Stochastic sampling methods are arguably the most direct and least intrusivemeans of incorporating parametric uncertainty into numerical simulations ofpartial differential equations with random inputs. However, to achieve anoverall error that is within a desired tolerance, a large number of samplesimulations may be required (to control the sampling error), each of which mayneed to be run at high levels of spatial fidelity (to control the spatialerror). Multilevel sampling methods aim to achieve the same accuracy astraditional sampling methods, but at a reduced computational cost, through theuse of a hierarchy of spatial discretization models. Multilevel algorithmscoordinate the number of samples needed at each discretization level byminimizing the computational cost, subject to a given error tolerance. They canbe applied to a variety of sampling schemes, exploit nesting when available,can be implemented in parallel and can be used to inform adaptive spatialrefinement strategies. We extend the multilevel sampling algorithm to sparsegrid stochastic collocation methods, discuss its numerical implementation anddemonstrate its efficiency both theoretically and by means of numericalexamples.
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