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

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

Nobile, Fabio, Raul Tempone, and Sören Wolfers. "Sparse approximation of multilinear problems with applications to kernel-based methods in UQ." Numerische Mathematik 139, no. 1 (2018): 247-280.​
Nobile, Fabio, Raul Tempone, and Sören Wolfers
kernel-based methods in UQ
2018
​We provide a framework for the sparse approximation of multilinear problems and show that several problems in uncertainty quantification fit within this framework. In these problems, the value of a multilinear map has to be approximated using approximations of different accuracy and computational work of the arguments of this map. We propose and analyze a generalized version of Smolyak’s algorithm, which provides sparse approximation formulas with convergence rates that mitigate the curse of dimension that appears in multilinear approximation problems with a large number of arguments. We apply the general framework to response surface approximation and optimization under uncertainty for parametric partial differential equations using kernel-based approximation. The theoretical results are supplemented by numerical experiments.​​