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Algebraic methods to study the dimension of supersmooth spline spaces

DeepeshToshniwal, Nelly Villamizar Orcid Logo

Advances in Applied Mathematics, Volume: 142

Swansea University Author: Nelly Villamizar Orcid Logo

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DOI (Published version): 10.1016/j.aam.2022.102412

Abstract

Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation theory, and computer aided geometric design. In this paper we...

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Published in: Advances in Applied Mathematics
ISSN: 0196-8858
Published: Elsevier 2023
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URI: https://cronfa.swan.ac.uk/Record/cronfa60699
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spelling 2022-09-07T14:23:13.9664884 v2 60699 2022-08-02 Algebraic methods to study the dimension of supersmooth spline spaces 41572bcee47da6ba274ecd1828fbfef4 0000-0002-8741-7225 Nelly Villamizar Nelly Villamizar true false 2022-08-02 SMA Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation theory, and computer aided geometric design. In this paper we address various challenges arising in the study of splines with enhanced mixed (super-)smoothness conditions at the vertices and across interior faces of the partition. Such supersmoothness can be imposed but can also appear unexpectedly on certain splines depending on the geometry of the underlying polyhedral partition. Using algebraic tools, a generalization of the Billera–Schenck–Stillman complex that includes the effect of additional smoothness constraints leads to a construction which requires the analysis of ideals generated by products of powers of linear forms in several variables. Specializing to the case of planar triangulations, a combinatorial lower bound on the dimension of splines with supersmoothness at the vertices is presented, and we also show that this lower bound gives the exact dimension in high degree. The methods are further illustrated with several examples. Journal Article Advances in Applied Mathematics 142 Elsevier 0196-8858 Spline functions; Superspline spaces on triangulations; Dimension of spline spaces; Supersmoothness; Intrisic supersmoothness 1 1 2023 2023-01-01 10.1016/j.aam.2022.102412 COLLEGE NANME Mathematics COLLEGE CODE SMA Swansea University SU Library paid the OA fee (TA Institutional Deal) EPSRC (New Investigator Award/EP/V012835/1); Dutch Research Council (Veni research programme/212.150) 2022-09-07T14:23:13.9664884 2022-08-02T10:01:06.9569063 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics DeepeshToshniwal 1 Nelly Villamizar 0000-0002-8741-7225 2 60699__25089__25b688480d5d4887a980159a78b5f904.pdf 60699_VoR.pdf 2022-09-07T14:20:29.8806100 Output 652923 application/pdf Version of Record true © 2022 The Author(s). This is an open access article under the CC-BY license true eng http://creativecommons.org/licenses/by/4.0/
title Algebraic methods to study the dimension of supersmooth spline spaces
spellingShingle Algebraic methods to study the dimension of supersmooth spline spaces
Nelly Villamizar
title_short Algebraic methods to study the dimension of supersmooth spline spaces
title_full Algebraic methods to study the dimension of supersmooth spline spaces
title_fullStr Algebraic methods to study the dimension of supersmooth spline spaces
title_full_unstemmed Algebraic methods to study the dimension of supersmooth spline spaces
title_sort Algebraic methods to study the dimension of supersmooth spline spaces
author_id_str_mv 41572bcee47da6ba274ecd1828fbfef4
author_id_fullname_str_mv 41572bcee47da6ba274ecd1828fbfef4_***_Nelly Villamizar
author Nelly Villamizar
author2 DeepeshToshniwal
Nelly Villamizar
format Journal article
container_title Advances in Applied Mathematics
container_volume 142
publishDate 2023
institution Swansea University
issn 0196-8858
doi_str_mv 10.1016/j.aam.2022.102412
publisher Elsevier
college_str Faculty of Science and Engineering
hierarchytype
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 Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation theory, and computer aided geometric design. In this paper we address various challenges arising in the study of splines with enhanced mixed (super-)smoothness conditions at the vertices and across interior faces of the partition. Such supersmoothness can be imposed but can also appear unexpectedly on certain splines depending on the geometry of the underlying polyhedral partition. Using algebraic tools, a generalization of the Billera–Schenck–Stillman complex that includes the effect of additional smoothness constraints leads to a construction which requires the analysis of ideals generated by products of powers of linear forms in several variables. Specializing to the case of planar triangulations, a combinatorial lower bound on the dimension of splines with supersmoothness at the vertices is presented, and we also show that this lower bound gives the exact dimension in high degree. The methods are further illustrated with several examples.
published_date 2023-01-01T04:19:01Z
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score 11.035634

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