Umeå universitets logga

umu.sePublikationer
Ändra sökning
RefereraExporteraLänk till posten
Permanent länk

Direktlänk
Referera
Referensformat
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Annat format
Fler format
Språk
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Annat språk
Fler språk
Utmatningsformat
  • html
  • text
  • asciidoc
  • rtf
Dyadic diagonalization of positive definite band matrices and efficient B-spline orthogonalization
Umeå universitet, Samhällsvetenskapliga fakulteten, Handelshögskolan vid Umeå universitet, Statistik.ORCID-id: 0000-0002-8143-8500
Cognitive Systems, Department of Applied Mathematics and Computer Science, Technical University of Denmark, Denmark.
Department of Statistics, Lund University, Sweden.
2022 (Engelska)Ingår i: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 414, artikel-id 114444Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

A dyadic algorithm for diagonalizing an arbitrary positive definite band matrix, referred to as a band Gramian, is obtained to efficiently orthogonalize the B-splines. The algorithm can be also used as a fast inversion method for a band Gramian characterized by remarkable sparsity of the diagonalizing matrix. There are two versions of the algorithm: the first one is more efficient and is applicable to a Toeplitz band Gramian while the second one is more general, works with any Gramian matrix, but is more computationally intensive. In the context of the B-splines, these two cases result in new symmetric orthogonalization procedures and correspond to equally and arbitrarily spaced knots, respectively. In the algorithm, the sparsity of a band Gramian is utilized to produce a natural dyadic net of orthogonal splines, rather than a sequence of them. Such a net is thus naturally referred to as a splinet. The splinets exploit “near-orthogonalization” of the B-splines and feature locality expressed through a small size of the total support set and computational efficiency that is a result of a small number of inner product evaluations needed for their construction. These and other efficiencies are formally quantified by upper bounds and asymptotic rates with respect to the number of splines in a splinet. An additional assessment is provided through numerical experiments. They suggest that the theoretical bounds are rather conservative and the method is even more efficient than the bounds indicate. The dyadic net-like structures and the locality bear some resemblance to wavelets but in fact, the splinets are fundamentally different because they do not aim at capturing the resolution scales. The orthogonalization method together with efficient spline algebra and calculus has been implemented in R-package Splinets available on CRAN.

Ort, förlag, år, upplaga, sidor
Elsevier, 2022. Vol. 414, artikel-id 114444
Nyckelord [en]
B-splines, Band matrices, Dyadic structure, Matrix inversion, Orthogonalization
Nationell ämneskategori
Beräkningsmatematik
Identifikatorer
URN: urn:nbn:se:umu:diva-203280DOI: 10.1016/j.cam.2022.114444ISI: 000811831800016Scopus ID: 2-s2.0-85131372657OAI: oai:DiVA.org:umu-203280DiVA, id: diva2:1728206
Forskningsfinansiär
Vetenskapsrådet, 2020-0516Tillgänglig från: 2023-01-18 Skapad: 2023-01-18 Senast uppdaterad: 2023-01-18Bibliografiskt granskad

Open Access i DiVA

fulltext(1589 kB)104 nedladdningar
Filinformation
Filnamn FULLTEXT01.pdfFilstorlek 1589 kBChecksumma SHA-512
0c78a75596dd5f7caf5dd20173815cf000d5004220940f0cecd138d6c30ccee25d0954855d0ee85aa30ebad8848f42361dceea9a69b9563695c40c76e2781fa6
Typ fulltextMimetyp application/pdf

Övriga länkar

Förlagets fulltextScopus

Person

Liu, Xijia

Sök vidare i DiVA

Av författaren/redaktören
Liu, Xijia
Av organisationen
Statistik
I samma tidskrift
Journal of Computational and Applied Mathematics
Beräkningsmatematik

Sök vidare utanför DiVA

GoogleGoogle Scholar
Totalt: 110 nedladdningar
Antalet nedladdningar är summan av nedladdningar för alla fulltexter. Det kan inkludera t.ex tidigare versioner som nu inte längre är tillgängliga.

doi
urn-nbn

Altmetricpoäng

doi
urn-nbn
Totalt: 126 träffar
RefereraExporteraLänk till posten
Permanent länk

Direktlänk
Referera
Referensformat
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Annat format
Fler format
Språk
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Annat språk
Fler språk
Utmatningsformat
  • html
  • text
  • asciidoc
  • rtf