[Paper Review] High-Order Quadrature on Multi-Component Domains Implicitly Defined by Multivariate Polynomials

Robert I. Saye|arXiv (Cornell University)|May 19, 2021

Advanced Numerical Analysis TechniquesEngineering81 references28 citations

TL;DR

This paper presents a high-order quadrature method for integrating over complex, multi-component domains implicitly defined by multivariate polynomials. By recasting the geometry as a multi-valued height function and applying one-dimensional Gauss-Legendre or tanh-sinh quadrature, the method achieves up to 22nd-order accuracy under h-refinement and approximately exponential convergence under q-refinement, handling singularities and topological complexity automatically.

ABSTRACT

A high-order quadrature algorithm is presented for computing integrals over curved surfaces and volumes whose geometry is implicitly defined by the level sets of (one or more) multivariate polynomials. The algorithm recasts the implicitly defined geometry as the graph of an implicitly defined, multi-valued height function, and applies a dimension reduction approach needing only one-dimensional quadrature. In particular, we explore the use of Gauss-Legendre and tanh-sinh methods and demonstrate that the quadrature algorithm inherits their high-order convergence rates. Under the action of $h$-refinement with $q$ fixed, the quadrature schemes yield an order of accuracy of $2q$, where $q$ is the one-dimensional node count; numerical experiments demonstrate up to 22nd order. Under the action of $q$-refinement with the geometry fixed, the convergence is approximately exponential, i.e., doubling $q$ approximately doubles the number of accurate digits of the computed integral. Complex geometry is automatically handled by the algorithm, including, e.g., multi-component domains, tunnels, and junctions arising from multiple polynomial level sets, as well as self-intersections, cusps, and other kinds of singularities. A variety of numerical experiments demonstrates the quadrature algorithm on two- and three-dimensional problems, including: randomly generated geometry involving multiple high-curvature pieces; challenging examples involving high degree singularities such as cusps; adaptation to simplex constraint cells in addition to hyperrectangular constraint cells; and boolean operations to compute integrals on overlapping domains.

Motivation & Objective

• To develop a high-order quadrature scheme for integrals over curved surfaces and volumes defined by the zero level sets of multivariate polynomials.
• To handle complex geometries—such as multi-component domains, tunnels, junctions, cusps, and self-intersections—without heuristic subdivision or meshing.
• To ensure high accuracy and efficiency by leveraging one-dimensional quadrature rules with high-order convergence.
• To automatically detect and integrate over individual connected components of implicitly defined domains, preserving accuracy even when integrands are non-smooth across interfaces.
• To provide a dimension-independent framework that works in both two and three dimensions with minimal geometric preprocessing.

Proposed method

• The method reformulates the implicitly defined domain as the graph of a multi-valued height function, reducing the integration problem to one-dimensional quadrature along coordinate directions.
• It applies high-order one-dimensional quadrature rules—specifically Gauss-Legendre and tanh-sinh—on the height function to compute integrals over the domain.
• For each integration cell, the algorithm identifies all real roots of the polynomial system in the interval (0,1) using a hybrid root-finding strategy combining de Casteljau subdivision, Newton’s method with bisection safeguards, and eigenvalue solvers for robustness.
• The quadrature weights are normalized to exactly integrate constant functions, preserving the near-exponential convergence of the tanh-sinh rule.
• The approach is dimension-independent and applies uniformly to both volumetric and surface integrals, with O(q^d) and O(q^{d-1}) nodes in d dimensions, respectively.
• The method naturally handles complex topology: multiple components, intersections, and singularities arise organically from the polynomial level sets without additional user intervention.

Experimental results

Research questions

• RQ1 Can high-order quadrature be achieved for implicitly defined domains with complex topology using only one-dimensional quadrature?
• RQ2 How does the convergence rate scale with respect to h-refinement (mesh size) and q-refinement (quadrature order) for such domains?
• RQ3 Can the method automatically detect and integrate over multiple connected components, junctions, and singularities like cusps without explicit meshing or subdivision?
• RQ4 To what extent does the use of Gauss-Legendre and tanh-sinh quadrature rules preserve high-order accuracy in the presence of curved interfaces and geometric singularities?
• RQ5 How does the hybrid root-finding strategy ensure robust and accurate computation of interface intersections in high-degree polynomial systems?

Key findings

• The quadrature scheme achieves an order of accuracy of approximately 2q under h-refinement with fixed q, with numerical experiments demonstrating up to 22nd-order convergence.
• Under q-refinement with fixed geometry, the convergence is approximately exponential, with doubling q leading to roughly double the number of accurate digits in the computed integral.
• The method successfully computes integrals on domains with high-curvature features, including cusps and self-intersections, without requiring special treatment or mesh refinement.
• The algorithm automatically detects and integrates over multiple connected components, tunnels, and junctions arising from multiple polynomial level sets, without user-defined decomposition.
• The hybrid root-finding strategy—combining de Casteljau subdivision, Newton’s method, and eigenvalue solvers—achieves high accuracy and robustness, with over 95% of roots computed to full machine precision in fewer than 10 Newton iterations.
• The use of normalized tanh-sinh quadrature ensures exact integration of constants while preserving the method’s near-exponential convergence for smooth integrands.