Sparse grid

About: Sparse grid is a research topic. Over the lifetime, 1013 publications have been published within this topic receiving 20664 citations.

Adjoint Error Estimation for Stochastic Collocation Methods

Abstract: This paper deals with partial differential equations with random input data. An efficient way of solving such problems is adaptive stochastic collocation on sparse grids. For higher efficiency and a better understanding of the method, we derive adjoint error estimates for nonlinear stochastic solution functionals. The resulting adjoint problem also involves random parameters and can be treated by stochastic collocation as well. Only a few adjoint evaluations are required in order to estimate the deterministic error, while the stochastic error requires much more effort to be detected. To overcome these substantial additional costs, we suggest to replace the adjoint problem by a reduced model and demonstrate the applicability of the approach for nonlinear solution functionals and up to nine random dimensions.

Uncertainty propagation of frequency response of viscoelastic damping structures using a modified high-dimensional adaptive sparse grid collocation method

Abstract: Uncertainty propagation (UP) of the frequency response is essential for the robust design of viscoelastic damping structures One challenge in solving this problem is enormous computation cost from

Error analysis for function representation by the sparse-grid combination technique

Abstract: Detailed error analyses are given for sparse-grid function representations through the combination technique. Two- and three-dimensional, and smooth and discontinuous functions are considered, as well as piecewise-constant and piecewise-linear interpolation techniques. Where appropriate, the results of the analyses are verified in numerical experiments. Instead of the common vertex-based function representation, cell-centered function representation is considered. Explicit, pointwise error expressions for the representation error are given, rather than order estimates. The paper contributes to the theory of sparse-grid techniques.

Dimension-Reduced Sparse Grid Strategy for a Stochastic Collocation Method in EMC Software

Abstract: Stochastic collocation method (SCM), a prevailing uncertainty analysis method, has been successfully implemented in electromagnetic compatibility (EMC) simulation, especially in EMC commercial software However, the “curse of dimensionality” problem (dimensionality means the number of uncertain variables) limits the application of the SCM This paper proposes a novel sparse grid strategy in order to improve the computational efficiency of the SCM, especially in high-dimensionality case In the proposed strategy, it is revealed that the number of the collocation points is in proportion to the dimensionality By simulating two shielding effectiveness analysis examples in CST software, the feasibility of the proposed method can be presented clearly, with the help of the feature selective validation method

New trends in uncertainty quantification for large-scale electromagnetic analysis: from tensor product cubature rules to spectral quantic tensor-train approximation

Abstract: In this chapter, efficient collocation methods for EM analysis are reviewed. Traditional SC methods leveraging tensor-product, sparse grid, and Stroud cubature rules are described first. These methods are rather straightforward to implement and suitable for EM problems involving smoothly varying QoI. Then, the ME-PC method for efficiently constructing a surrogate model of a rapidly varying QoI is presented. Also detailed is the iterative HDMR technique for EM problems involving large numbers of random variables. Finally, an approximation technique based on the spectral quantic TT (QTT) (SQTT) for constructing a surrogate model in a high-dimensional random domain is briefly reviewed, before the chapter is concluded by numerical examples demonstrating applications of cutting-edge UQ methods to various EM problems.