Optimized Gauss-Hermite Quadrature: A Refined Approach

For nonlinear problems involving Gaussian inte-grals, the Gauss-Hermite quadrature is preferred. The optimized Gauss-Hermite quadrature can outperform the conventional Gauss-Hermite quadrature through the optimal quadrature points and weights. However, obtaining the optimal quadrature points and weights presents challenges, mainly due to the complexity of constrained nonlinear least-squares fitting in an adaptive sampling. This paper introduces a novel approach to optimizing the Gauss-Hermite quadrature, distinguishing itself from prior works by addressing the optimization as an uncon-strained nonlinear least-squares problem. This modification not only simplifies the optimization problem but also enables the derivation of an exact gradient formula, significantly improving the employment of gradient-based optimization algorithms. Furthermore, we develop a deterministic grid strategy to bolster the method's efficacy. The superiority of the new approach is demonstrated via two numerical examples, showing its accuracy and efficiency.