Piecewise-polynomial interpolations and quadratures for parametric PDEs with log-Laplace random inputs
This work addresses uncertainty quantification in computational science, providing theoretical foundations for efficient numerical methods, but it appears incremental as it builds on existing piecewise-polynomial chaos expansions.
The paper tackles the problem of approximating solutions to parametric elliptic PDEs with log-Laplace random inputs by establishing sparsity in coefficients and deriving convergence rates for semi-discrete approximations, achieving results based on $\ell_p$-summability and weighted $\ell_2$-summability.
We establish a sparsity in terms of $\ell_p$-summability and weighted $\ell_2$-summability for the coefficients of the Laguerre generalized piecewise-polynomial chaos expansion of solutions to parametric elliptic PDEs with log-Laplace random inputs. From the sparsity, we derive convergence rates for semi-discrete approximations with respect to parametric variables. These rates are valid for sparse-grid, piecewise-polynomial interpolations and the generated quadratures, and to related extended least-squares approximations and generated quadratures.