用蒙特卡洛方法计算主动子空间
Computing active subspaces with Monte Carlo
AI总结：
本文研究了用蒙特卡洛方法近似主动子空间矩阵的特征对，结合理论分析和实证方法，探讨了梯度样本数量对精度的影响及子空间估计的理论与实际准确性。
摘要：
主动子空间能有效降低高维参数研究的维度，使昂贵模拟的实验变得可行。主动子空间方法的关键组件是正定矩阵的特征向量，其元素是模拟输入输出映射的偏导数的平均乘积。本文研究了一种蒙特卡洛方法来近似该矩阵的特征对。我们基于最近的非渐近随机矩阵理论提供了理论结果，并基于自助法提出了实用方法。我们扩展分析到梯度被有限差分近似的情况。我们的目标是为主动子空间中出现的两个问题提供指导：(i) 需要多少梯度样本才能准确近似特征值和子空间？(ii) 估计的子空间的精度在理论上和实践中如何？我们在简单的二次函数和具有100个变量的参数化PDE上测试了该方法。
英文摘要：
Active subspaces can effectively reduce the dimension of high-dimensional parameter studies enabling otherwise infeasible experiments with expensive simulations. The key components of active subspace methods are the eigenvectors of a symmetric, positive semidefinite matrix whose elements are the average products of partial derivatives of the simulation's input/output map. We study a Monte Carlo method for approximating the eigenpairs of this matrix. We offer both theoretical results based on recent non-asymptotic random matrix theory and a practical approach based on the bootstrap. We extend the analysis to the case when the gradients are approximated, for example, with finite differences. Our goal is to provide guidance for two questions that arise in active subspaces: (i) How many gradient samples does one need to accurately approximate the eigenvalues and subspaces? (ii) What can be said about the accuracy of the estimated subspace, both theoretically and practically? We test the approach on both simple quadratic functions where the active subspace is known and a parameterized PDE with 100 variables characterizing the coefficients of the differential operator.
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