Sparse grid

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

Multilevel Adaptive Stochastic Collocation with Dimensionality Reduction

Abstract: We present a multilevel stochastic collocation (MLSC) with a dimensionality reduction approach to quantify the uncertainty in computationally intensive applications. Standard MLSC typically employs grids with predetermined resolutions. Even more, stochastic dimensionality reduction has not been considered in previous MLSC formulations. In this paper, we design an MLSC approach in terms of adaptive sparse grids for stochastic discretization and compare two sparse grid variants, one with spatial and the other with dimension adaptivity. In addition, while performing the uncertainty propagation, we analyze, based on sensitivity information, whether the stochastic dimensionality can be reduced. We test our approach in two problems. The first one is a linear oscillator with five or six stochastic inputs. The dimensionality is reduced from five to two and from six to three. Furthermore, the dimension-adaptive interpolants proved superior in terms of accuracy and required computational cost. The second test case is a fluid-structure interaction problem with five stochastic inputs, in which we quantify the uncertainty at two instances in the time domain. The dimensionality is reduced from five to two and from five to four.

Stochastic Optimization of Economic Dispatch With Wind and Photovoltaic Energy Using the Nested Sparse Grid-Based Stochastic Collocation Method

Abstract: Due to the increasing uncertainty brought about by renewable energy, conventional deterministic dispatch approaches have not been very applicative. This paper investigates a nested sparse grid-based stochastic collocation method (NS-SCM) as a possible solution for stochastic economic dispatch (SED) problems. The SCM was used to simplify the scenario-based optimization model; specifically, a finite-order expansion using the generalized polynomial chaos (gPC) theory was applied to approximate random variables as a more facile approach compared to using complicated optimization models. Furthermore, a nested sparse grid-based approach was adopted to reduce the number of collocation points while still satisfying the nested property, thereby alleviating and effectively eliminating the need for computation. The proposed approach can be directly applied to the SED optimization problem. Lastly, simulations on the modified IEEE 39-bus system and a practical 1009-bus power system were provided to verify the accuracy, effectiveness, and practicality of the proposed algorithm.

A Sparse Grid PDE Solver; Discretization, Adaptivity, Software Design and Parallelization

Abstract: Sparse grids are an efficient approximation method for functions, especially in higher dimensions d≥3 Compared to regular, uniform grids of a mesh parameter h, which contain h −d points in d dimensions, sparse grids require only h −1| log h| d−1 points due to a truncated, tensor-product multi-scale basis representation The purpose of this paper is to survey some activities for the solution of partial differential equations with method based sparse grids Furthermore some aspects of sparse grids are discussed such as adaptive grid refinement, parallel computing, a space-time discretization scheme and the structure of a code to implement these methods

A Graph Learning Algorithm Based On Gaussian Markov Random Fields And Minimax Concave Penalty

Abstract: This paper presents a graph learning framework to produce sparse and accurate graphs from network data. While our formulation is inspired by the graphical lasso, a key difference is the use of a nonconvex alternative of the l 1 norm as well as a quadratic term to ensure overall convexity. Specifically, the weakly-convex minimax concave penalty (MCP) is used, which is given by subtracting the Huber function from the l 1 norm, inducing a less-biased sparse solution than l 1 . In our framework, the graph Laplacian is represented by a linear transform of the vector corresponding to its upper triangular part. Via a reformulation relying on the Moreau decomposition, the problem can be solved by the primal-dual splitting method. An admissible choice of parameters for provable convergence is presented. Numerical examples show that the proposed method significantly outperforms its l 1 -based counterpart for sparse grid graphs.

Uncertainty Quantification Using Parameter Space Partitioning

Abstract: A new method is presented for high-dimensional variability analysis based on two main concepts, namely, node tearing for parameter partitioning and sparse grid interpolation. Node tearing is used to localize the parameters and, thus, reducing the number of stochastic parameters within the subcircuits and sparse grids reduce the required number of samples for a targeted accuracy. MC analysis of the overall circuit is carried out using interface equations of a much smaller dimension than the original circuit equations. Pertinent computational results are presented to validate the efficiency and accuracy of the proposed method.