Sparse grid

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

Sparse grids for the Schrödinger equation

Abstract: We present a sparse grid/hyperbolic cross discretization for many-particle problems. It involves the tensor product of a one-particle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, different variants of sparse grid techniques for many-particle spaces can be derived that, in the best case, result in complexities and error estimates which are independent of the number of particles. Furthermore we introduce an additional constraint which gives antisymmetric sparse grids which are suited to fermionic systems. We apply the antisymmetric sparse grid discretization to the electronic Schrodinger equation and compare costs, accuracy, convergence rates and scalability with respect to the number of electrons present in the system.

Dimension-Adaptive Sparse Grid Interpolation for Uncertainty Quantification in Modern Power Systems: Probabilistic Power Flow

Abstract: In this paper, the authors firstly present the theoretical foundation of a state-of-the-art uncertainty quantification method, the dimension-adaptive sparse grid interpolation (DASGI), for introducing it into the applications of probabilistic power flow (PPF), specifically as discussed herein. It is well-known that numerous sources of uncertainty are being brought into the present-day electrical grid, by large-scale integration of renewable, thus volatile, generation, e.g., wind power, and by unprecedented load behaviors. In presence of these added uncertainties, it is imperative to change traditional deterministic power flow (DPF) calculation to take them into account in the routine operation and planning. However, the PPF analysis is still quite challenging due to two features of the uncertainty in modern power systems: high dimensionality and presence of stochastic interdependence. Both are traditionally addressed by the Monte Carlo simulation (MCS) at the cost of cumbersome computation; in this paper instead, they are tackled with the joint application of the DASGI and Copula theory (especially advantageous for constructing nonlinear dependence among various uncertainty sources), in order to accomplish the dependent high-dimensional PPF analysis in an accurate and faster manner. Based on the theory of DASGI, its combination with Copula and the DPF for the PPF is also introduced systematically in this work. Finally, the feasibility and the effectiveness of this methodology are validated by the test results of two standard IEEE test cases.

Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems

Abstract: We develop in this paper some efficient algorithms which are essential to implementations of spectral methods on the sparse grid by Smolyak's construction based on a nested quadrature. More precisely, we develop a fast algorithm for the discrete transform between the values at the sparse grid and the coefficients of expansion in a hierarchical basis; and by using the aforementioned fast transform, we construct two very efficient sparse spectral-Galerkin methods for a model elliptic equation. In particular, the Chebyshev-Legendre-Galerkin method leads to a sparse matrix with a much lower number of nonzero elements than that of low-order sparse grid methods based on finite elements or wavelets, and can be efficiently solved by a suitable sparse solver. Ample numerical results are presented to demonstrate the efficiency and accuracy of our algorithms.