Computational Methods for Multivariate Orthogonal Polynomials

University Of Utah, Salt Lake City UT

Investigators

Abstract

Much of computational mathematics relies on the task of constructing computer-based models that predict a physical property, such as expected capacitance of an eletrochemical battery or power output of a wind turbine. A computationally stable way to construct such a computer model is to build new mathematical functions. This project aims to provide rigorous mathematical foundations and robust computational tools for generating polynomials and subsequently devise efficient algorithms for using these polynomials to build functions. The objectives of this project provide insight into the related mathematical fields of analysis and applied approximation theory, and aid computational scientists in building robust simulation models. Approximations built from an expansion in orthogonal polynomials are classical tools in applied mathematics. These approximations are frequently the bedrock of algorithms for computationally solving differential and integral equations. The generation and manipulation of such expansions in one variable has been the subject of a great deal of theoretical and computational research in past decades, and there are constructive algorithms for most problems of interest. Far less is known in the multivariate non-tensorial case, for which there are only a few restrictive computational tools for generation of multivariate orthogonal polynomials. The research of this project focuses on theoretical development and computational implementation of methods for generating multivariate orthogonal polynomials on non-tensorial domains with non-tensorial weights. Mathematical investigations of this project involve fundamental contributions to the theory of multivariate orthogonal polynomials, and design of robust algorithms for generation of multivariate orthogonal polynomials. From a practical standpoint the algorithms and methodologies in this project produce expansions in an orthogonal series and will be useful in various engineering design, optimization, and reliability contexts.