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Sparse grid

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


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Journal ArticleDOI
TL;DR: A sparse grid stochastic collocation method combined with discontinuous Galerkin method is developed for solving convection dominated diffusion optimal control problem with random coefficients, which leads to the solution of uncoupled deterministic problems.
Abstract: A sparse grid stochastic collocation method combined with discontinuous Galerkin method is developed for solving convection dominated diffusion optimal control problem with random coefficients By

7 citations

Proceedings ArticleDOI
08 Sep 2014
TL;DR: This talk presents the GOSGrid optimization algorithm based on sparse grids and compares it with other derivative free global algorithms based on a parameter estimation problem in reservoir engineering.
Abstract: ive function are in general the output of a complex simulator for which we don’t have any explicit expression. And the absence of any information on the function gradient narrows the resolution field to algorithms using no first or second order derivatives. There exists many different approaches in derivative free optimization, among which the most popular are space partitioning methods like DIRECT, direct search methods like Nelder Mead or MADS but also evolutionary algorithms like genetic algorithms, evolution strategies or other similar methods. The main drawback for all these methods is that they can suffer from a poor convergence rate and a high computational cost, especially for high dimensional cases. However, they can succeed in finding a global optimum where all other classical methods fail. In this work we consider surrogate based optimization which has already been widely used for many years. A surrogate model is a framework used to minimize a function by sequentially building and minimizing a simpler model (surrogate) of the original function. In this work we build a new surrogate model by using the sparse grid interpolation method. Basically, the sparse grid approach is a grid error-controlled hierarchical approximation method which neglects the basis functions with the smallest supports. This approach was introduced in 1963 by Smolyak in order to evaluate integrals in high dimensions. It was then applied for PDE approximations but also for optimization and more recently for sensitivity analysis. Compared to the first optimization algorithm based on sparse grids proposed by Klimke et al, a local refinement step is constructed here in order to explore the more promising regions. Moreover, no optimization steps are performed over the objective function, which reduces significantly the number of function evaluations employed. In this talk we present our GOSGrid optimization algorithm based on sparse grids. We also compare it with other derivative free global algorithms. The comparison is based on a parameter estimation problem in reservoir engineering.

7 citations

Journal ArticleDOI
TL;DR: This method provides an interpolation approach to approximate eigenvalues and eigenvectors’ functional dependencies on uncertain parameters by repetitively evaluating the deterministic solutions at the pre-selected nodal set to construct a high-dimensional interpolation formula of the result.
Abstract: In this paper, the eigenvalue problem with multiple uncertain parameters is analyzed by the sparse grid stochastic collocation method. This method provides an interpolation approach to approximate eigenvalues and eigenvectors’ functional dependencies on uncertain parameters. This method repetitively evaluates the deterministic solutions at the pre-selected nodal set to construct a high-dimensional interpolation formula of the result. Taking advantage of the smoothness of the solution in the uncertain space, the sparse grid collocation method can achieve a high order accuracy with a small nodal set. Compared with other sampling based methods, this method converges fast with the increase of the number of points. Some numerical examples with different dimensions are presented to demonstrate the accuracy and efficiency of the sparse grid stochastic collocation method.

7 citations

Book ChapterDOI
01 Jan 2019
TL;DR: It is shown that a level 1 sparse grid can be used for the propagation of manufacturing uncertainties and that a surface reconstruction accuracy of 99% seems necessary for the purpose of UQ studies on manufacturing variability.
Abstract: An industry-ready uncertainty quantification tool chain is developed and successfully applied to both simultaneous operational and geometrical uncertainties and uncertainties resulting from manufacturing variability, which are characterized by correlations of the measured coordinates. The non-intrusive probabilistic collocation method is combined with a sparse grid approach to drastically reduce the computational cost. This is one of the key features that make UQ in industrial applications feasible. A second required element is the automatization of the entire simulation chain, from uncertainty definition, simulation setup, post-processing and in case of geometrical uncertainties, geometry modification, and re-meshing. This process is fully automated including the post-processing of the UQ simulations, which consists of output PDF reconstruction and the calculation of scaled sensitivity derivatives. This tool chain is applied to the rotor 37 configuration with imposed uncertainties, demonstrating its capability of handling many simultaneous operational and geometrical or correlated manufacturing uncertainties in turnaround times significantly below the UMRIDA quantitative objectives of less than 1000CPUh for 10 simultaneous uncertainties. It is found that a level 1 sparse grid approach is sufficient if the mean and variance of output quantities are needed and a level 2 sparse grid is sufficient for the reconstructed PDF shape for most engineering applications. For manufacturing uncertainties, it is shown that a level 1 sparse grid can be used for the propagation of manufacturing uncertainties and that a surface reconstruction accuracy of 99% seems necessary for the purpose of UQ studies on manufacturing variability.

7 citations

01 Jan 2009
TL;DR: In this paper, a sparse grid-based global polynomial cross-section parameterization method is presented for MTR fuel and reactor designs, which is based on the analysis of variance (ANOVA) technique.
Abstract: The advent of more advanced MTR (Material Test Reactor) fuel and reactor designs necessitate a corresponding improvement in the cross-section representation models applied to few group homogenized diffusion parameters. This problem may be decomposed into the construction of an appropriate cross-section representation model (the definition of relevant state parameters and associated basis functions), and the subsequent determination of approximation coefficients. In this work a sparse grid - based global polynomial cross-section parameterization method is presented which addresses both these aspects. A technique known as analysis of variance (ANOVA) is applied to construct the model and an efficient sparse grid integration scheme is implemented to determine the expansion coefficients. The method has a number of advantages, such as: built-in error control; consistent treatment and importance evaluation of all state parameters, including burn-up; and identification and approximation of all cross-section dependencies, including cross-terms. The method is applied to different MTR fuel designs and compared to more traditional MTR parameterization methods. The usefulness of the method as both model buildingand few-group, cross-section reconstruction tool, is illustrated.

7 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202314
202242
202157
202040
201960
201872