Open Access

ARTICLE

Multidirectional Gaussian Mixture Models for Nonlinear Uncertainty Propagation

V. Vittaldev¹, R. P. Russell²

1 Ph.D. Candidate, The University of Texas at Austin, Austin, TX, USA. E-mail: v.vittaldev@utexas.edu
2 Associate Professor, The University of Texas at Austin, Austin, TX, USA. E-mail: ryan.russell@utexas.edu

Computer Modeling in Engineering & Sciences 2016, 111(1), 83-117. https://doi.org/10.3970/cmes.2016.111.083

Abstract

Monte Carlo simulations are an accurate but computationally expensive procedure for approximating the resultant non-Gaussian probability density function (PDF) after propagation of an initial Gaussian PDF through a nonlinear function. Univariate splitting libraries for Gaussian Mixture Models (GMMs) exist with up to five elements in the literature. The number of splits are extended in the present work by generating three homoscedastic univariate splitting libraries with up to 39 elements. Mulitvariate GMMs are typically handled with splits along a single direction. Instead, we generate a regular multidirectional grid over the initial multivariate Gaussian distribution by recursively applying the splitting library along multiple directions. The splitting direction is arbitrary and no longer limited to directions parallel to the columns of the square-root of the covariance matrix. A second order Stirling's interpolation of the nonlinear function evaluated at the mean of the initial Gaussian distribution is used to quantify nonlinearity along candidate splitting directions. The directions with the highest nonlinearity benefit most from splitting. The Multidirectional GMM (MGMM) has applications for uncertainty quantification with computationally intensive nonlinear functions. The variable number of splits in each direction allows for a spectrum of models in the accuracy versus compute time design space, filling the gap between expensive Monte Carlos and fast linearized models. The multidirectional method is demonstrated with four test cases, including an orbit uncertainty propagation case, to illustrate the benefit of splitting along multiple directions and of ranking the splitting directions.

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