Uncertainty quantification in kinetic and hyperbolic problems

Quasi-optimal methods for deterministic and stochastic parameterized PDEs

Clayton Webster

Oak Ridge National Lab

In this talk, we present a generalized analytic framework for quasi-optimal polynomial and interpolation approximations, applicable to a wide class of parameterized PDEs with both deterministic and stochastic inputs. Such quasi-optimal methods construct an index set that corresponds to the “best M-terms,” based on sharp estimates of the polynomial coefficients. In particular, we consider several cases of N dimensional affine and non-affine coefficients, and prove analytic dependence of the PDE solution map in a polydisc or polyellipse of the multi-dimensional complex plane respectively. The framework we propose for analyzing asymptotic truncation errors of quasi-optimal methods is based on an extension of the underlying multi-index set into a continuous domain, and then an approximation of the cardinality (number of integer multi-indices) by its Lebesgue measure. Several types of isotropic and anisotropic (weighted) multi-index sets are explored, and rigorous proofs reveal sharp asymptotic error estimates in which we achieve sub-exponential convergence rates with respect to the total number of degrees of freedom. Through several theoretical examples, we explicitly derive the the rate constant and use the resulting sharp bounds to illustrate the effectiveness of our approach, as well as compare our rates of convergence with current published results. Finally, computational evidence complements the theory and shows the advantage of our generalized methodology compared to previously developed estimates.