# Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs

## TL;DR

This paper develops sparse-grid polynomial interpolation and quadrature methods for efficiently solving parametric and stochastic elliptic PDEs with lognormal inputs, providing explicit convergence rates and improved accuracy over existing methods.

## Contribution

It introduces fully discrete sparse-grid polynomial interpolation and quadrature methods for elliptic PDEs with lognormal inputs, with proven convergence rates and explicit construction.

## Key findings

- Convergence rates of the proposed methods are explicitly derived.
- Sparse-grid collocation methods effectively approximate solutions and integrals.
- The methods outperform previous approaches in accuracy and efficiency.

## 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. D\~ung 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 $L_1({\mathbb R}^\infty,V,\gamma)$ norm of the generating methods where $\gamma$ 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.