We consider a stand ard elliptic PDE model with uncertain coefficients . Such models are simple\, but are well understood theoretically and so serve as a canonical class o f problems on which to compare different numerical schemes (computer models).

Approximation s which take the form of polynomial chaos (PC) exp ansions have been widely used in applied mathemati cs and can be used as surrogate models in UQ studi es. When the coefficients of the approximation are computed using a Galerkin method\, we use the ter m &lsquo\;Stochastic Galerkin approximation&rsquo\ ;. In statistics\, the term &lsquo\;intrusive PC a pproximation&rsquo\; is also often used. In the Ga lerkin approach\, the resulting PC approximation i s optimal in that the energy norm of the error bet ween the true model solution and the PC approximat ion is minimised. This talk will focus on how to b uild the approximation space (in a computer code) in a computationally efficient way while also guar anteeing accuracy.

In the stochastic Galerkin finite element (SGFEM) approach\, an app roximation is sought in a space which is defined t hrough a chosen set of spatial finite element basi s functions and a set of orthogonal polynomials in the parameters that define the uncertain PDE coef ficients. When the number of parameters is too hig h\, the dimension of this space becomes unmanageab le. One remedy is to use &lsquo\;adaptivity&rsquo\ ;. First\, we generate an approximation in a low-d imensional approximation space (which is cheap) an d then use a computable a posteriori error estimat or to decide whether the current approximation is accurate enough or not. If not\, we enrich the app roximation space\, estimate the error again\, and so on\, until the final approximation is accurate enough. This allows us to design problem-specific polynomial approximations. We describe an error es timation procedure\, outline the computational cos ts\, and illustrate its use through numerical resu lts. An improved multilevel implem entation will b e outlined in a poster given by Adam Crowder. LOCATION:Seminar Room 1\, Newton Institute CONTACT:INI IT END:VEVENT END:VCALENDAR