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A lower bound for the dimension of tetrahedral splines in large degree

Michael DiPasquale, Nelly Villamizar Orcid Logo

Constructive Approximation, Volume: 59, Issue: 1, Pages: 1 - 30

Swansea University Author: Nelly Villamizar Orcid Logo

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DOI (Published version): 10.1007/s00365-023-09625-5

Abstract

Splines are piecewise polynomial functions which are continuously differentiable to some order r. For a fixed integer d the space of splines of degree at most d is a finite dimensional vector space, and a largely open problem in numerical analysis is to determine its dimension. While considerable at...

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Published in: Constructive Approximation
ISSN: 0176-4276 1432-0940
Published: Springer Science and Business Media LLC 2024
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URI: https://cronfa.swan.ac.uk/Record/cronfa63205
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spelling 2024-06-06T12:40:26.3550973 v2 63205 2023-04-19 A lower bound for the dimension of tetrahedral splines in large degree 41572bcee47da6ba274ecd1828fbfef4 0000-0002-8741-7225 Nelly Villamizar Nelly Villamizar true false 2023-04-19 MACS Splines are piecewise polynomial functions which are continuously differentiable to some order r. For a fixed integer d the space of splines of degree at most d is a finite dimensional vector space, and a largely open problem in numerical analysis is to determine its dimension. While considerable attention has been given to this problem in the bivariate setting, the literature on trivariate splines is less conclusive. In particular, the dimension of generic trivariate splines is not known even in large degree when r>1. In this paper we use a bound we previously derived for splines on vertex stars to compute a new lower bound on the dimension of trivariate splines in large enough degree. We illustrate in several examples that our formula gives the exact dimension of the spline space in large enough degree if vertex positions are generic. In contrast, for splines continuously differentiable of order r>1, every lower bound in the literature diverges (often significantly) in large degree from the dimension of the spline space in these examples. We derive the bound using commutative and homological algebra. Journal Article Constructive Approximation 59 1 1 30 Springer Science and Business Media LLC 0176-4276 1432-0940 Trivariate spline spaces, tetrahedral partitions, dimension of spline spaces. 1 2 2024 2024-02-01 10.1007/s00365-023-09625-5 COLLEGE NANME Mathematics and Computer Science School COLLEGE CODE MACS Swansea University SU Library paid the OA fee (TA Institutional Deal) Swansea University. EPSRC (EP/V012835/1) 2024-06-06T12:40:26.3550973 2023-04-19T14:42:34.3017512 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Michael DiPasquale 1 Nelly Villamizar 0000-0002-8741-7225 2 63205__27909__2028ae8b97394672b8348632ea125d6e.pdf 63205.pdf 2023-06-21T12:23:25.7417908 Output 703184 application/pdf Version of Record true © The Author(s) 2023. Distributed under the terms of a Creative Commons Attribution 4.0 License (CC BY 4.0). true eng https://creativecommons.org/licenses/by/4.0/
title A lower bound for the dimension of tetrahedral splines in large degree
spellingShingle A lower bound for the dimension of tetrahedral splines in large degree
Nelly Villamizar
title_short A lower bound for the dimension of tetrahedral splines in large degree
title_full A lower bound for the dimension of tetrahedral splines in large degree
title_fullStr A lower bound for the dimension of tetrahedral splines in large degree
title_full_unstemmed A lower bound for the dimension of tetrahedral splines in large degree
title_sort A lower bound for the dimension of tetrahedral splines in large degree
author_id_str_mv 41572bcee47da6ba274ecd1828fbfef4
author_id_fullname_str_mv 41572bcee47da6ba274ecd1828fbfef4_***_Nelly Villamizar
author Nelly Villamizar
author2 Michael DiPasquale
Nelly Villamizar
format Journal article
container_title Constructive Approximation
container_volume 59
container_issue 1
container_start_page 1
publishDate 2024
institution Swansea University
issn 0176-4276
1432-0940
doi_str_mv 10.1007/s00365-023-09625-5
publisher Springer Science and Business Media LLC
college_str Faculty of Science and Engineering
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hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description Splines are piecewise polynomial functions which are continuously differentiable to some order r. For a fixed integer d the space of splines of degree at most d is a finite dimensional vector space, and a largely open problem in numerical analysis is to determine its dimension. While considerable attention has been given to this problem in the bivariate setting, the literature on trivariate splines is less conclusive. In particular, the dimension of generic trivariate splines is not known even in large degree when r>1. In this paper we use a bound we previously derived for splines on vertex stars to compute a new lower bound on the dimension of trivariate splines in large enough degree. We illustrate in several examples that our formula gives the exact dimension of the spline space in large enough degree if vertex positions are generic. In contrast, for splines continuously differentiable of order r>1, every lower bound in the literature diverges (often significantly) in large degree from the dimension of the spline space in these examples. We derive the bound using commutative and homological algebra.
published_date 2024-02-01T02:38:47Z
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score 11.04748

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