Sökning: id:"swepub:oai:DiVA.org:umu-203280" >
Dyadic diagonalizat...
Dyadic diagonalization of positive definite band matrices and efficient B-spline orthogonalization
-
- Liu, Xijia (författare)
- Umeå University,Umeå universitet,Statistik
-
- Nassar, Hiba (författare)
- Cognitive Systems, Department of Applied Mathematics and Computer Science, Technical University of Denmark, Denmark
-
- Podgórski, Krzysztof (författare)
- Lund University,Lunds universitet,Statistiska institutionen,Ekonomihögskolan,Department of Statistics,Lund University School of Economics and Management, LUSEM
-
(creator_code:org_t)
- Elsevier, 2022
- 2022
- Engelska.
-
Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 414
- Relaterad länk:
-
https://doi.org/10.1...
-
visa fler...
-
https://umu.diva-por... (primary) (Raw object)
-
http://dx.doi.org/10... (free)
-
https://urn.kb.se/re...
-
https://doi.org/10.1...
-
https://lup.lub.lu.s...
-
visa färre...
Abstract
Ämnesord
Stäng
- 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.
Ämnesord
- NATURVETENSKAP -- Matematik -- Beräkningsmatematik (hsv//swe)
- NATURAL SCIENCES -- Mathematics -- Computational Mathematics (hsv//eng)
- TEKNIK OCH TEKNOLOGIER -- Elektroteknik och elektronik -- Reglerteknik (hsv//swe)
- ENGINEERING AND TECHNOLOGY -- Electrical Engineering, Electronic Engineering, Information Engineering -- Control Engineering (hsv//eng)
Nyckelord
- B-splines
- Band matrices
- Dyadic structure
- Matrix inversion
- Orthogonalization
- B-splines
- Band matrices
- Dyadic structure
- Matrix inversion
- Orthogonalization
Publikations- och innehållstyp
- ref (ämneskategori)
- art (ämneskategori)
Hitta via bibliotek
Till lärosätets databas