B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness

Let $\xi = \{x^j\}_{j=1}^n$ be a grid of $n$ points in the $d$-cube {\II}^d:=[0,1]^d, and $\Phi = \{\phi_j\}_{j =1}^n$ a family of $n$ functions on {\II}^d. We define the linear sampling algorithm $L_n(\Phi,\xi,\cdot)$ for an approximate recovery of a continuous function $f$ on {\II}^d from the sampled values $f(x^1), ..., f(x^n)$, by $$L_n(\Phi,\xi,f)\ := \ \sum_{j=1}^n f(x^j)\phi_j$$. For the Besov class $B^\alpha_{p,\theta}$ of mixed smoothness $\alpha$ (defined as the unit ball of the Besov space \MB), to study optimality of $L_n(\Phi,\xi,\cdot)$ in L_q({\II}^d) we use the quantity $$r_n(B^\alpha_{p,\theta})_q \ := \ \inf_{H,\xi} \ \sup_{f \in B^\alpha_{p,\theta}} \, \|f - L_n(\Phi,xi,f)\|_q$$, where the infimum is taken over all grids $\xi = \{x^j\}_{j=1}^n$ and all families $\Phi = \{\phi_j\}_{j=1}^n$ in L_q({\II}^d). We explicitly constructed linear sampling algorithms $L_n(\Phi,\xi,\cdot)$ on the grid \xi = \ G^d(m):= \{(2^{-k_1}s_1,...,2^{-k_d}s_d) \in \II^d : \ k_1 + ... + k_d \le m\}, with $\Phi$ a family of linear combinations of mixed B-splines which are mixed tensor products of either integer or half integer translated dilations of the centered B-spline of order $r$. The grid $G^d(m)$ is of the size $2^m m^{d-1}$ and sparse in comparing with the generating dyadic coordinate cube grid of the size $2^{dm}$. For various $0<p,q,\theta \le \infty$ and $1/p < \alpha < r$, we proved upper bounds for the worst case error $\sup_{f \in B^\alpha_{p,\theta}} \, \|f - L_n(\Phi,\xi,f)\|_q$ which coincide with the asymptotic order of $r_n(B^\alpha_{p,\theta})_q$ in some cases. A key role in constructing these linear sampling algorithms, plays a quasi-interpolant representation of functions $f \in B^\alpha_{p,\theta}$ by mixed B-spline series

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

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