Archimedes, Gauss and Stochastic computation: A new (old) approach to Fast Algorithms for the evaluation of transcendental functions of generalized Polynomial Chaos Expansions

In this paper, we extend the work of Debusschere et al. (2004) by introducing a new approach to evaluating transcendental functions of generalized polynomial chaos expansions. We derive the elementary algebraic operations for the generalized PC expansions and show how these operations can be extended to polynomial and rational functions of PC expansions. We introduce and implement the Borchardt-Gauss Algorithm, an Arithmetic-Geometric Mean (AGM)-type method to derive the arctangent for the Jacobi-Chaos expansion. We compare numerically the BG Algorithm versus the Line Integral Method of Debusschere et al. and the Non-intrusive Spectral Projection (NISP) Method. We present the future direction of our research, including incorporating more efficient AGM-type methods proposed by Carlson (1972) and Brent (1976) to calculate the arctangent and other transcendental functions.