Sparse Quadrature for High-Dimensional Integration with Gaussian Measure

In this work we analyze the dimension-independent convergence property of an abstract sparse quadrature scheme for numerical integration of functions of high-dimensional parameters with Gaussian measure. Under certain assumptions of the exactness and the boundedness of univariate quadrature rules as well as the regularity of the parametric functions with respect to the parameters, we obtain the convergence rate $O(N^{-s})$, where $N$ is the number of indices, and $s$ is independent of the number of the parameter dimensions. Moreover, we propose both an a-priori and an a-posteriori schemes for the construction of a practical sparse quadrature rule and perform numerical experiments to demonstrate their dimension-independent convergence rates

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

On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion

Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting

On the convergence of adaptive stochastic collocation for elliptic partial differential equations with affine diffusion

Convergence of an adaptive collocation method for the stationary parametric diffusion equation with finite-dimensional affine coefficient is shown. The adaptive algorithm relies on a recently introduced residual-based reliable a posteriori error estimator. For the convergence proof, a strategy recently used for a stochastic Galerkin method with an hierarchical error estimator is transferred to the collocation setting. Extensions to other variants of adaptive collocation methods (including the classical one proposed in the paper "Dimension-adaptive tensor-product quadratuture" Computing (2003) by T. Gerstner and M. Griebel) is explored.Comment: 24 pages, 1 figur

Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations

We analyze the convergence of compressive sensing based sampling techniques for the efficient evaluation of functionals of solutions for a class of high-dimensional, affine-parametric, linear operator equations which depend on possibly infinitely many parameters. The proposed algorithms are based on so-called "non-intrusive" sampling of the high-dimensional parameter space, reminiscent of Monte-Carlo sampling. In contrast to Monte-Carlo, however, a functional of the parametric solution is then computed via compressive sensing methods from samples of functionals of the solution. A key ingredient in our analysis of independent interest consists in a generalization of recent results on the approximate sparsity of generalized polynomial chaos representations (gpc) of the parametric solution families, in terms of the gpc series with respect to tensorized Chebyshev polynomials. In particular, we establish sufficient conditions on the parametric inputs to the parametric operator equation such that the Chebyshev coefficients of the gpc expansion are contained in certain weighted $\ell_p$-spaces for $0<p\leq 1$. Based on this we show that reconstructions of the parametric solutions computed from the sampled problems converge, with high probability, at the $L_2$, resp. $L_\infty$ convergence rates afforded by best $s$-term approximations of the parametric solution up to logarithmic factors.Comment: revised version, 27 page

IGA-based Multi-Index Stochastic Collocation for random PDEs on arbitrary domains

This paper proposes an extension of the Multi-Index Stochastic Collocation (MISC) method for forward uncertainty quantification (UQ) problems in computational domains of shape other than a square or cube, by exploiting isogeometric analysis (IGA) techniques. Introducing IGA solvers to the MISC algorithm is very natural since they are tensor-based PDE solvers, which are precisely what is required by the MISC machinery. Moreover, the combination-technique formulation of MISC allows the straight-forward reuse of existing implementations of IGA solvers. We present numerical results to showcase the effectiveness of the proposed approach.Comment: version 3, version after revisio