Sparse grid

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

Uncertainty quantification of EM-circuit systems using stochastic polynomial chaos method

Abstract: Uncertainties in realistic lumped and distributive circuit systems are of great importance to today’s high yield manufacture demand. However, evaluating the stochastic effect in the time domain for the hybrid electromagnetics (EM)-circuit system was seldom done, especially when Monte Carlo is too expensive to be feasible. In this work, an adaptive hierarchical sparse grid collocation (ASGC) method is presented to quantify the impacts of stochastic inputs on hybrid electromagnetics (EM)-circuit or EM scattering systems. The ASGC method approximates the stochastic observables of interest using interpolation functions over series collocation points. Instead of employing a full-tensor product sense, the collocation points in ASGC method are hierarchically marched with interpolation depth based upon Smolyaks construction algorithm. To further reduce the collocation points, an adaptive scheme is employed by using hierarchical surplus of each collocation point as the error indicator. With the proposed method, the number of collocation points is significantly deduced. To verify the effectiveness and robustness of the proposed stochastic solver, hybrid EM-circuit systems are quantified by a full-wave EM-circuit simulator based upon discontinuous Galerkin time domain (DGTD) method and modified nodal analysis (MNA). The time domain influences of uncertainty inputs such as geometrical information and electrical material properties are thereby benchmarked and demonstrated through this paper

The application of sparse grid quadrature in solving stochastic optimisation problems

Abstract: Stochastic optimisation problems minimise expectations of random cost functions. Thus they require accurate quadrature methods in order to evaluate the objective. Promising methods based on sparse grids were shown to display high quadrature accuracy for smooth integrands. But they have negative quadrature weights which potentially destroy the convexity of the objective and thus may lead to totally wrong results. We prove here that, due to their high accuracy, sparse grids maintain the convexity of the objective for sufficiently fine grids. An application to optimal control demonstrates the superiority of sparse grids over Monte Carlo and product rule based approaches. References D. P. Bertsekas. Dynamic programming and optimal control. Vol. I . Athena Scientific, 2005. URL http://athenasc.com/dpbook.html . H.-J. Bungartz and M. Griebel. Sparse grids. Acta Numer. , 13:147–269, 2004. doi:10.1017/S0962492904000182 . M. Chen, S. Mehrotra, and D. Papp. Scenario generation for stochastic optimization problems via the sparse grid method. Comput. Optim. Appl. , 62(3):669–692, 2015. doi:10.1007/s10589-015-9751-7 . C. W. Clenshaw and A. R. Curtis. A method for numerical integration on an automatic computer. Numer. Math. , 2:197–205, 1960. doi:10.1007/BF01386223 . P. J. Davis and P. Rabinowitz. Methods of numerical integration . Computer Science and Applied Mathematics. Academic Press, 1984. doi:10.1016/C2013-0-10566-1 . T. Gerstner and M. Griebel. Numerical integration using sparse grids. Numer. Algorithms , 18:209–232, 1998. doi:10.1023/A:1019129717644 . M. Holtz. Sparse grid quadrature in high dimensions with applications in finance and insurance , volume 77 of Lecture Notes in Computational Science and Engineering . Springer-Verlag, 2011. doi:10.1007/978-3-642-16004-2 . B. Oksendal. Stochastic differential equations . Springer-Verlag, 1998. doi:10.1007/978-3-662-03620-4 . T. N. L. Patterson. The optimum addition of points to quadrature formulae. Math. Comp. , 22:847–856, 1968. doi:10.2307/2004583 . S. W. Wallace and W. T. Ziemba. Applications of stochastic programming . MOS-SIAM Series on Optimization. SIAM, 2005. doi:10.1137/1.9780898718799 .

Comparing multi-index stochastic collocation and multi-fidelity stochastic radial basis functions for forward uncertainty quantification of ship resistance

Abstract: Abstract This paper presents a comparison of two multi-fidelity methods for the forward uncertainty quantification of a naval engineering problem. Specifically, we consider the problem of quantifying the uncertainty of the hydrodynamic resistance of a roll-on/roll-off passenger ferry advancing in calm water and subject to two operational uncertainties (ship speed and payload). The first four statistical moments (mean, variance, skewness, and kurtosis), and the probability density function for such quantity of interest (QoI) are computed with two multi-fidelity methods, i.e., the Multi-Index Stochastic Collocation (MISC) and an adaptive multi-fidelity Stochastic Radial Basis Functions (SRBF). The QoI is evaluated via computational fluid dynamics simulations, which are performed with the in-house unsteady Reynolds-Averaged Navier–Stokes (RANS) multi-grid solver $$\chi$$ χ navis. The different fidelities employed by both methods are obtained by stopping the RANS solver at different grid levels of the multi-grid cycle. The performance of both methods are presented and discussed: in a nutshell, the findings suggest that, at least for the current implementation of both methods, MISC could be preferred whenever a limited computational budget is available, whereas for a larger computational budget SRBF seems to be preferable, thanks to its robustness to the numerical noise in the evaluations of the QoI.