On a Sparse Representation of an n-Dimensional Laplacian in Wavelet Coordinates

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F46747885%3A24510%2F16%3A00003742" target="_blank" >RIV/46747885:24510/16:00003742 - isvavai.cz</a>

Výsledek na webu

<a href="http://link.springer.com/article/10.1007/s00025-015-0488-5" target="_blank" >http://link.springer.com/article/10.1007/s00025-015-0488-5</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00025-015-0488-5" target="_blank" >10.1007/s00025-015-0488-5</a>

Jazyk výsledku

angličtina

Název v původním jazyce

On a Sparse Representation of an n-Dimensional Laplacian in Wavelet Coordinates

Popis výsledku v původním jazyce

Important parts of adaptive wavelet methods are well-conditioned wavelet stiffness matrices and an efficient approximate multiplication of quasi-sparse stiffness matrices with vectors in wavelet coordinates. Therefore it is useful to develop a well-conditioned wavelet basis with respect to which both the mass and stiffness matrices are sparse in the sense that the number of nonzero elements in each column is bounded by a constant. Consequently, the stiffness matrix corresponding to the n-dimensional Laplacian in the tensor product wavelet basis is also sparse. Then a matrix-vector multiplication can be performed exactly with linear complexity. In this paper, we construct a wavelet basis based on Hermite cubic splines with respect to which both the mass matrix and the stiffness matrix corresponding to a one-dimensional Poisson equation are sparse. Moreover, a proposed basis is well-conditioned on low decomposition levels. Small condition numbers for low decomposition levels and a sparse structure of stiffness matrices are kept for any well-conditioned second order partial differential equations with constant coefficients; furthermore, they are independent of the space dimension.

Název v anglickém jazyce

On a Sparse Representation of an n-Dimensional Laplacian in Wavelet Coordinates

Popis výsledku anglicky

Important parts of adaptive wavelet methods are well-conditioned wavelet stiffness matrices and an efficient approximate multiplication of quasi-sparse stiffness matrices with vectors in wavelet coordinates. Therefore it is useful to develop a well-conditioned wavelet basis with respect to which both the mass and stiffness matrices are sparse in the sense that the number of nonzero elements in each column is bounded by a constant. Consequently, the stiffness matrix corresponding to the n-dimensional Laplacian in the tensor product wavelet basis is also sparse. Then a matrix-vector multiplication can be performed exactly with linear complexity. In this paper, we construct a wavelet basis based on Hermite cubic splines with respect to which both the mass matrix and the stiffness matrix corresponding to a one-dimensional Poisson equation are sparse. Moreover, a proposed basis is well-conditioned on low decomposition levels. Small condition numbers for low decomposition levels and a sparse structure of stiffness matrices are kept for any well-conditioned second order partial differential equations with constant coefficients; furthermore, they are independent of the space dimension.

Druh

J_x - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)

CEP obor

BA - Obecná matematika

OECD FORD obor

—

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2016

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Results in Mathematics

ISSN

1422-6383

e-ISSN

—

Svazek periodika

69

Číslo periodika v rámci svazku

1-2

Stát vydavatele periodika

CH - Švýcarská konfederace

Počet stran výsledku

19

Strana od-do

225-243

Kód UT WoS článku

000376102900012

EID výsledku v databázi Scopus

2-s2.0-84955634906