Sparsity for parametric PDEs with log-gamma random inputs and applications
This addresses computational efficiency in uncertainty quantification for PDEs with specific random inputs, representing an incremental advance in sparsity analysis.
The paper tackles the problem of establishing sparsity in coefficients for parametric elliptic PDEs with log-gamma random inputs, quantifying it through ℓp-summability and weighted ℓ2-summability, and derives convergence rates for semi-discrete approximations. It also improves the sufficient condition for ℓp-summability in log-normal cases compared to prior work.
We propose a novel method for establishing the sparsity of the coefficients of the Laguerre generalized polynomial chaos expansion of solutions to parametric elliptic PDEs with log-gamma inputs on $\mathbb{R}_+^\infty$. The established sparsity is quantified by $\ell_p$-summability and weighted $\ell_2$-summability of the coefficients. Building on these sparsity results, we derive convergence rates for semi-discrete approximations in the parametric variables. These rates apply to sparse-grid polynomial interpolations, extended least-squares approximations and the associated semi-discrete quadrature rules. Moreover, a counterpart of our method for parametric elliptic PDEs with log-normal inputs yields a significant improvement in the sufficient condition for $\ell_p$-summability when the component functions in the log-normal representation of the parametric diffusion coefficients have global support, compared with results obtained in prior works.