Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs
Dinh Dũng
TL;DR
The paper addresses high-dimensional uncertainty quantification for parametric and stochastic elliptic PDEs with lognormal inputs by developing non-adaptive linear methods built on truncated Hermite GPC expansions and sparse-grid polynomial interpolation, together with corresponding weighted quadrature for integration. It creates a framework that ties spatial-approximation properties to parametric summability, yielding explicit convergence rates in the approximation dimension $n$, and shows that non-adaptive schemes can attain the same $\min(\alpha,\beta)$ rates as adaptive best-$n$-term methods, with improved integration rates such as $2/q-1/2$ in the Gaussian setting. The results extend to affine inputs via Jacobi expansions and provide practical, provably convergent schemes for both interpolation and integration in Bochner spaces ${\mathcal L}_p(\mathbb R^\infty,X,\gamma)$. Overall, the work advances non-adaptive, scalable approaches for sparse-grid Hermite and Jacobi GPC-based PDE solvers under lognormal and affine input models.
Abstract
By combining a certain approximation property in the spatial domain, and weighted $\ell_2$-summability of the Hermite polynomial expansion coefficients in the parametric domain obtained in [M. Bachmayr, A. Cohen, R. DeVore and G. Migliorati, ESAIM Math. Model. Numer. Anal. $\bf 51$(2017), 341-363] and [M. Bachmayr, A. Cohen, D. Dũng and C. Schwab, SIAM J. Numer. Anal. $\bf 55$(2017), 2151-2186], we investigate linear non-adaptive methods of fully discrete polynomial interpolation approximation as well as fully discrete weighted quadrature methods of integration for parametric and stochastic elliptic PDEs with lognormal inputs. We explicitly construct such methods and prove corresponding convergence rates in $n$ of the approximations by them, where $n$ is a number characterizing computation complexity. The linear non-adaptive methods of fully discrete polynomial interpolation approximation are sparse-grid collocation methods. Moreover, they generate in a natural way discrete weighted quadrature formulas for integration of the solution to parametric and stochastic elliptic PDEs and its linear functionals, and the error of the corresponding integration can be estimated via the error in the Bochner space $L_1({\mathbb R}^\infty,V,γ)$ norm of the generating methods where $γ$ is the Gaussian probability measure on ${\mathbb R}^\infty$ and $V$ is the energy space. We also briefly consider similar problems for parametric and stochastic elliptic PDEs with affine inputs, and by-product problems of non-fully discrete polynomial interpolation approximation and integration. In particular, the convergence rate of non-fully discrete obtained in this paper improves the known one.