Sparse grid

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

Scenario generation for stochastic optimization problems via the sparse grid method

Abstract: We study the use of sparse grids in the scenario generation (or discretization) problem in stochastic programming problems where the uncertainty is modeled using a continuous multivariate distribution. We show that, under a regularity assumption on the random function involved, the sequence of optimal objective function values of the sparse grid approximations converges to the true optimal objective function values as the number of scenarios increases. The rate of convergence is also established. We treat separately the special case when the underlying distribution is an affine transform of a product of univariate distributions, and show how the sparse grid method can be adapted to the distribution by the use of quadrature formulas tailored to the distribution. We numerically compare the performance of the sparse grid method using different quadrature rules with classic quasi-Monte Carlo (QMC) methods, optimal rank-one lattice rules, and Monte Carlo (MC) scenario generation, using a series of utility maximization problems with up to 160 random variables. The results show that the sparse grid method is very efficient, especially if the integrand is sufficiently smooth. In such problems the sparse grid scenario generation method is found to need several orders of magnitude fewer scenarios than MC and QMC scenario generation to achieve the same accuracy. It is indicated that the method scales well with the dimension of the distribution--especially when the underlying distribution is an affine transform of a product of univariate distributions, in which case the method appears scalable to thousands of random variables.

A Causality Free Computational Method for HJB Equations with Application to Rigid Body Satellites

Abstract: Solving Hamilton-Jacobi-Bellman (HJB) equations is essential in feedback optimal control. Using the solution of HJB equations, feedback optimal control laws can be implemented in real-time with minimum computational load. However, except for systems with two or three state variables, numerically solving HJB equations for general nonlinear systems is unfeasible due to the curse of dimensionality. In this paper, we develop a new computational method of solving HJB equations. The method is causality free, which enjoys the advantage of perfect parallelism on a sparse grid. Compared with dense grids, a sparse grid has a significantly reduced size which is feasible for systems with relatively high dimensions, such as 6-D HJB equations for the attitude control of rigid bodies. The method is applied to the optimal attitude control of a satellite system using momentum wheels. The accuracy of the numerical solution is verified at a set of randomly selected sample points.

Face authentication with sparse grid Gabor information

Abstract: This paper investigates the application of statistical pattern recognition methods in the framework of the dynamic link matching approach. This method describes objects by means of local frequency information on nodes of a sparse grid. Matching of an input image with a reference is achieved by displacement and deformation of the grid. This method is applied here to the authentication of human faces in a cooperative scenario where candidates claim an identity that is to be checked. The matching error is not powerful enough to provide satisfying results in this case. We introduce an automatic weighting of the nodes according to their significance. Results show that for regular grids, this weighting leads to a significant improvement of the performance.

Sparse Grid Quadrature

Abstract: This chapter is concerned with sparse grid (SG) quadrature methods These methods are constructed using certain combinations of tensor products of one-dimensional quadrature rules They can exploit the smoothness of f, overcome the curse of dimension to a certain extent and profit from low effective dimensions, see, eg, [16, 44, 45, 57, 116, 146]

Addressing global uncertainty and sensitivity in first-principles based microkinetic models by an adaptive sparse grid approach.

Abstract: In the last decade, first-principles-based microkinetic modeling has been developed into an important tool for a mechanistic understanding of heterogeneous catalysis. A commonly known, but hitherto barely analyzed issue in this kind of modeling is the presence of sizable errors from the use of approximate Density Functional Theory (DFT). We here address the propagation of these errors to the catalytic turnover frequency (TOF) by global sensitivity and uncertainty analysis. Both analyses require the numerical quadrature of high-dimensional integrals. To achieve this efficiently, we utilize and extend an adaptive sparse grid approach and exploit the confinement of the strongly non-linear behavior of the TOF to local regions of the parameter space. We demonstrate the methodology on a model of the oxygen evolution reaction at the Co3O4 (110)-A surface, using a maximum entropy error model that imposes nothing but reasonable bounds on the errors. For this setting, the DFT errors lead to an absolute uncertainty of several orders of magnitude in the TOF. We nevertheless find that it is still possible to draw conclusions from such uncertain models about the atomistic aspects controlling the reactivity. A comparison with derivative-based local sensitivity analysis instead reveals that this more established approach provides incomplete information. Since the adaptive sparse grids allow for the evaluation of the integrals with only a modest number of function evaluations, this approach opens the way for a global sensitivity analysis of more complex models, for instance, models based on kinetic Monte Carlo simulations.