Sparse grid

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

Numerical Methods for Uncertainty Quantification and Bayesian Update in Aerodynamics

Abstract: In this work we research the propagation of uncertainties in parameters and airfoil geometry to the solution. Typical examples of uncertain parameters are the angle of attack and the Mach number. The discretisation techniques which we used here are the Karhunen-Loeve and the polynomial chaos expansions. To integrate high-dimensional integrals in probabilistic space we used Monte Carlo simulations and collocation methods on sparse grids. To reduce storage requirement and computing time, we demonstrate an algorithm for data compression, based on a low-rank approximation of realisations of random fields. This low-rank approximation allows us an efficient postprocessing (e.g. computation of the mean value, variance, etc) with a linear complexity and with drastically reduced memory requirements. Finally, we demonstrate how to compute the Bayesian update for updating a priori probability density function of uncertain parameters. The Bayesian update is also used for incorporation of measurements into the model.

An adaptive sparse-grid iterative ensemble Kalman filter approach for parameter field estimation

Abstract: A new sparse-grid (SG) iterative ensemble Kalman filter (IEnKF) approach is proposed for estimating spatially varying parameters. The adaptive high-order hierarchical sparse-grid (aHHSG) method is adopted to discretize the unknown parameter field. An IEnKF is used to explore the parameter space and estimate the surpluses of the aHHSG interpolant at each SG level. Moreover, the estimated aHHSG interpolant on coarser levels is employed to provide a good initial guess of the IEnKF solver for the approximation on the finer levels. The method is demonstrated in estimating permeability field in flows through porous media.

Application of sparse grid combination techniques to low temperature plasmas particle-in-cell simulations. I. Capacitively coupled radio frequency discharges

Abstract: The use of a particle-in-cell (PIC) algorithm with an explicit scheme to model low temperature plasmas is challenging due to computational time constrains related to resolving both the electron Debye length in space and the inverse of a fraction of the plasma frequency in time. One recent publication [Ricketson and Cerfon, Plasma Phys. Control. Fusion 59, 024002 (2017)] has demonstrated the interest of using a sparse grid combination technique to accelerate the explicit PIC model. Simplest plasma conditions were considered. This paper is the demonstration of the capability and the effectiveness of the sparse grid combination technique embedded in the PIC algorithm (hereafter called “sparse PIC”) to self-consistently model capacitively coupled radio frequency discharges. For two-dimensional calculations, the sparse PIC approach is shown to accurately reproduce the plasma profiles and the energy distribution functions compared to the standard PIC model. The plasma parameters obtained by these two numerical methods differ by less than 5%, while a speed up in the executable time between 2 and 5 is obtained depending on the setup.

A Sparse Grid Stochastic Collocation Method for Elliptic Interface Problems with Random Input

Abstract: In this paper, numerical solutions of elliptic partial differential equations with both random input and interfaces are considered. The random coefficients are piecewise smooth in the physical space and moderately depend on a large number of random variables in the probability space. To relieve the curse of dimensionality, a sparse grid collocation algorithm based on the Smolyak construction is used. The numerical method consists of an immersed finite element discretization in the physical space and a Smolyak construction of the extreme of Chebyshev polynomials in the probability space, which leads to the solution of uncoupled deterministic problems as in the Monte Carlo method. Numerical experiments on two-dimensional domains are also presented. Convergence is verified and compared with the Monte Carlo simulations.