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Gauss-Hermite filter
Published in Shovan Bhaumik, Paresh Date, Nonlinear Estimation, 2019
sparse-grid GHFWe have seen in the previous discussion that the GHF suffers from the curse of dimensionality problemcurse of dimensionality problem. This means that the number of quadrature points increases exponentially with the dimension of the system. More specifically, the problem arises due to the multiplicative association between the 1D quadrature points. Instead of a multiplicative association, if we use the Smolyak’s rule to generate multidimensional quadrature points, almost similar accuracy could be obtained from a smaller number of support points. When an estimator uses Smolyak’s rule Smolyak’s rule [53, 66] to generate quadrature points, the filtering algorithm is known as a Sparse-grid Gauss-Hermite filter (SGHF) [79, 80]. As the number of quadrature points is smaller with Smolyak’s rule (compared to the multiplicative rule) the execution time of SGHF is accordingly less than that of the GHF.
Application Uncertainty Propagation
Published in Bin Jia, Ming Xin, Grid-based Nonlinear Estimation and Its Applications, 2019
The multi-element grid (MEG) is inspired by the multi-element generalized polynomial chaos method (ME-gPC) (Wan and Karniadakis 2006). The MEG is one of the grid-based propagation methods that is essentially different from the ME-gPC. In the MEG, the support space S is decomposed into many subspaces Sk, i.e., S=⋃k=1NdSk, where Sk = (ak,1, bk,1) × ⋯ × (ak,n, bk,n) and Sk1⋂Sk2=∅, if k1 ≠ k2. Note that k, k1, k2 = 1, ⋯, Nd. Sk is referred to as ‘element’ and Nd is the number of elements. The quadrature points, as well as the corresponding weights, are obtained for each subspace Sk and directly used in uncertainty propagation. Compared with the ME-gPC, MEG is straightforward and easy to implement. After decomposition, the points are generated for each element by the tensor product rule or the sparse-grid method when the univariate quadrature points/weights are available. We assume the number of decompositions for dimension m is denoted as Nm. Then the total number of elements is Nd=∏m=1nNm, if the tensor product rule is used.
Stochastic coordinate-exchange optimal designs with complex constraints
Published in Quality Engineering, 2019
Many design methods, such as factorial experimental designs, orthogonal arrays, and Latin hypercube designs, discretize the design space. But it is not ideal for the non-box constrained design space. One obvious drawback is that it usually leads to a sub-optimal solution. Moreover, discretizing the non-box shaped space is not straightforward, especially when the design space is not a polyhedron. A direct solution is to ignore the non-box constraints and discretize the space using mesh grid, and then check the feasibility of the grid points. The optimal design points are to be selected from the candidate set of the feasible grid points. As simple as this solution is, users have to face the dilemma of choosing between a dense grid and a sparse grid. Using a dense grid leads to a better solution by paying the price of more computational costs, whereas a sparse grid requires less computation, but the design can be far from optimal.
Impact of Kinetic Uncertainties on Accurate Prediction of NO Concentrations in Premixed Alkane-Air Flames
Published in Combustion Science and Technology, 2019
Antoine Durocher, Philippe Versailles, Gilles Bourque, Jeffrey M. Bergthorson
Sparse grids were proposed by Smolyak (1963) for use in high-dimensional problems to provide similar accuracy as full tensor-product expansions, while requiring significantly fewer quadrature points. By removing multivariate high-order terms, only a subset of the full tensor-product expansion is retained. Figure 3(c) illustrates the sparsity of the approach in two dimensions by comparison to the full tensor-product expansion presented in Figure 3(b). The quadrature rule used for sparse grids at the level is defined by
Nonlinear magnetoquasistatic interface problem in a permanent-magnet machine with stochastic partial differential equation constraints
Published in Engineering Optimization, 2019
In this work, for the approximative computation of the coefficient functions , the stochastic collocation method is used. This technique represents a non-intrusive approach, which allows a deterministic solver to be reused. In general, it employs a sampling scheme (Morokoff and Caflisch 1995; Xiu and Hesthaven 2005; Babuška, Nobile, and Tempone 2007) or a multi-dimensional quadrature formula (Xiu 2007) in order to approximate the probabilistic integrals (15), i.e. In fact, the application of any quadrature formula requires the definition of a set of quadrature points and a set of weights , for . For this reason, the Gauss quadrature technique, the sparse grid approach or cubature formulas can be applied, see for example Xiu (2010). Then, the approximation of (19) reads as In the pseudo-spectral approach of Xiu (2007) used in this work, the coefficient functions of (18) are calculated using (20) with the Stroud formulas (Xiu and Hesthaven 2005) for the discrete projections of provided solutions at collocation nodes, i.e. onto the basis polynomials . Here, denotes the differential operator given by (11), while a random source term is denoted by . Specifically, the type of cubature formulas used is exact for multivariate polynomials up to the degree , e.g.K=2Q for and for . They are highly efficient, especially in high-dimensional spaces () (Xiu and Hesthaven 2005). However, their accuracy is fixed and cannot be improved. Instead, a sparse grids approach based on the Smolyak algorithm can be applied (Nobile, Tempone, and Webster 2008).