Numerical Integration in Multiple Dimensions with Designed Quadrature

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.

Cited by in corpus (14)

• Polynomial chaos expansions for dependent random variables
• Tensors in computations
• Parametric Topology Optimization with Multi-Resolution Finite Element Models
• Multifidelity uncertainty quantification with models based on dissimilar parameters
• Stable high-order cubature formulas for experimental data
• Efficient sampling for polynomial chaos-based uncertainty quantification and sensitivity analysis using weighted approximate Fekete points
• Schrödinger Approach to Optimal Control of Large-Size Populations
• Construction and application of provable positive and exact cubature formulas
• Generating nested quadrature rules with positive weights based on arbitrary sample sets
• Variational Inference for Nonlinear Inverse Problems via Neural Net Kernels: Comparison to Bayesian Neural Networks, Application to Topology Optimization
• Graph-accelerated non-intrusive polynomial chaos expansion using partially tensor-structured quadrature rules for uncertainty quantification
• An Efficient Solver for Cumulative Density Function-based Solutions of Uncertain Kinematic Wave Models
• Efficient numerical evaluation of thermodynamic quantities on infinite (semi-)classical chains
• Efficient Numerical Evaluation of Triple Integral Using the Euler's Method and Richardson's Extrapolation