Sparse approximation of multilinear problems with applications to kernel-based methods in UQ
For researchers in uncertainty quantification, this provides a method to reduce computational cost in multilinear problems with many arguments, though the approach is an extension of existing Smolyak-type algorithms.
The paper proposes a generalized Smolyak algorithm for sparse approximation of multilinear problems, mitigating the curse of dimensionality in uncertainty quantification tasks like response surface approximation and optimization under uncertainty for parametric PDEs. Numerical experiments confirm the theoretical convergence rates.
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.