Sparse grid

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

A multigrid algorithm for higher order finite elements on sparse grids.

Abstract: For most types of problems in numerical mathematics, efficient discretization techniques are of crucial importance. This holds for tasks like how to define sets of points to approximate, interpolate, or integrate certain classes of functions as accurate as possible as well as for the numerical solution of differential equations. Introduced by Zenger in 1990 and based on hierarchical tensor product approximation spaces, sparse grids have turned out to be a very efficient approach in order to improve the ratio of invested storage and computing time to the achieved accuracy for many problems in the areas mentioned above. Concerning the sparse grid finite element discretization of elliptic partial differential equations, recently, the class of problems that can be tackled has been enlarged significantly. First, the tensor product approach led to the formulation of unidirectional algorithms which are essentially independent of the number d of dimensions. Second, techniques for the treatment of the general linear elliptic differential operator of second order have been developed, which, with the help of domain transformation, enable us to deal with more complicated geometries, too. Finally, the development of hierarchical polynomial bases of piecewise arbitrary degree p has opened the way to a further improvement of the order of approximation. In this paper, we discuss the construction and the main properties of a class of hierarchical polynomial bases and present a symmetric and an asymmetric finite element method on sparse grids, using the hierarchical polynomial bases for both the approximation and the test spaces or for the approximation space only, resp., with standard piecewise multilinear hierarchical test functions. In both cases, the storage requirement at a grid point does not depend on the local polynomial degree p, and p and the resulting representations of the basis functions can be handled in an efficient and adaptive way. An advantage of the latter approach, however, is the fact that it allows the straightforward implementation of a multigrid solver for the resulting system which is discussed, too.

Optimization of black-box problems using Smolyak grids and polynomial approximations

Abstract: A surrogate-based optimization method is presented, which aims to locate the global optimum of box-constrained problems using input–output data. The method starts with a global search of the n-dimensional space, using a Smolyak (Sparse) grid which is constructed using Chebyshev extrema in the one-dimensional space. The collected samples are used to fit polynomial interpolants, which are used as surrogates towards the search for the global optimum. The proposed algorithm adaptively refines the grid by collecting new points in promising regions, and iteratively refines the search space around the incumbent sample until the search domain reaches a minimum hyper-volume and convergence has been attained. The algorithm is tested on a large set of benchmark problems with up to thirty dimensions and its performance is compared to a recent algorithm for global optimization of grey-box problems using quadratic, kriging and radial basis functions. It is shown that the proposed algorithm has a consistently reliable performance for the vast majority of test problems, and this is attributed to the use of Chebyshev-based Sparse Grids and polynomial interpolants, which have not gained significant attention in surrogate-based optimization thus far.

Uncertainty Quantification for Electromagnetic Systems Using ASGC and DGTD Method

Abstract: In this paper, an adaptive hierarchical sparse grid collocation (ASGC) method combined with the discontinuous Galerkin time-domain method is leveraged to quantify the impacts of random parameters on the electromagnetics systems. The ASGC method approximates the stochastic observables of interest using interpolation functions over a set of collocation points determined by the Smolyak's algorithm integrated with an adaptive strategy. Instead of resorting to a full-tensor product sense, the Smolyak's algorithm constructs the collocation points in a hierarchical scheme with the interpolation level. Enhanced by an adaptive strategy, the Smolyak's algorithm will sample more points along important dimensions with sharp variations or discontinuities, resulting in a nonuniform sampling scheme. To flexibly handle different stochastic systems, either piecewise linear or Lagrange polynomial basis functions are applied. With these strategies, the number of collocation points is significantly reduced. The statistical knowledge of stochastic observables including the expected value, variance, probability density function, and cumulative distribution function are presented. The accuracy and robustness of the algorithm are demonstrated by various examples.

Sparse Grids for the Vlasov–Poisson Equation

Abstract: The Vlasov–Poisson equation models the evolution of a plasma in an external or self-consistent electric field. The model consists of an advection equation in six dimensional phase space coupled to Poisson’s equation. Due to the high dimensionality and the development of small structures the numerical solution is quite challenging. For two or four dimensional Vlasov problems, semi-Lagrangian solvers have been successfully applied. Introducing a sparse grid, the number of grid points can be reduced in higher dimensions. In this paper, we present a semi-Lagrangian Vlasov–Poisson solver on a tensor product of two sparse grids. In order to defeat the problem of poor representation of Gaussians on the sparse grid, we introduce a multiplicative delta-f method and separate a Gaussian part that is then handled analytically. In the semi-Lagrangian setting, we have to evaluate the hierarchical surplus on each mesh point. This interpolation step is quite expensive on a sparse grid due to the global nature of the basis functions. In our method, we use an operator splitting so that the advection steps boil down to a number of one dimensional interpolation problems. With this structure in mind we devise an evaluation algorithm with constant instead of logarithmic complexity per grid point. Results are shown for standard test cases and in four dimensional phase space the results are compared to a full-grid solution and a solution on the four dimensional sparse grid.