  This work proposes a method for sparse polynomial chaos (PC) approximation of
high-dimensional stochastic functions based on non-adapted random sampling. We
modify the standard l1 -minimization algorithm, originally proposed in the
context of compressive sampling, using a priori information about the decay of
the PC coefficients and refer to the resulting algorithm as weighted l1
-minimization. We provide conditions under which we may guarantee recovery
using this weighted scheme. Numerical tests are used to compare the weighted
and non-weighted methods for the recovery of solutions to two differential
equations with high-dimensional random inputs: a boundary value problem with a
random elliptic operator and a 2-D thermally driven cavity flow with random
boundary condition.
