Uncertainty Modeling Using Fuzzy Arithmetic Based on Sparse Grids: Applications to Dynamic Systems

Klimke, Andreas; Willner, Kai; Wohlmuth, Barbara

doi:10.1142/s0218488504003181

Cited by 90 publications

(173 citation statements)

References 12 publications

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“…Note that I(w) in (13) may not satisfy the admissibility condition (12), that has to be explicitely enforced. This criterion can be implemented in an adaptive procedure [14,19] that explores the space of hierarchical surpluses and adds to I(w) the most profitable according to (13). As an alternative, in [4] we have detailed an apriori/a-posteriori procedure to detect I(w) based on estimates of ∆E(i) and ∆W (i).…”

Section: Let Us Now Denotementioning

confidence: 99%

A Quasi-optimal Sparse Grids Procedure for Groundwater Flows

Beck

Nobile

Tamellini

et al. 2013

Lecture Notes in Computational Science and Engineering

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In this work we explore the extension of the quasi-optimal sparse grids method proposed in our previous work "On the optimal polynomial approximation of stochastic PDEs by Galerkin and Collocation methods" to a Darcy problem where the permeability is modeled as a lognormal random field. We propose an explicit a-priori/a-posteriori procedure for the construction of such quasi-optimal grid and show its effectivenenss on a numerical example. In this approach, the two main ingredients are an estimate of the decay of the Hermite coefficients of the solution and an efficient nested quadrature rule with respect to the Gaussian weight.

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Section: Let Us Now Denotementioning

confidence: 99%

A Quasi-optimal Sparse Grids Procedure for Groundwater Flows

Beck

Nobile

Tamellini

et al. 2013

Lecture Notes in Computational Science and Engineering

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“…In this paper, they are defined in advance but the domains in partitions may be learned. In general, fuzzy set is mathematically defined as follows [49]: Definition 1 If X is a space with generic elements of x, and μe Y : X → M ⊆ [0, 1] is the characteristic function that maps X to membership space M. Then the following set of pairs uniquely represents a fuzzy set.…”

Section: Applied Fuzzy Setmentioning

confidence: 99%

Toward learning autonomous pallets by using fuzzy rules, applied in a Conwip system

Mehrsai

Karimi

Scholz‐Reiter

2012

Int J Adv Manuf Technol

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Nowadays, material planning and control strategies are becoming continuously complex tasks spanning from individual plants to logistic networks. In fact, this is the consequence of increasing intricacy in product variants and their respective convolution in networks' structures. Customers ask for specific products with individual characteristics that force companies for more clever performances by more flexibility. For doing so, the existing planning and control systems, which work based on central monitoring and controlling, show some limitations for organizing every operation on time or in the right time. Therefore, in the recent decade, a great attention is put on decentralized control and, to some extent, autonomy. This paper tries to investigate the possibility of combining this new research paradigm with existing strategies in production logistics, in order to improve material handling and control task according to material flow criteria. To show this, an exemplary plant after decoupling point out of a logistic network is considered for simulation and analysis. This combines Conwip system with learning autonomous pallets' concept in a discrete event simulation model. Several decentralized control scenarios are experimented and compared together. Here, the learn methodology is brought to pallets based on fuzzy rules and advantage of closed loop systems.

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“…We also compare our results with the dimension adaptive algorithm proposed in [11], in the implementation of [14], available at http://www.ians.uni-stuttgart.de/spinterp. This is an adaptive algorithm that given a sparse grid S I explores all neighbour multi-indices and adds to I the most profitable ones.…”

Section: Numerical Tests On Sparse Gridsmentioning

confidence: 99%

“…This is an adaptive algorithm that given a sparse grid S I explores all neighbour multi-indices and adds to I the most profitable ones. The algorithm implemented in [14] has a tunable parameter ω that allows one to move continuously from the classical Smolyak formula ( ω = 0) to the fully adaptive algorithm ( ω = 1). Following [14], in the present work we have set ω = 0.9, that numerically has been proved to be a good performing choice.…”

Section: Numerical Tests On Sparse Gridsmentioning

confidence: 99%

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Implementation of optimal Galerkin and Collocation approximations of PDEs with Random Coefficients

et al. 2011

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Abstract. In this work we first focus on the Stochastic Galerkin approximation of the solution u of an elliptic stochastic PDE. We rely on sharp estimates for the decay of the coefficients of the spectral expansion of u on orthogonal polynomials to build a sequence of polynomial subspaces that features better convergence properties compared to standard polynomial subspaces such as Total Degree or Tensor Product.We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new effective class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids.Key words: Uncertainty Quantification, PDEs with random data, elliptic equations, multivariate polynomial approximation, Best M -Terms approximation, Stochastic Galerkin methods, Smolyak approximation, Sparse grids, Stochastic Collocation methods.AMS Subject Classification: 41A10, 65C20, 65N12, 65N35 IntroductionMany works have been recently devoted to the analysis and the improvement of the Stochastic Galerkin and Collocation techniques for Uncertainty Quantification for PDEs with random input data. These methods are promising since they can exploit the possible regularity of the solution with respect to the stochastic parameters to achieve much faster convergence than sampling methods like Monte Carlo.Stochastic Galerkin and Collocation methods can be classified as parametric techniques, since both expand u, the solution of the PDE of interest, as a summation over suitable deterministic basis functions in probability space, typically polynomials or piecewise polynomials. Stochastic Galerkin is a projection technique over a set of orthogonal polynomials with respect to the probability measure at hand (see e.g. [1,12,15,20]), while Stochastic Collocation is a sum of Lagrangian interpolants over the probability space (see e.g. [2,10,22] 11The comparison between performances of these deterministic methods is a matter of study (see e.g.[3]). However, both suffer the so-called "Curse of Dimensionality": using naive projections/interpolations over tensor product polynomials spaces/tensor grids leads to computational costs that grow exponentially fast with the number of input random variables. In such a case, careful construction of approximation spaces/sparse grids is needed in order to retain accuracy while keeping computational work acceptably low.In a Stochastic Galerkin setting this requirement can be translated to the implementation of algorithms able to compute what is known as "Best M -Terms approximation". In other words, the method should be able to establish a-priori the set of the M most fruitful multivariate polynomials in the spectral approximation of the solution u, and to compute only those terms.Important contributions in the study of the Best M -Terms approximation have been given by Cohen, DeVore and Schwab: estimates on the decay of the coefficients of the spectral expansi...

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Uncertainty Modeling Using Fuzzy Arithmetic Based on Sparse Grids: Applications to Dynamic Systems

Cited by 90 publications

References 12 publications

A Quasi-optimal Sparse Grids Procedure for Groundwater Flows

A Quasi-optimal Sparse Grids Procedure for Groundwater Flows

Toward learning autonomous pallets by using fuzzy rules, applied in a Conwip system

Implementation of optimal Galerkin and Collocation approximations of PDEs with Random Coefficients

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