New techniques for calculating cubature rules on the triangle

Numerical integration is a fundamental technique of numerical analysis, with applications across science and engineering. This talk starts with a general overview of cubature (multivariate numerical integration) and then proceeds to describe three recent techniques to obtain cubature rules on the triangle. The first technique exploits the two types of invariance in fully symmetric rules to reduce the problem to one amenable to analytical solution. The two other techniques are numerical, involving either minimal orthonormal bases or variable projection. In all three cases, the new techniques have allowed the computation of new rules of practical and theoretical interest.