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About 1,736 papers for “Multivariate interpolation”

Multivariate interpolation

Igor Aleksandrovich Pakhnutov · Interactive science · 2017

The paper deals with iterative interpolation methods in forms of similar recursive procedures defined by a sort of simple functions (interpolation basis) not necessarily real valued. These basic functions are kind of arbitrary type being defined just by wish and considerations of user. The studied interpolant construction shows virtue of versatility: it may be used in a wide range of vector spaces endowed with scalar product, no dimension restrictions, both in Euclidean and Hilbert spaces. The choice of basic interpolation functions is as wide as possible since it is subdued nonessential restrictions. The interpolation method considered in particular coincides with traditional polynomial interpolation (mimic of Lagrange method in real unidimensional case) or rational, exponential etc. in other cases. The interpolation as iterative process, in fact, is fairly flexible and allows one procedure to change the type of interpolation, depending on the node number in a given set. Linear interpolation basis options (perhaps some nonlinear ones) allow to interpolate in noncommutative spaces, such as spaces of nondegenerate matrices, interpolated data can also be relevant elements of vector spaces over arbitrary numeric field. By way of illustration, the author gives the examples of interpolation on the real plane, in the separable Hilbert space and the space of square matrices with vektorvalued source data.

Cited by 6Open access

Multivariate Perpendicular Interpolation

Peter Alfeld · SIAM Journal on Numerical Analysis · 1985

An approach to multivariate interpolation is described. The algorithm is applicable in arbitrary dimensions, and can generate surfaces of arbitrary smoothness. This is accomplished by tesselating the (polyhedral) domain into simplices and using one-dimensional algorithms to construct interpolants first on edges and then successively on higher order faces by blending methods. The result is a piecewise rational function of a high degree which has the prescribed global smoothness and interpolates to the original data. The interpolants are local, i.e. their evaluation at a point requires only data on the simplex that the point resides in. The schemes require data of the same degree as the degree of global smoothness. The degree of polynomial precision is greater than or equal to the degree of smoothness. The approach derives its power and simplicity from the fact that derivatives in directions perpendicularly across faces are incorporated directly as data.

Cited by 19

Algebraic multivariate interpolation

Giménez Rodilla, José · Dipòsit Digital de la Universitat de Barcelona (Universitat de Barcelona) · 2022

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Carlos D'Andrea

Cited by 0

Generalized recursive multivariate interpolation

Earl McKinney · Mathematics of Computation · 1972

A generalized recursive interpolation technique for a set of linear functionals over a set of general univariate basis functions has been previously developed. This paper extends these results to restricted multivariate interpolation over a set of general multivariate basis functions. When the data array is a suitable configuration (e.g., an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional simplex), minimal degree multivariate interpolating polynomials are produced by this recursive interpolation scheme. By using product rules, recursive univariate interpolation applied to each variable singly produces multivariate interpolating polynomials (not of minimal degree) when the data are arranged in a hyper-rectangular array. By proper ordering of points in a data array, multivariate polynomial interpolation is accomplished over other arrays such as diamonds and truncated diamonds in two dimensions and their counterparts in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> dimensions.

Cited by 5Open access

Generalized Recursive Multivariate Interpolation

Earl McKinney · Mathematics of Computation · 1972

A generalized recursive interpolation technique for a set of linear functionals over a set of general univariate basis functions has been previously developed. This paper extends these results to restricted multivariate interpolation over a set of general multivariate basis functions. When the data array is a suitable configuration (e.g., an $n$-dimensional simplex), minimal degree multivariate interpolating polynomials are produced by this recursive interpolation scheme. By using product rules, recursive univariate interpolation applied to each variable singly produces multivariate interpolating polynomials (not of minimal degree) when the data are arranged in a hyper-rectangular array. By proper ordering of points in a data array, multivariate polynomial interpolation is accomplished over other arrays such as diamonds and truncated diamonds in two dimensions and their counterparts in $n$ dimensions.

