用于岭近似的一个近稳态子空间
A near-stationary subspace for ridge approximation
AI总结：
本文提出一种近稳态子空间用于高维非线性模型的响应面近似，通过理论分析和实例验证了该方法的有效性。
摘要：
本文提出了一种近稳态子空间用于高维非线性模型的响应面近似，通过理论分析和实例验证了该方法的有效性。响应面常用于工程分析中替代昂贵的计算机模拟，但随着模型输入数量的增加，拟合准确响应面的成本呈指数增长，这使得在高维非线性模型中构建响应面变得不可行。我们描述了在多个变量中拟合响应面的岭近似方法。岭函数在其域内沿几个方向是常数，因此拟合发生在输入空间的低维子空间的坐标上。我们回顾了岭近似的基本理论——例如最佳均方逼近和最优低维子空间——并证明了基于梯度的主动子空间对于定义最优子空间的最小二乘问题是近稳态的。受理论启发，我们提出了一种计算启发式方法，利用估计的主动子空间作为岭近似拟合问题的初始猜测。我们展示了一个简单的例子，其中该启发式方法失效，这揭示了一种该方法不适用的函数类型。随后，我们提出了一种简单的交替启发式方法用于拟合岭函数，并展示了主动子空间初始猜测应用于机翼模型的阻力作为其18个形状参数函数时的有效性。
英文摘要：
Response surfaces are common surrogates for expensive computer simulations in engineering analysis. However, the cost of fitting an accurate response surface increases exponentially as the number of model inputs increases, which leaves response surface construction intractable for high-dimensional, nonlinear models. We describe ridge approximation for fitting response surfaces in several variables. A ridge function is constant along several directions in its domain, so fitting occurs on the coordinates of a low-dimensional subspace of the input space. We review essential theory for ridge approximation---e.g., the best mean-squared approximation and an optimal low-dimensional subspace---and we prove that the gradient-based active subspace is near-stationary for the least-squares problem that defines an optimal subspace. Motivated by the theory, we propose a computational heuristic that uses an estimated active subspace as an initial guess for a ridge approximation fitting problem. We show a simple example where the heuristic fails, which reveals a type of function for which the proposed approach is inappropriate. We then propose a simple alternating heuristic for fitting a ridge function, and we demonstrate the effectiveness of the active subspace initial guess applied to an airfoil model of drag as a function of its 18 shape parameters.
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