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-based adaptive noise reduction strategy for particle-in-cell schemes

Abstract: We propose a sparse grid-based adaptive noise reduction strategy for electrostatic particle-in-cell (PIC) simulations. By projecting the charge density onto sparse grids we reduce the high-frequency particle noise. Thus, we exploit the ability of sparse grids to act as a multidimensional low-pass filter in our approach. Thanks to the truncated combination technique [1] , [2] , [3] , we can reduce the larger grid-based error of the standard sparse grid approach for non-aligned and non-smooth functions. The truncated approach also provides a natural framework for minimizing the sum of grid-based and particle-based errors in the charge density. We show that our approach is, in fact, a filtering perspective for the noise reduction obtained with the sparse PIC schemes first introduced in [4] . This enables us to propose a heuristic based on the formal error analysis in [4] for selecting the optimal truncation parameter that minimizes the total error in charge density at each time step. Hence, unlike the physical and Fourier domain filters typically used in PIC codes for noise reduction, our approach automatically adapts to the mesh size, number of particles per cell, smoothness of the density profile and the initial sampling technique. It can also be easily integrated into high performance large-scale PIC code bases, because we only use sparse grids for filtering the charge density. All other operations remain on the regular grid, as in typical PIC codes. We demonstrate the efficiency and performance of our approach with two test cases: the diocotron instability in two dimensions and the three-dimensional electron dynamics in a Penning trap. Our run-time performance studies indicate that our approach can provide significant speedup and memory reduction to PIC simulations for achieving comparable accuracy in the charge density.

A Third Order Hierarchical Basis WENO Interpolation for Sparse Grids with Application to Conservation Laws with Uncertain Data

Abstract: In this paper, we introduce a third order hierarchical basis WENO interpolation, which possesses similar accuracy and stability properties as usual WENO interpolations. The main motivation for the hierarchical approach is the direct applicability on sparse grids. This is for instance of large practical interest in the numerical solution of conservation laws with uncertain data, where discontinuities in the physical domain often carry over to the (potentially high-dimensional) stochastic domain. For this, we apply the introduced hierarchical basis WENO interpolation within a non-intrusive collocation method and present first results on 2- and 3-dimensional sparse grids.

Sparse tensor spherical harmonics approximation in radiative transfer

Abstract: The stationary monochromatic radiative transfer equation is a partial differential transport equation stated on a five-dimensional phase space. To obtain a well-posed problem, boundary conditions have to be prescribed on the inflow part of the domain boundary. We solve the equation with a multi-level Galerkin FEM in physical space and a spectral discretization with harmonics in solid angle and show that the benefits of the concept of sparse tensor products, known from the context of sparse grids, can also be leveraged in combination with a spectral discretization. Our method allows us to include high spectral orders without incurring the ''curse of dimension'' of a five-dimensional computational domain. Neglecting boundary conditions, we find analytically that for smooth solutions, the convergence rate of the full tensor product method is retained in our method up to a logarithmic factor, while the number of degrees of freedom grows essentially only as fast as for the purely spatial problem. For the case with boundary conditions, we propose a splitting of the physical function space and a conforming tensorization. Numerical experiments in two physical and one angular dimension show evidence for the theoretical convergence rates to hold in the latter case as well.

Towards Goal-Oriented Stochastic Design Employing Adaptive Collocation Methods

Abstract: Non-intrusive polynomial chaos expansion (NIPCE) methods based on orthogonal polynomials and stochastic collocation (SC) methods based on Lagrange interpolation polynomials are attractive techniques for uncertainty quantification (UQ) due to their strong mathematical basis and ability to produce functional representations of stochastic dependence. Both techniques reside in the collocation family, in that they sample the response metrics of interest at selected locations within the random domain without intrusion to simulation software. In this work, we explore the use of polynomial order refinement (prefinement) approaches, both uniform and adaptive, in order to automate the assessment of UQ convergence and improve computational efficiency. In the first class of p-refinement approaches, we employ a general-purpose metric of response covariance to control the uniform and adaptive refinement processes. For the adaptive case, we detect anisotropy in the importance of the random variables as determined through variance-based decomposition and exploit this decomposition through anisotropic tensor-product and anisotropic sparse grid constructions. In the second class of p-refinement approaches, we move from anisotropic sparse grids to generalized sparse grids and employ a goal-oriented refinement process using statistical quantities of interest. Since these refinement goals can frequently involve metrics that are not analytic functions of the expansions (i.e., beyond low order response moments), we additionally explore efficient mechanisms for accurately and efficiently estimated tail probabilities from the expansions based on importance sampling.