Cited by 2

Minterpy - multivariate polynomial interpolation

Hernandez Acosta, Uwe, Thekke Veettil, Sachin Krishnan, Wicaksono, Damar Canggih et al. · RODARE · 2025

<code>minterpy</code> is an open-source Python package for a multivariate generalization of the classical Newton and Lagrange interpolation schemes as well as related tasks. It is based on an optimized re-implementation of the multivariate interpolation prototype algorithm (<em>MIP</em>) by Hecht et al.1 and thereby provides software solutions that lift the curse of dimensionality from interpolation tasks. While interpolation occurs as the bottleneck of most computational challenges, <code>minterpy</code> aims to free empirical sciences from their computational limitations.

Cited by 0Open access

Minterpy - multivariate polynomial interpolation

Hernandez Acosta, Uwe, Wicaksono, Damar Canggih, Thekke Veettil, Sachin Krishnan et al. · RODARE · 2024

<code>minterpy</code> is an open-source Python package for a multivariate generalization of the classical Newton and Lagrange interpolation schemes as well as related tasks. It is based on an optimized re-implementation of the multivariate interpolation prototype algorithm (<em>MIP</em>) by Hecht et al.1 and thereby provides software solutions that lift the curse of dimensionality from interpolation tasks. While interpolation occurs as the bottleneck of most computational challenges, <code>minterpy</code> aims to free empirical sciences from their computational limitations.

Cited by 0Open access

Interpolation by multivariate splines

Charles K. Chui, Harvey Diamond, Louise A. Raphael · Mathematics of Computation · 1988

A general interpolation scheme by multivariate splines at regular sample points is introduced. This scheme guarantees the local optimal order of approximation to sufficiently smooth data functions. A discussion on numerical implementation is included.

Cited by 21

Minterpy - multivariate polynomial interpolation

Hernandez Acosta, Uwe, Thekke Veettil, Sachin Krishnan, Wicaksono, Damar Canggih et al. · RODARE · 2021

<code>minterpy</code> is an open-source Python package for a multivariate generalization of the classical Newton and Lagrange interpolation schemes as well as related tasks. It is based on an optimized re-implementation of the multivariate interpolation prototype algorithm (<em>MIP</em>) by Hecht et al.1 and thereby provides software solutions that lift the curse of dimensionality from interpolation tasks. While interpolation occurs as the bottleneck of most computational challenges, <code>minterpy</code> aims to free empirical sciences from their computational limitations.

Cited by 0Open access

Interpolation by Multivariate Splines

Charles K. Chui, Harvey Diamond, Louise A. Raphael · Mathematics of Computation · 1988

A general interpolation scheme by multivariate splines at regular sample points is introduced.This scheme guarantees the local optimal order of approximation to sufficiently smooth data functions.A discussion on numerical implementation is included. Introduction.In this paper we introduce a very general interpolation scheme by multivariate splines.Based on the quasi-interpolation formulas developed in [3], we show that the interpolating multivariate splines so obtained give the optimal order of approximation to sufficiently smooth functions.Let (j) be a nonnegative locally supported piecewise polynomial function symmetric with respect to the origin, and S the linear span of all the translates (j>(--j), j 6 Zs, of 4>.Hence, S is a multivariate spline space on a certain grid partition A, with certain smoothness joining conditions, and of certain total degree, induced by <>.Denote by 0 the Fourier transform of <>.We assume that (j> is normalized, that is, X>oe = i, j and that S contains ttp, the space of all polynomials in Rs of total degree p.This is equivalent to the assumption that the commutator of <f> is of degree p (cf. [4]), or that 4> satisfies the Strang and Fix condition of degree p (cf. [6]):f CO) = i.O^jeZ*, \a\<p.The equivalence of these conditions is shown in [4] (see also [1]).The interpolation problem we are going to study can be stated as follows: Let K be a compact set in Rs and / C(K).For any h > 0, consider the sample points Ph = {iheK:jeZs} on K.The problem is to find an s/^ in Sh = {s(-/h):seS} such that(2)_ 8f,k(y) = /(y), y G Ph,

Cited by 5Open access