Uncertainty Quantification for Spectral Fractional Diffusion: Sparsity Analysis of Parametric Solutions

In bounded, polygonal domains $D \subset \Bbb{R}^d$, we analyze solution regularity and sparsity for computational uncertainty quantification for spectral fractional diffusion. Several types of uncertainty are considered: (i) uncertain, parametric diffusion coefficients; (ii) uncertain fractional order; and (iii) uncertain physical domains $D$. For any of these problem classes, we analyze sparsity of countably parametric solution families. A principal novel technical contribution of the paper is a sparsity analysis for operator equations with distributed uncertain inputs; in particular, the parametric input may be given as a generalized polynomial chaos (gpc) representation, generalizing earlier results which required an affine-parametric representation. The summability results established here imply best $N$-term approximation rate bounds as well as dimension-independent convergence rates of numerical approximation methods, such as stochastic collocation, Smolyak and quasi--Monte Carlo integration methods, compressed sensing, and least-squares approximation.