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To be or not to be intrusive? The solution of parametric and stochastic equations – the “plain vanilla” Galerkin case. (English) Zbl 1310.65132

Summary: 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).

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65J15 Numerical solutions to equations with nonlinear operators
60H25 Random operators and equations (aspects of stochastic analysis)
65C30 Numerical solutions to stochastic differential and integral equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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