Computing periods of rational integrals

A period of a rational integral is the result of integrating, with respect to one or several variables, a rational function over a closed path. This work focuses particularly on periods depending on a parameter: in this case the period under consideration satisfies a linear differential equation, the Picard-Fuchs equation. I give a reduction algorithm that extends the Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs equations. The resulting algorithm is elementary and has been successfully applied to problems that were previously out of reach.Comment: To appear in Math. comp. Supplementary material at http://pierre.lairez.fr/supp/periods

The algebraic method in quadrature for uncertainty quantification

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 nonnegative 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 are made with methods such as Gaussian quadrature and Smolyak grids. Application to UQ is examined in the context of polynomial chaos expansion and the probabilistic collocation method, where solution statistics are estimated