Hierarchical preconditioning for the stochastic Galerkin method: Upper bounds to the strengthened CBS constants

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F16%3A10334597" target="_blank" >RIV/00216208:11320/16:10334597 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/68407700:21110/16:00235797

Výsledek na webu

<a href="http://dx.doi.org/10.1016/j.camwa.2016.01.006" target="_blank" >http://dx.doi.org/10.1016/j.camwa.2016.01.006</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.camwa.2016.01.006" target="_blank" >10.1016/j.camwa.2016.01.006</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Hierarchical preconditioning for the stochastic Galerkin method: Upper bounds to the strengthened CBS constants

Popis výsledku v původním jazyce

One of the popular methods for numerical solution of partial differential equations with uncertain data is the stochastic Galerkin method, where function spaces used for discretized problems are tensor products of finite element spaces of spatial variables and of sets of polynomials of random variables. Related systems of linear equations are thus usually huge. Studying relations between subspaces of the solution spaces is important for obtaining efficient preconditioning. For a hierarchy of polynomials of random variables we introduce upper bounds to the strengthened Cauchy-Bunyakowsky-Schwarz (CBS) constants with respect to the scalar product defined by the operator of the weak form of the underlying equation. Small values of the CBS constant indicate that certain additive or multiplicative two-by-two block preconditioners reduce enough the condition number of the system. Moreover, we show that a recursive multiplicative two-by-two block preconditioning can be used, resulting in the algebraic multilevel iterative (AMLI) method. We present the conditions under which the AMLI preconditioning is of an optimal order. Numerical experiments confirm the introduced theoretical estimates.

Název v anglickém jazyce

Hierarchical preconditioning for the stochastic Galerkin method: Upper bounds to the strengthened CBS constants

Popis výsledku anglicky

One of the popular methods for numerical solution of partial differential equations with uncertain data is the stochastic Galerkin method, where function spaces used for discretized problems are tensor products of finite element spaces of spatial variables and of sets of polynomials of random variables. Related systems of linear equations are thus usually huge. Studying relations between subspaces of the solution spaces is important for obtaining efficient preconditioning. For a hierarchy of polynomials of random variables we introduce upper bounds to the strengthened Cauchy-Bunyakowsky-Schwarz (CBS) constants with respect to the scalar product defined by the operator of the weak form of the underlying equation. Small values of the CBS constant indicate that certain additive or multiplicative two-by-two block preconditioners reduce enough the condition number of the system. Moreover, we show that a recursive multiplicative two-by-two block preconditioning can be used, resulting in the algebraic multilevel iterative (AMLI) method. We present the conditions under which the AMLI preconditioning is of an optimal order. Numerical experiments confirm the introduced theoretical estimates.

Druh

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

CEP obor

BA - Obecná matematika

OECD FORD obor

—

Projekt

<a href="/cs/project/LL1202" target="_blank" >LL1202: Materiály s implicitními konstitutivními vztahy: Od teorie přes redukci modelů k efektivním numerickým metodám</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)<br>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

Computers and Mathematics with Applications

ISSN

0898-1221

e-ISSN

—

Svazek periodika

71

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

16

Strana od-do

949-964

Kód UT WoS článku

000371940000005

EID výsledku v databázi Scopus

2-s2.0-84956860005