The Algebraic Method in Quadrature for Uncertainty Quantification

Abstract

Abstract. A general method of quadrature for uncertainty quantification (UQ) is introduced based on the algebraic method in experimental design. This is a method based on the theory of zero dimensional algebraic varieties. It allows quadrature of polynomials or polynomial approximands for quite general sets of quadrature points, here called 'designs'. The method goes some way to explaining when quadrature weights are non-negative and gives exact quadrature for monomials in the quotient ring defined by the algebraic method. The relationship to the classical methods based on zeros of orthogonal polynomials is discussed and numerical comparisons made with methods such as Gaussian quadrature and Smolyak grids. Application to UQ is examined in the context of polynomial chaos expansion and probabilistic collocation method where solution statistics are estimated.