On the general theory of data interpolation

XXIII Congreso de Métodos Numéricos y sus Aplicaciones (ENIEF) (La Plata, noviembre 2017), 2017

Nested cartesian grid systems by design require interpolation of solution fields from coarser

2018 AIAA Aerospace Sciences Meeting, 2018

Nested cartesian grid systems by design require interpolation of solution fields from coarser

Cambridge University Press eBooks, 2019

The basic geological description of a petroleum reservoir or an aquifer system will typically consist of two sets of surfaces. Geological horizons are lateral surfaces describing the bedding planes that delimit the rock strata, whereas faults are vertical or inclined surfaces along which the strata may have been displaced by geological processes. In this chapter, we discuss how to turn the basic geological description into a discrete model that can be used to formulate various computational methods, e.g., for solving the equations that describe fluid flow. A grid is a tessellation of a planar or volumetric object by a set of contiguous simple shapes referred to as cells. Grids can be described and distinguished by their geometry, which gives the shape of the cells that form the grid, and their topology that tells how the individual cells are connected. In 2D, a cell is in general a closed polygon for which the geometry is defined by a set of vertices and a set of edges that connect pairs of vertices and define the interface between two neighboring cells. In 3D, a cell is a closed polyhedron for which the geometry is defined by a set of vertices, a set of edges that connect pairs of vertices, and a set of faces (surfaces delimited by a subset of the edges) that define the interface between two different cells; see Figure 3.1. Herein, we assume that all cells are nonoverlapping, so that each point in the planar/volumetric object represented by the grid is either inside a single cell, lies on an interface or edge, or is a vertex. Two cells that share a common face are said to be connected. Likewise, one can also define connections based on edges and vertices. The topology of a grid is defined by the total set of connections, which is sometimes also called the connectivity of the grid. When implementing grids in modeling software, one always has the choice between generality and efficiency. To represent an arbitrary grid, it is necessary to explicitly store the geometry of each cell in terms of vertices, edges, and faces, as well as storing the connectivity among cells, faces, edges, and vertices. However, as we will see later, huge simplifications can be made for particular classes of grids by exploiting regularity in the geometry and structures in the topology. Consider, for instance, a planar grid consisting of rectangular cells of equal size. Here, the topology can be represented by two indices, and one only needs to specify a reference point and the two side lengths of the rectangle to describe the geometry. This way, one ensures minimal memory usage and optimal efficiency when accessing the grid. On the other hand, exploiting the simplified description 55

Applications of Mathematics

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz 38 (1993)

arXiv (Cornell University), 2023

In this work, we study the Hermite interpolation on n-dimensional non-equally spaced, rectilinear grids over a field k of characteristic zero, given the values of the function at each point of the grid and the partial derivatives up to a maximum degree. First, we prove the uniqueness of the interpolating polynomial, and we further obtain a compact closed form that uses a single summation, irrespective of the dimensionality, which is algebraically simpler than the only alternative closed form for the n-dimensional classical Hermite interpolation [1]. We provide the remainder of the interpolation in integral form; we derive the ideal of the interpolation and express the interpolation remainder using only polynomial divisions, in the case of interpolating a polynomial function. Moreover, we prove the continuity of Hermite polynomials defined on adjacent n−dimensional grids, thus establishing spline behavior. Finally, we perform illustrative numerical examples to showcase the applicability and high accuracy of the proposed interpolant, in the simple case of few points, as well as hundreds of points on 3D-grids using a spline-like interpolation, which compares favorably to state-of-the-art spline interpolation methods.

Numerische Mathematik, 1996

Dedicated to Jean Descloux on his 60 th birthday Summary. In adapting a grid for a Computational Fluid Dynamics problem one uses a mapping from the unit square onto itself that is the solution of an elliptic partial differential equation with rapidly varying coefficients. For a regular discretization this mapping has to be invertible. We will show that such result holds for general elliptic operators (in two dimensions). The Carleman-Hartman-Wintner Theorem will be fundamental in our proof. We will also explain why such a general result cannot be expected to hold for the (three-dimensional) cube.

2006

Solving discrete boundary value problems with the help of an appropriate multigrid method necessitates the construction of a sequence of coarse grids with corresponding coarse grid approximations for the given fine grid discretization. Popular choices in this context are the Galerkin coarse grid approximation (GCA) and the use of the same discretization on the coarser grids as on the fine grid with properly adjusted mesh sizes (DCA). In this paper we propose an alternative strategy to select the required coarse grid discretizations within a multigrid solution method. It can be applied to fine grid operators that can be locally represented by a stencil. The coarse grid approximations are constructed by minimizing a certain low-frequency L 2 -norm. More precisely, a coarse grid discretization is chosen in such a way, that its Fourier symbol is a best approximation (w.r.t. low frequencies) of the Fourier symbol of the fine grid operator. This strategy is abbreviated by FCA since the design of coarse grid approximations is based on local Fourier analysis . The entries of the coarse grid stencils are simply given by linear combinations of the fine grid entries. As a consequence, FCA can be considered as a black-box method to construct coarse grid operators. This method has been successfully applied to (anisotropic) diffusion equations, operators with mixed derivatives, problems with dominant convection, and operators involving jumping coefficients. For nicely elliptic examples, FCA resembles the DCA approach, whereas for more difficult applications (w.r.t. an efficient multigrid treatment) it behaves similarly to GCA based on operator-dependent transfers.

Applied Numerical Mathematics, 2010

Radial basis functions are popular for interpolation on a scattered or irregular grid. However, theory for an irregular grid is mostly limited to proofs of convergence. Here, we present theory and numerical experiments for two specific cases. The first is an otherwise uniform grid of spacing h in which one point is shifted by an amount sh. The second is a uniform grid with one point omitted. We discuss Gaussian, hyperbolic secant, and inverse quadratic RBFs.

Applied Mathematics Letters, 2013

We show that Hermite functions are successful for interpolation on a finite interval, even on a uniform grid where polynomial interpolation fails. The Runge Phenomenon is not completely abolished, but is greatly diminished. Finite interval Hermite interpolation is ill-conditioned and therefore limited to a maximum of about 250 interpolation points on a uniform univariate grid, but it is still far superior to the approach using Gaussian radial basis functions (RBFs). The Hermite functions are a complete spectral basis for the infinite interval; the motivation for employing them here derives from a careful study that showed, for small RBF shape parameter α and small number of points N, that Gaussian RBF cardinal functions can be accurately approximated by the product of a Gaussian with the usual polynomial cardinal function. Direct comparisons show that when N and α are not both small, Hermite interpolation is greatly superior in accuracy, condition number and efficiency to the RBF methods that inspired it.