Gaussian quadrature and its applications

Songquan Liu, Purdue University

Abstract

Integration is a frequently used operator in mathematical models of economic systems. When these integrals do not have closed form solutions, it is necessary to resort to numerical methods to approximate them. The theory of Gaussian quadrature approximation of a single-fold integral is known in detail and is well developed using the theory of orthogonal polynomials. Although several integration formulas for multivariate integrals are known, the theory of these formulas is far from complete. The first goal of this dissertation is to develop general, practical algorithms to generate Gaussian quadrature formulas, which display desirable mathematical properties. In the simplest case of the computation of an expectation with respect to random variables which are independent and have symmetric density function over a hypercube domain, Degree 2 and 3 Gaussian quadrature formulas, which require virtually no computational effort, for any dimension n are constructed. Also, a degree 5 Gaussian quadrature formula for lower dimensions $(n < 16)$ is constructed for this case. Next, the assumption of symmetry is relaxed and a practical, general algorithm based on Cartesian products of univariate Gaussian quadrature and linear programming is developed. Finally, the most general case, where the probability density function may be dependent and the domain may be an arbitrary region within $R\sp{n}$, is considered. We develop a heuristic algorithm based on linear programming and the Monte Carlo method to generate Gaussian quadrature formulas, and for the case where integrand evaluations are particularly costly, another heuristic algorithm is developed which further reduces the number of points. The second part of this dissertation focuses on the application of Gaussian quadrature to two practical problems. The first application is known in the manufacturing management literature as the design centering problem. Design centering is an iterative process for determining settings for the design parameters of a manufacturing process in order to maximize product yield. By using the Gaussian quadrature method, we can efficiently and accurately estimate yield. This makes it tractable to search the feasible region of design parameters to obtain an optimal system of design parameters. The second application focuses on developing a systematic scheme to recover probability distributions as part of the sensitivity analysis of model results by using a combination of the Gaussian quadrature and Maximum Entropy methods. In the problem of estimation of the moments of model results where variability is driven by uncertainty regarding model parameters, DeVuyst (1992) finds that Gaussian quadrature is an effective tool. In this thesis, a technique is developed to recover the distribution of model results based on the moments estimated via DeVuyst's approach. This will further assist modelers in evaluating the sensitivity of results from their mathematical models and provide policy makers with additional qualitative information regarding the likely effect of policy alternatives.

Degree

Ph.D.

Advisors

Preckel, Purdue University.

Subject Area

Agricultural economics|Mathematics|Statistics

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