Van Kien Nguyen

Dinh Dũng, Van Kien Nguyen, Viet Ha Hoang

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.

Dinh DÅ©ng, Helmut Harbrecht, Van Kien Nguyen et al.

In this article, we consider the solution to elliptic diffusion problems on a class of random domains obtained by log-Gaussian random homothety of the unit disk respectively an annulus. We model the problem under consideration and verify the existence and uniqueness of the random solution by path-wise pullback to the nominal unit disk respectively annulus. We prove the analytic regularity of the solution with respect to the random input parameter. We consider the numerical approximation of the random diffusion problem by means of continuous, piecewise linear Lagrangian Galerkin Finite Elements with numerical quadrature in the nominal domain, and by sparse grid interpolation and quadrature of Gauss-Hermite Smolyak and Quasi-Monte Carlo type in the parameter domain. The theoretical findings are complemented by numerical results.

Van Kien Nguyen

This paper investigates the numerical approximation of integrals for functions in fractional Gaussian Sobolev spaces $W^s_{p}(\mathbb{R}^d,γ)$ with dominating mixed smoothness defined via kernel related to the fractional Ornstein-Uhlenbeck operator. Building upon quadrature rules for fractional Sobolev spaces on the unit cube $[-\tfrac{1}{2}, \tfrac{1}{2}]^d$, we construct quadrature schemes on $\mathbb{R}^d$ that achieve the same rate of convergence. As a consequence, we establish the optimal asymptotic order of the integration error in the regime $1 < p < \infty$ and $s > \frac{1}{p}$. Furthermore, we show that the fractional Gaussian Sobolev spaces $W^s_{2}(\mathbb{R}^d,γ)$ coincide with Hermite spaces $\mathcal{H}^s(\mathbb{R}^d,γ)$ characterized by the weighted $\ell_2$-summability of their Fourier-Hermite coefficients. From this, we derive the optimal asymptotic order of the integration error for functions in these spaces for all $s > \frac{1}{2}$. We also establish the corresponding optimal asymptotic order for functions in fractional Sobolev spaces $W^s_{p,G}(\mathbb{R}^d,γ)$ defined via the Gagliardo seminorm.