Quasi-optimal polynomial approximations for parameterized PDEs with deterministic and stochastic coefficients

We present a generalized methodology for analyzing the convergence of quasi-optimal Taylor and Legendre approximations, applicable to a wide class of parameterized elliptic PDEs with both deterministic and stochastic inputs. Such approximations construct an index set that corresponds to the best M-terms based on sharp estimates of the polynomial coefficients. Several types of isotropic and anisotropic (weighted) multi-index sets are explored and computational evidence shows the advantage of our methodology compared to previously developed estimates.