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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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TL;DR: In this paper, a sparse grid collocation method is proposed to solve an optimal control problem involving an elliptic partial differential equation with random coefficients and forcing terms, where the input data are assumed to be dependent on a finite number of random variables.
Abstract: In this article, we propose and analyse a sparse grid collocation method to solve an optimal control problem involving an elliptic partial differential equation with random coefficients and forcing terms. The input data are assumed to be dependent on a finite number of random variables. We prove that an optimal solution exists, and derive an optimality system. A Galerkin approximation in physical space and a sparse grid collocation in the probability space is used. Error estimates for a fully discrete solution using an appropriate norm are provided, and we analyse the computational efficiency. Computational evidence complements the present theory, to show the effectiveness of our stochastic collocation method.

2 citations

Posted Content
TL;DR: This work proposes a scheme for significantly reducing the computational complexity of discretized problems involving the non-smooth forward propagation of uncertainty by combining the adaptive hierarchical sparse grid stochastic collocation method with a hierarchy of successively finer spatial discretizations of the underlying deterministic problem.
Abstract: This work proposes a scheme for significantly reducing the computational complexity of discretized problems involving the non-smooth forward propagation of uncertainty by combining the adaptive hierarchical sparse grid stochastic collocation method (ALSGC) with a hierarchy of successively finer spatial discretizations (e.g. finite elements) of the underlying deterministic problem. To achieve this, we build strongly upon ideas from the Multilevel Monte Carlo method (MLMC), which represents a well-established technique for the reduction of computational complexity in problems affected by both deterministic and stochastic error contributions. The resulting approach is termed the Multilevel Adaptive Sparse Grid Collocation (MLASGC) method. Preliminary results for a low-dimensional, non-smooth parametric ODE problem are promising: the proposed MLASGC method exhibits an error/cost-relation of $\varepsilon \sim t^{-0.95}$ and therefore significantly outperforms the single-level ALSGC ($\varepsilon \sim t^{-0.65}$) and MLMC methods ($\varepsilon \lesssim t^{-0.5}$).

2 citations

DOI
21 Dec 2005
TL;DR: It turns out that even smaller granularity parallelism can be exploited effectively in the problems considered, and the development is illustrated by four examples of nonparametric regression techniques.
Abstract: Parallel computing enables the analysis of very large data sets using large collections of flexible models with many variables The computational methods are based on ideas from computational linear algebra and can draw on the extensive research on parallel algorithms in this area Many algorithms for the direct and iterative solution of penalised least squares problems and for updating can be applied Both methods for dense and sparse problems are applicable An important property of the algorithms is their scalability, ie, their ability to solve larger problems in the same time using hardware which grows linearly with the problem size While in most cases large granularity parallelism is to be preferred, it turns out that even smaller granularity parallelism can be exploited effectively in the problems considered The development is illustrated by four examples of nonparametric regression techniques In a first example, additive models are considered While the backfitting method contains dependencies which inhibit parallel execution it turns out that parallelisation over the data leads to a viable method, akin to the bagging algorithm without replacement which is known to have superior statistical properties in many cases The second example considers radial basis function fitting with thin plate splines Here the direct approach turns out to be non-scalable but an approximation with finite elements is shown to be scalable and parallelises well One of the most popular algorithms in data mining is MARS (Multivariate Adaptive Regression Splines) This is discussed in the third example MARS has been modified to use a multiscale approach and a parallel algorithm with a small granularity has been seen to give good results The final example considers the current research area of sparse grids Sparse grids take up many ideas from the previous examples and, in fact, can be considered as a generalisation of MARS and additive models They are naturally parallel when the combination technique is used We discuss limitations and improvements of the combination technique

2 citations

Posted Content
TL;DR: The results show the proposed approach can effectively detect and extract the periodical oscillatory features and is compared to some to some state-of-the-art methods.
Abstract: This paper addresses the problem of extracting periodic oscillatory features in vibration signals for detecting faults in rotation machinery. To extract the feature, we propose an approach in the short-time Fourier transform (STFT) domain, where the periodic oscillatory feature manifests itself as a relatively sparse grid.To estimate the sparse grid, we formulate an optimization problem using customized binary weights in the regularizer, where the weights are formulated to promote periodicity. As examples, the proposed approach is applied to simulated data, and used as a tool for diagnosing faults in bearings and gearboxes for real data, and compared to some to some state-of-the-art methods. The results show the proposed approach can effectively detect and extract the periodical oscillatory features.

2 citations

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