An adaptive minimum spanning tree multielement method for uncertainty quantification of smooth and discontinuous responses

A novel approach for nonintrusive uncertainty propagation is proposed. Our approach overcomes the limitation of many traditional methods, such as generalized polynomial chaos methods, which may lack sufficient accuracy when the quantity of interest depends discontinuously on the input parameters. As a remedy we propose an adaptive sampling algorithm based on minimum spanning trees combined with a domain decomposition method based on support vector machines. The minimum spanning tree determines new sample locations based on both the probability density of the input parameters and the gradient in the quantity of interest. The support vector machine efficiently decomposes the random space in multiple elements, avoiding the appearance of Gibbs phenomena near discontinuities. On each element, local approximations are constructed by means of least orthogonal interpolation, in order to produce stable interpolation on the unstructured sample set. The resulting minimum spanning tree multielement method does not require initial knowledge of the behavior of the quantity of interest and automatically detects whether discontinuities are present. We present several numerical examples that demonstrate accuracy, efficiency, and generality of the method.

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techReport An adaptive minimum spanning tree multi-element method for uncertainty quantification of smooth and discontinuous responses Y. van Halder (Yous), B. Sanderse (Benjamin) and B. Koren (Barry)