation processes based on the zeros of anti-Gauss Jacobi polynomials
University of Cagliari（卡利亚里大学）；Insubria University（因斯布鲁克大学）；University of Basilicata（巴西利卡塔大学）；Istituto per le Applicazioni del Calcolo “Mauro Picone”, Naples branch, C.N.R. National Research Council of Italy（意大利国家研究委员会计算应用研究所“毛罗·皮切内”那不勒斯分部）
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.
Bivariate Quasi-Tower Sets and Their Associated Lagrange Interpolation Bases
School of Mathemathics, Key Lab. of Symbolic Computation and Knowledge Engineering (Ministry of Education), Jilin University（吉林大学数学学院，符号计算与知识工程教育部重点实验室）
In the error analysis of finite element methods, the shape regularity assumption on triangulations is typically imposed to obtain a priori error estimations. In practical computations, however, very thin or degenerated elements that violate the shape regularity assumption may appear when we use adaptive mesh refinement. In this manuscript, we attempt to establish an error analysis approach without the shape regularity assumption on triangulations. We have presented several papers on the error analysis of finite element methods on non-shape regular triangulations. The main points in these paper…
Image resizing is a basic tool in image processing and in literature we have many methods, based on different approaches, which are often specialized in only upscaling or downscaling. In this paper, independently of the (reduced or enhanced) size we aim to get, we approach the problem at a continuous scale where the underlying continuous image is globally approximated by the tensor product Lagrange polynomial interpolating at a suitable grid of first kind Chebyshev zeros. This is a well-known approximation tool that is widely used in many applicative yields, due to the optimal behavior of the …
The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vallée Poussin filters. These polynomials can be an useful device for many theoretical and applicative problems since they combine the advantages of the classical Lagrange interpolation, with the uniform convergence in spaces of locally continuous functions equipped with suitable, Jacobi--weighted, uniform norms. The uniform boundedness of the related Lebesgue constants, which equals to the uniform convergence and is missing from Lagrange interpolation, has…
In this paper we study the error propagation of numerical schemes for the advection equation in the case where high precision is desired. The numerical methods considered are based on the fast Fourier transform, polynomial interpolation (semi-Lagrangian methods using a Lagrange or spline interpolation), and a discontinuous Galerkin semi-Lagrangian approach (which is conservative and has to store more than a single value per cell). We demonstrate, by carrying out numerical experiments, that the worst case error estimates given in the literature provide a good explanation for the error propagati…
We present a unified framework to tie overlapping meshes in solid mechanics applications. This framework is a combination of the X-FEM method and the mortar method, which uses Lagrange multipliers to fulfill the tying constraints. As known, mixed formulations are prone to mesh locking which manifests itself by the emergence of spurious oscillations in the vicinity of the tying interface. To overcome this inherent difficulty, we suggest a new coarse-grained interpolation of Lagrange multipliers. This technique consists in selective assignment of Lagrange multipliers on nodes of the mortar side …
