一种用于一维ridge函数近似与积分的Lanczos-Stieltjes方法
A Lanczos-Stieltjes method for one-dimensional ridge function approximation and integration
AI总结：
本文提出了一种用于一维ridge函数近似与积分的Lanczos-Stieltjes方法，通过重复卷积近似输入线性组合的单变量密度，并利用Lanczos-Stieltjes方法构造正交多项式和高斯求积公式来加速计算。
摘要：
许多计算科学中出现的输入参数到输出量的映射都表现出令人惊讶的低维结构，其中输出主要沿高维输入空间中的少数方向变化。这种结构可通过ridge函数建模，即低维线性变换与非线性函数的组合。如果目标是计算输出的统计特性（例如在不确定性量化或鲁棒设计中），则应利用这种低维结构来加速计算。我们开发了用于一维ridge函数的高斯求积及其相关的多项式近似。该方法的关键要素包括（i）通过重复卷积近似给定输入线性组合的单变量密度，以及（ii）利用Lanczos-Stieltjes方法构造正交多项式和高斯求积公式。
英文摘要：
Many of the input-parameter-to-output-quantity-of-interest maps that arise in computational science admit a surprising low-dimensional structure, where the outputs vary primarily along a handful of directions in the high-dimensional input space. This type of structure is well modeled by a ridge function, which is a composition of a low-dimensional linear transformation with a nonlinear function. If the goal is to compute statistics of the output (e.g., as in uncertainty quantification or robust design) then one should exploit this low-dimensional structure, when present, to accelerate computations. We develop Gaussian quadrature and the associated polynomial approximation for one-dimensional ridge functions. The key elements of our method are (i) approximating the univariate density of the given linear combination of inputs by repeated convolutions and (ii) a Lanczos-Stieltjes method for constructing orthogonal polynomials and Gaussian quadrature.
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