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{\large A. D. Gordon \\
{\bf Solving Stochastic Collocation Systems with Algebraic Multigrid}}

School of Mathematics \\ University of Manchester \\ Oxford Road \\ Manchester \\ M13 9PL \\ UK
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{\tt Andrew.Gordon-2@postgrad.manchester.ac.uk}\\
C. E. Powell\end{center}

Stochastic collocation methods facilitate the numerical solution of
partial differential equations (PDEs) with random data and give rise to
long sequences of similar discrete linear systems. When elliptic PDEs
with random diffusion coefficients are discretized with standard finite
element methods in the physical domain, the resulting collocation systems
can be solved iteratively with the conjugate gradient method and
algebraic multigrid (amg) is a highly robust preconditioner. When mixed
finite element methods are applied, amg is also a key tool for solving
the resulting sequence of saddle point systems via the preconditioned
minimal residual method. In both cases, the stochastic collocation
systems are trivial to solve when considered individually. The challenge
lies in exploiting the systems' similarities to recycle information and
minimize the cost of solving the entire sequence.

In this talk, we consider full tensor and sparse grid stochastic
collocation schemes applied to a model stochastic elliptic problem and
discretize in physical space using standard piecewise linear finite
elements and lowest order Raviart-Thomas mixed finite elements. We
propose efficient solvers for the resulting sequences of linear systems
and show, in particular, that it is feasible to use finely-tuned amg
preconditioning for each system if key set-up information is reused.
Crucially, the preconditioners are robust with respect to variations in
the discretization and statistical parameters for both stochastically
linear and nonlinear diffusion coefficients.


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