Numerical Integration in Multiple Dimensions with Designed Quadrature
For researchers in numerical integration and computational science, this provides a systematic method for designing quadrature rules on arbitrary domains with stability guarantees.
This work presents a computational framework for generating positive quadrature rules in multiple dimensions on general geometries, using moment-matching and Gauss-Newton optimization. The method achieves accurate integration in up to 100 dimensions and outperforms sparse grids and quasi-Monte Carlo in linear elasticity and topology optimization problems.
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.