An Adaptive Sparse Grid Algorithm for Elliptic PDEs with Lognormal Diffusion Coefficient

Dimension-adaptive sparse grid interpolation is a powerful tool to obtain surrogate functions of smooth, medium to high-dimensional objective models. In case of expensive models, the efficiency of the sparse grid algorithm is governed by the time required for the function evaluations. In this paper, we first briefly analyze the inherent parallelism of the standard dimension-adaptive algorithm. Then, we present an enhanced version of the standard algorithm that permits, in each step of the algorithm, a specified number (equal to the number of desired processes) of function evaluations to be executed in parallel, thereby increasing the parallel efficiency.

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

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2021017 ◽

2021 ◽

Author(s):

Dũng Đinh

Keyword(s):

Convergence Rates ◽

Polynomial Interpolation ◽

Hermite Polynomials ◽

Adaptive Methods ◽

Sparse Grid ◽

Collocation Methods ◽

Quadrature Formulas ◽

Elliptic Pdes ◽

Fully Discrete ◽

Weighted Quadrature

By combining a certain approximation property in the spatial domain, and weighted summability of the Hermite polynomial expansion coefficients in the parametric domain, 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 convergence rates of the approximations by them. The linear non-adaptive methods of fully discrete polynomial interpolation approximation are sparse-grid collocation methods which are certain sums taken over finite nested Smolyak-type indices sets of mixed tensor products of dyadic scale successive differences of spatial approximations of particular solvers, and of successive differences of their parametric Lagrange interpolating polynomials. The Smolyak-type sparse interpolation grids in the parametric domain are constructed from the roots of Hermite polynomials or their improved modifications. Moreover, they generate in a natural way fully 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 Bochner space. We also briefly consider similar problems for parametric and stochastic elliptic PDEs with affine inputs, and problems of non-fully discrete polynomial interpolation approximation and integration. In particular, the convergence rates of non-fully discrete polynomial interpolation approximation and integration obtained in this paper significantly improve the known ones.

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