Optimal Point Sets for Total Degree Polynomial Interpolation in Moderate Dimensions

Aretha L Teckentrup; Max Gunzburger

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Optimal Point Sets for Total Degree Polynomial Interpolation in Moderate Dimensions

Max Gunzburger, Aretha L Teckentrup

Numerical Analysis · Submitted 11 Jul 2014

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We perform a numerical study of the growth of the minimal Lebesgue constant with respect to the degree of the polynomials and the dimension, and report the lowest values known as yet of the Lebesgue constant in the unit cube and the unit ball in up to 10 dimensions.

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arXiv ID1407.3291 (version 1)

Pith ID3GVJ3VNR, Pith's permanent record of this paper and its review

CitationAretha L Teckentrup, Max Gunzburger. "Optimal Point Sets for Total Degree Polynomial Interpolation in Moderate Dimensions." arXiv:1407.3291 (2014). https://pith.science/paper/3GVJ3VNR

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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,690 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.14 · states limitation 0.12 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.48 · compares with prior results 0.64 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

This paper is concerned with Lagrange interpolation by total degree polynomials in moderate dimensions. In particular, we are interested in characterising the optimal choice of points for the interpolation problem, where we define the optimal interpolation points as those which minimise the Lebesgue constant. We give a novel algorithm for numerically computing the location of the optimal points, which is independent of the shape of the domain and does not require computations with Vandermonde matrices. We perform a numerical study of the growth of the minimal Lebesgue constant with respect to the degree of the polynomials and the dimension, and report the lowest values known as yet of the Lebesgue constant in the unit cube and the unit ball in up to 10 dimensions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	This paper is concerned with Lagrange interpolation by total degree polynomials in moderate dimensions.	background 0.99	0.24	0.03	asserted 0.94
s1	In particular, we are interested in characterising the optimal choice of points for the interpolation problem, where we define the optimal interpolation points as those which minimise the Lebesgue constant.	background 0.83	0.31	0.10	asserted 0.96
s2	We give a novel algorithm for numerically computing the location of the optimal points, which is independent of the shape of the domain and does not require computations with Vandermonde matrices.	method 0.67	0.95	0.15	asserted 1.00
s3	We perform a numerical study of the growth of the minimal Lebesgue constant with respect to the degree of the polynomials and the dimension, and report the lowest values known as yet of the Lebesgue constant in the unit cube and the unit ball in up to 10 dimensions.	result 0.87	0.94	0.54	asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.99 · evidence from simulation or computation 1.00.

What was examined, and what was not

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arXiv listing abstract v1

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full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

This paper is concerned with Lagrange interpolation by total degree polynomials in moderate dimensions. In particular, we are interested in characterising the optimal choice of points for the interpolation problem, where we define the optimal interpolation points as those which minimise the Lebesgue constant. We give a novel algorithm for numerically computing the location of the optimal points, which is independent of the shape of the domain and does not require computations with Vandermonde matrices. We perform a numerical study of the growth of the minimal Lebesgue constant with respect to the degree of the polynomials and the dimension, and report the lowest values known as yet of the Lebesgue constant in the unit cube and the unit ball in up to 10 dimensions.

Human review

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How the literature uses it

5 arXiv papers cite this in 8 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 3 sentences without a confident label

Uses its method or tool 1 paper

We have used both the ideas from [ref] and the direct evaluation of [ref] in a fine grid (definitely, a much slower procedure).

Optimal sampling patterns for Zernike polynomials · Experimental results · 0.98

Compares with it 1 paper

The Lebesgue numbers corresponding to the OCS interpolation nodes are also larger but comparable to those reported in [ref] (see Table 3 and Figure 8 therein).

Optimal sampling patterns for Zernike polynomials · Experimental results · 0.99

Cites it as background 2 papers

The interpolation and approximation of functions in this setting using adaptive function sampling have been studied in [ref] , including bounds on the Lebesgue constant for different basis sets in the multidimensional cases [ref] .

Function approximation using gradient information with application to parametric and stochastic differential equations · Related works · 0.94
These problems have been treated in many papers and monographs, see, e.g., [ref] .

Optimal Lagrange Interpolation Projectors and Legendre Polynomials · Introduction · 1.00

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