Compressed sensing Petrov-Galerkin approximations for parametric PDEs

摘要

We consider the computation of parametric solution families of high-dimensional stochastic and parametric PDEs. We review theoretical results on sparsity of polynomial chaos expansions of parametric solutions, and on compressed sensing based collocation methods for their efficient numerical computation. With high probability, these randomized approximations realize best N-term approximation rates afforded by solution sparsity and are free from the curse of dimensionality, both in terms of accuracy and number of samples evaluations (i.e. PDE solves). Through various examples we illustrate the performance of Compressed Sensing Petrov-Galerkin (CSPG) approximations of parametric PDEs, for the computation of (functionals of) solutions of intregral and differential operators on high-dimensional parameter spaces. The CSPG approximations reduce the number of PDE solves, as compared to Monte-Carlo methods, while being likewise nonintrusive, and being “embarassingly parallel”, unlike dimension-adaptive collocation or Galerkin methods.