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We provide a framework for the sparse approximation of multilinear problemsand show that several problems in uncertainty quantification fit within thisframework. In these problems, the value of a multilinear map has to beapproximated using approximations of different accuracy and computational workof the arguments of this map. We propose and analyze a generalized version ofSmolyak's algorithm, which provides sparse approximation formulas withconvergence rates that mitigate the curse of dimension that appears inmultilinear approximation problems with a large number of arguments. We applythe general framework to response surface approximation and optimization underuncertainty for parametric partial differential equations using kernel-basedapproximation. The theoretical results are supplemented by numericalexperiments.

Sparse approximation of multilinear problems with applications to kernel-based methods in UQ