Sparse grid

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

Sparse Grid Interpolation

Abstract: For the approximation of multidimensional functions, using classical numerical discretization schemes such as full grids suffers the curse of dimensionality which is still a roadblock for the numerical treatment of high-dimensional problems. The number of basis functions or nodes (grid points) have to be stored and processed depend exponentially on the number of dimensions, where efficient computation are challenging in the implementation. Recently, the technique of sparse grids has been introduced to significantly reduce the cost to approximate high-dimensional functions under certain regularity conditions. In this thesis, we present the classical sparse grid where the problem is discretized and solved on a certain sequence of conventional grids with uniform mesh sizes in each coordinate direction. Furthermore, the different types of sparse grids,i.e. Clenshaw Curtis sparse grid, have been taken into consideration to compare the accuracy and complexity of these algorithms. We then describe the sparse grid combination technique to demonstrate that it is competitive to the classical sparse grid approaches with respect to quality and run time and give proof that the interpolation by using combination approach is the classical sparse grid. We give details on the basic features of sparse grids and we consider several test problems up to dimensions. The results of numerical experiments report on the quality of approximation generated by the sparse grids, and, finally, employ the sparse grid interpolation for a real-world case to reduce a computationally expensive simulation model. We aim to obtain an efficient surrogate approximation based on a small number of simulations.

A sparse grid method for the Navier–Stokes equations based on hyperbolic cross†

Abstract: A sparse grid method for the time-dependent Navier–Stokes equations based on hyperbolic cross approximation is considered in this article. Subsequent truncation of the associated series expansion results in a sparse grid discretization. Stability and convergence of the fully discrete sparse grid method are established. Finally, the numerical experiment is presented to show the effectiveness of this sparse grid method. Copyright © 2013 John Wiley & Sons, Ltd.

Multivariate function approximation using sparse grids and High Dimensional Model Representation – a comparison

Abstract: In many areas of science and technology, there is a need for effective procedures for approximating multivariate functions. Sparse grids and cut-HDMR (High Dimensional Model Representation) are two alternative approaches to such multivariate approximations. It is therefore interesting to compare these two methods. Numerical experiments performed in this study indicate that the sparse grid approximation is more accurate than the cut-HDMR approximation that uses a comparable number of known values of the approximated function unless the approximated function can be expressed as a sum of high order polynomials of one or two variables.

An adaptive hierarchical sparse grid collocation method for stochastic scattering systems analysis

Abstract: To quantify the impacts of random inputs on hybrid electromagnetics (EM)-circuit systems or EM scattering from objects, an adaptive hierarchical sparse grid collocation (ASGC) algorithm combined with discontinuous Galerkin time-domain (DGTD) method is presented in this work. As a stochastic polynomial chaos modality, the ASGC method approximates the interested stochastic observables using interpolation functions over a set of collocation points. Instead of employing a full-tensor product sense, the collocation points in ASGC method are hierarchically marched with interpolation level based on Smolyak's algorithm. To further reduce the collocation points, the hierarchical surplus is used as the error indicator for each collocation point to achieve adaptivity. To handle different stochastic systems, both piecewise linear and Lagrange polynomial basis functions are applied. More specifically, the locally supported piecewise linear basis functions based on Newton-Cotes grid are particularly suitable to attack sharp variations and discontinuities in stochastic observables, while the Lagrange polynomial basis functions based on Clenshaw-Curtis grid are more favorable for smoothly stochastic observables due to its global property. With these strategies, the number of collocation points is significantly reduced with exponential convergence rate. To characterize the far-field scattering properties of objects, the radar-cross-section (RCS) of perfectly electrical conducting (PEC) and dielectric spheres are investigated under the influence of geometrical and material uncertainties such as radius R and relative electrical permittivity ∊ r . With this stochastic simulation algorithm, statistical information including the mean values, variances, probability density functions (pdfs) and cumulative distribution functions (cdfs) can be acquired conveniently.

Adaptive Point Block Methods

Abstract: We present adaptive algorithms for the solution of elliptic PDEs that are based on the so-called sparse grid discretization technique and hierarchical error indicators. We discuss both the approximation qualities of the adaptively generated sparse grids and the efficient iterative solution of the arising linear systems by means of certain grid- and point-oriented multilevel methods.