In the paper, the planar polynomial geometric interpolation of data points is revisited. Simple sufficient geometric conditions that imply the existence of the interpolant are derived in general. They require data points to be convex in a certain discrete sense. Since the geometric interpolation is based precisely on the known data only, one may consider it as the parametric counterpart to the polynomial function interpolation. The established result confirms the Höllig-Koch conjecture on the existence and the approximation order in the planar case for parametric polynomial curves of any degree stated quite a while ago.
A mixed interpolation-regression method for function approximation on certain planar domains
La Laguna University, Spain（拉帕尔马大学）；IMAG and Departamento de Matemática Aplicada, Universidad de Granada, Spain（格拉纳达大学）；Departamento de Matemáticas, Universidad Carlos III de Madrid, Spain（马德里卡洛斯三世大学）
A novel static algorithm is proposed for numerical reparametrization of periodic planar curves. The method identifies a monitor function of the arclength variable with the true curvature of an open planar curve and considers a simple interpolation between the object and the unit circle at the curvature level. Since a convenient formula is known for tangential velocity that maintains the equidistribution rule with curvature-type monitor functions, the strategy enables to compute the correspondence between the arclength and another spatial variable by evolving the interpolated curve. With a cert…
We present precise anisotropic interpolation error estimates for smooth functions using a new geometric parameter and derive inverse inequalities on anisotropic meshes. In our theory, the interpolation error is bounded in terms of the diameter of a simplex and the geometric parameter. Imposing additional assumptions makes it possible to obtain anisotropic error estimates. This paper also includes corrections to an error in Theorem 2 of our previous paper, "General theory of interpolation error estimates on anisotropic meshes" (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-…
We present precise Raviart-Thomas interpolation error estimates on anisotropic meshes. The novel aspect of our theory is the introduction of a new geometric parameter of simplices. It is possible to obtain new anisotropic Raviart-Thoma error estimates using the parameter. We also include corrections to an error in "General theory of interpolation error estimates on anisotropic meshes" (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-191), in which Theorem 3 was incorrect.
The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and have not been studied yet in the literature. Parametric polynomial curves of low degree are used and a geometric continuity is prescribed at the boundary points of the circular arc. A general theory about the existence and the uniqueness of the optimal approximant is presented and a rigorous analysis is done for some special cases for which the degree of the polynomial curve and the order of the geometric smoothness differ by two. This in…
We describe a general approach for constructing a broad class of operators approximating high-dimensional curves based on geometric Hermite data. The geometric Hermite data consists of point samples and their associated tangent vectors of unit length. Extending the classical Hermite interpolation of functions, this geometric Hermite problem has become popular in recent years and has ignited a series of solutions in the 2D plane and 3D space. Here, we present a method for approximating curves, which is valid in any dimension. A basic building block of our approach is a Hermite average - a notio…
We propose a general theory of estimating interpolation error for smooth functions in two and three dimensions. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the two-dimensional case, our geometric parameter is equivalent to the circumradius of a triangle. In the three-dimensional case, our geometric parameter also represents the flatness of a tetrahedron. Through the introduction of the geometric parameter, the error estimates newly obtained can be applied to cases that violate the maximum-angle condition.
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.
For fourth-order geometric evolution equations for planar curves with the dissipation of the bending energy, including the Willmore and the Helfrich flows, we consider a numerical approach. In this study, we construct a structure-preserving method based on a discrete variational derivative method. Furthermore, to prevent the vertex concentration that may lead to numerical instability, we discretely introduce Deckelnick's tangential velocity. Here, a modification term is introduced in the process of adding tangential velocity. This modified term enables the method to reproduce the equations' pr…
The present paper is focused on the comparison of the efficiency of two important meshless interpolants for gravity acceleration interpolation. Compactly-supported spherical radial basis functions and interpolating moving least squares are used to interpolate actual gravity accelerations in southern Africa. Interpolated values are compared with actual values, gathered by observation. A thorough analysis is presented for the standard deviation of the differences between interpolated and actual values. Three different class of spherical radial basis functions-Poisson, singularity, and logarithmi…
When balanced truncation is used for model order reduction, one has to solve a pair of Lyapunov equations for two Gramians and uses them to construct a reduced-order model. Although advances in solving such equations have been made, it is still the most expensive step of this reduction method. Parametric model order reduction aims to determine reduced-order models for parameter-dependent systems. Popular techniques for parametric model order reduction rely on interpolation. Nevertheless, the interpolation of Gramians is rarely mentioned, most probably due to the fact that Gramians are symmetri…
Stable High-Order Interpolation on the Grassmann Manifold by Maximum-Volume Coordinates and Arnoldi Orthogonalization
School of Mathematics and Physics, Xi’an Jiaotong-Liverpool University（西交利物浦大学数理学院）；Department of Mathematical Sciences, University of Liverpool（利物浦大学数学科学系）
One approach to parametric and adaptive model reduction is via the interpolation of orthogonal bases, subspaces or positive definite system matrices. In all these cases, the sampled inputs stem from matrix sets that feature a geometric structure and thus form so-called matrix manifolds. This work will be featured as a chapter in the upcoming Handbook on Model Order Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the numerical treatment of the most important matrix manifolds that arise in the context of…
Factorization of the Gaussian RBF kernel is developed for free-mesh interpolation in the flat, polynomial limit corresponding to Taylor expansion and the Vandermonde basis of geometric moments. With this spectral approximation, a top-down octree-scoping of an interpolant is found by recursively decomposing the residual, similar to the work of Driscoll and Heryudono (2007), except that in the current approach the grid is decoupled from the low rank approximation, allowing partial separation of sampling errors (the mesh) from representation errors (the polynomial order). Then, it is possible to …
A geometric numerical method for simulating suspensions of spherical and non-spherical particles with Stokes drag is proposed. The method combines divergence-free matrix-valued radial basis function interpolation of the fluid velocity field with a splitting method integrator that preserves the sum of the Lyapunov spectrum while mimicking the centrifuge effect of the exact solution. We discuss how breaking the divergence-free condition in the interpolation step can erroneously affect how the volume of the particulate phase evolves under numerical methods. The methods are tested on suspensions o…
