Performance of cubature formulae in probabilistic model analysis and optimization

Abstract

In probabilistic model analysis and optimization, expected values of a model output f(x) in face of continuous random inputs x are estimated through n-dimensional integrals, where n=dim(x). Cubature formulae are approximations of these integrals by a weighted sum of function evaluations at carefully chosen points. When each function evaluation corresponds to a heavy computational simulation, and particularly in optimization problems, one needs very efficient formulae with few integration points, even though only having modest accuracy. In this paper, we evaluate the performance of several cubature formulae with few points, including Smolyak type formulae, also known as sparse grid integration, and recently proposed thinned cubatures, constructed using orthogonal arrays. Tests are made for a wide family of smooth and non-oscillatory functions f(x), possibly with significant anisotropy, and covering both normal and uniform input probability distributions. Two practical case studies are also presented, one of analysis of a large scale mass transfer model with uncertain parameters and a second one of optimal production planning under uncertain market conditions. Results clearly indicate that cubatures with large negative weights, including Smolyak type formulae, are not reliable, contrary to positive thinned cubatures that produce very reasonable estimates up to dimension 24. These thinned cubatures may also surpass quasi-Monte Carlo methods also up to dimension 24.