TY - JOUR

T1 - To Be or Not to Be Intrusive? The Solution of Parametric and Stochastic Equations---the “Plain Vanilla” Galerkin Case

AU - Giraldi, Loïc

AU - Litvinenko, Alexander

AU - Liu, Dishi

AU - Matthies, Hermann G.

AU - Nouy, Anthony

N1 - KAUST Repository Item: Exported on 2020-10-01

PY - 2014/1

Y1 - 2014/1

N2 - In parametric equations---stochastic equations are a special case---one may want to approximate the solution such that it is easy to evaluate its dependence on the parameters. Interpolation in the parameters is an obvious possibility---in this context often labeled as a collocation method. In the frequent situation where one has a “solver” for a given fixed parameter value, this may be used “nonintrusively” as a black-box component to compute the solution at all the interpolation points independently of each other. By extension, all other methods, and especially simple Galerkin methods, which produce some kind of coupled system, are often classed as “intrusive.” We show how, for such “plain vanilla” Galerkin formulations, one may solve the coupled system in a nonintrusive way, and even the simplest form of block-solver has comparable efficiency. This opens at least two avenues for possible speed-up: first, to benefit from the coupling in the iteration by using more sophisticated block-solvers and, second, the possibility of nonintrusive successive rank-one updates as in the proper generalized decomposition (PGD).

AB - In parametric equations---stochastic equations are a special case---one may want to approximate the solution such that it is easy to evaluate its dependence on the parameters. Interpolation in the parameters is an obvious possibility---in this context often labeled as a collocation method. In the frequent situation where one has a “solver” for a given fixed parameter value, this may be used “nonintrusively” as a black-box component to compute the solution at all the interpolation points independently of each other. By extension, all other methods, and especially simple Galerkin methods, which produce some kind of coupled system, are often classed as “intrusive.” We show how, for such “plain vanilla” Galerkin formulations, one may solve the coupled system in a nonintrusive way, and even the simplest form of block-solver has comparable efficiency. This opens at least two avenues for possible speed-up: first, to benefit from the coupling in the iteration by using more sophisticated block-solvers and, second, the possibility of nonintrusive successive rank-one updates as in the proper generalized decomposition (PGD).

UR - http://hdl.handle.net/10754/555669

UR - http://epubs.siam.org/doi/abs/10.1137/130942802

UR - http://www.scopus.com/inward/record.url?scp=84919666940&partnerID=8YFLogxK

U2 - 10.1137/130942802

DO - 10.1137/130942802

M3 - Article

VL - 36

SP - A2720-A2744

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

SN - 1064-8275

IS - 6

ER -