Numerical Integration in Multiple Dimensions with Designed Quadrature

TL;DR

This paper introduces a systematic computational framework for generating positive quadrature rules in multiple dimensions, addressing geometric constraints and providing stability bounds, with applications to high-dimensional problems and comparisons to existing methods.

Contribution

A novel direct moment-matching approach with penalty methods and quadratic minimization for constructing positive multivariate quadrature rules on general geometries.

Findings

Effective quadrature rules for polynomial subspaces

Analysis of Lebesgue constants and rule stability

Successful high-dimensional quadrature in 100 dimensions

Abstract

We present a systematic computational framework for generating positive quadrature rules in multiple dimensions on general geometries. A direct moment-matching formulation that enforces exact integration on polynomial subspaces yields nonlinear conditions and geometric constraints on nodes and weights. We use penalty methods to address the geometric constraints, and subsequently solve a quadratic minimization problem via the Gauss-Newton method. Our analysis provides guidance on requisite sizes of quadrature rules for a given polynomial subspace, and furnishes useful user-end stability bounds on error in the quadrature rule in the case when the polynomial moment conditions are violated by a small amount due to, e.g., finite precision limitations or stagnation of the optimization procedure. We present several numerical examples investigating optimal low-degree quadrature rules, Lebesgue constants, and 100-dimensional quadrature. Our capstone examples compare our quadrature approach to popular alternatives, such as sparse grids and quasi-Monte Carlo methods, for problems in linear elasticity and topology optimization.