QuadratureOnImplicitRegions

This package implements a quadrature method on implicitly defined regions following the algorithm in:

R. I. Saye, High-Order Quadrature Methods for Implicitly Defined Surfaces and Volumes in Hyperrectangles, SIAM Journal on Scientific Computing, 37(2), A993-A1019 (2015)..

---

Examples:

Let $\Omega=[0,1]^2$ and let $\psi(x,y)=x^2+y^2-1$. Our goal is to create quadrature nodes and weights on the subdomains:

$$\Omega_1=\left\{(x,y)\in \Omega : \psi(x,y)<0\right\},\qquad \Omega_2=\left\{(x,y)\in \Omega : \psi(x,y)>0\right\}.$$

using QuadratureOnImplicitRegions

ψ(x)= x'*x-1.0 
a,b=zeros(2), ones(2) #the unit interval. 
quad_order=10

#the nodes and weights on Ω₁
xy1,w1=algoim_nodes_weights(ψ,-1.0, a,b,quad_order)
#the nodes and weights on Ω₂
xy2,w2=algoim_nodes_weights(ψ,+1.0, a,b,quad_order)

To plot the nodes, please see this tutorial.

The same syntax can be used for higher dimensional regions. For example, in the case of the intersection of the unit sphere and unit cube, we only need to adjust a and b:

using QuadratureOnImplicitRegions

ψ(x)= x'*x-1.0 
a,b=zeros(3), ones(3) #the unit cube. 
quad_order=5 

xyz1,w1=algoim_nodes_weights(ψ,-1.0, a,b,quad_order)

For the outer region, we only need to change -1.0 to 1.0

---

Roadmap

In the future, I plan on adding the following features/improvements. Suggestions and/or contributions are appreciated.

• Add support for regions defined as $\psi(\mathbf{x})=0$ (e.g. sphere in $\mathbb{R}^3$).
• Performance improvement: See the v0.2.0 release notes for a comparison with the previous release. This is still far from optimal.
• Add the option to integrate a function $f$ without the need to store the integration nodes. This is useful when the nodes are not re-used.
• Adaptivity? It would be cool to have a QuadGK-like function, where the error is estimated using a Gauss-Konrod rule.