BEGIN:VCALENDAR VERSION:2.0 PRODID:-//Talks.cam//talks.cam.ac.uk// X-WR-CALNAME:Talks.cam BEGIN:VEVENT SUMMARY:Adaptive Stochastic Galerkin Finite Element Approximation for Elli ptic PDEs with Random Coefficients - Catherine Powell (University of Manch ester) DTSTART:20180205T113000Z DTEND:20180205T123000Z UID:TALK99862@talks.cam.ac.uk CONTACT:INI IT DESCRIPTION:Co-author: Adam Crowder (University of Manchester)

We consider a standard elliptic PDE model with uncertai n coefficients. Such models are simple\, but are well understood theoretic ally and so serve as a canonical class of problems on which to compare dif ferent numerical schemes (computer models).

Approximations which take the form of polynomial chaos (PC) expansions have been widely used in applied mathematics and can be used as surrogate models in UQ studies. Wh en the coefficients of the approximation are computed using a Galerkin met hod\, we use the term &lsquo\;Stochastic Galerkin approximation&rsquo\;. I n statistics\, the term &lsquo\;intrusive PC approximation&rsquo\; is also often used. In the Galerkin approach\, the resulting PC approximation is optimal in that the energy norm of the error between the true model soluti on and the PC approximation is minimised. This talk will focus on how to b uild the approximation space (in a computer code) in a computationally eff icient way while also guaranteeing accuracy.

In the stochast ic Galerkin finite element (SGFEM) approach\, an approximation is sought i n a space which is defined through a chosen set of spatial finite element basis functions and a set of orthogonal polynomials in the parameters that define the uncertain PDE coefficients. When the number of parameters is t oo high\, the dimension of this space becomes unmanageable. One remedy is to use &lsquo\;adaptivity&rsquo\;. First\, we generate an approximation in a low-dimensional approximation space (which is cheap) and then use a com putable a posteriori error estimator to decide whether the current approxi mation is accurate enough or not. If not\, we enrich the approximation spa ce\, estimate the error again\, and so on\, until the final approximation is accurate enough. This allows us to design problem-specific polynomial a pproximations. We describe an error estimation procedure\, outline the com putational costs\, and illustrate its use through numerical results. An im proved multilevel implem entation will be outlined in a poster given by Ad am Crowder. LOCATION:Seminar Room 1\, Newton Institute END:VEVENT END:VCALENDAR