Speaker: Dr. Grady Wright (Boise State University)

 

Title: A new framework for numerical integration 

 

 

Abstract: Numerical integration, or quadrature, is ubiquitous in mathematics, statistics, science, and engineering, with a history dating back to the ancient Babylonians. A standard approach to generating quadrature formulas is to pick a "nice" vector space of functions for which the formulas are exact, such as algebraic or trigonometric polynomials. For integration over intervals, this approach gives rise to Newton-Cotes and Gaussian quadrature rules. However, for geometrically complex domains in higher dimensions, this exactness approach can be challenging, if not impossible since it requires being able to exactly integrate basis functions for the vector space over the domains (or some collection of subdomains). Another challenge with determining good quadrature formulas arises when the integrand is not given everywhere over the domain, but only as samples at predefined, possibly "scattered" points (i.e., a point cloud), which is common in applications involving experimental measurements or when quadrature is a secondary operation to some larger endeavor. In this talk we introduce a new framework for generating quadrature formulas that bypasses these challenges. The framework only relies on numerical approximations of certain Laplace operators and on linear algebra. We show how several classic univariate quadrature formulas can arise from this framework and demonstrate its applicability to generating accurate quadrature formulas for geometrically complex domains (including surfaces) discretized with point clouds.

 

 

Host: Ash Abebe