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Sci. Comput."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>We introduce a numerical method, multilevel designed quadrature for computing the statistical solution of partial differential equations with random input data. Similar to multilevel Monte Carlo methods, our method relies on hierarchical spatial approximations in addition to a parametric\/stochastic sampling strategy. A key ingredient in multilevel methods is the relationship between the spatial accuracy at each level and the number of stochastic samples required to achieve that accuracy. Our sampling is based on flexible quadrature points that are designed for a prescribed accuracy, which can yield less overall computational cost compared to alternative multilevel methods. We propose a constrained optimization problem that determines the number of samples to balance the approximation error with the computational budget. 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