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

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Title: Archimedes, Gauss and Stochastic computation: A new (old) approach to Fast Algorithms for the evaluation of transcendental functions of generalized Polynomial Chaos Expansions
Author: Mckale, Kaleb D.
Abstract: 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 .
URI: http : / /hdl .handle .net /2346 /ETD -TTU -2011 -05 -1480
Date: 2011-05

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Archimedes, Gauss and Stochastic computation: A new (old) approach to Fast Algorithms for the evaluation of transcendental functions of generalized Polynomial Chaos Expansions. Master's thesis, Texas Tech University. Available electronically from http : / /hdl .handle .net /2346 /ETD -TTU -2011 -05 -1480 .

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