自适应 Smolyak 伪谱逼近
Adaptive Smolyak Pseudospectral Approximations
AI总结：
提出一种基于广义稀疏网格的 Smolyak 自适应伪谱逼近方法，通过避免内部混叠和贪心启发式自适应细化，有效利用稀疏函数评估，提高计算密集型模型的不确定性量化精度。
摘要：
计算密集型模型的多项式逼近是不确定性量化的核心。本文描述了一种基于 Smolyak 算法和广义稀疏网格的非侵入式伪谱逼近的自适应方法。我们严格分析并扩展了文献[6]中提出的非自适应方法，并将其与另一种常用的稀疏网格构造多项式逼近方法——直接求积法进行了比较。对直接求积法的分析表明，由于内部混叠，O(1)误差是该方法的某些配置的内在属性。我们基于所选多项式基和求积规则，给出了这种混叠误差发生的精确条件。随后，我们建立了 Smolyak 伪谱逼近精度的理论结果，并表明 Smolyak 逼近避免了内部混叠，且更有效地利用了稀疏函数评估。这些结果适用于广泛的求积规则和广义稀疏网格。利用这种灵活性，我们引入了一种贪心启发式方法，用于伪谱逼近的自适应细化。我们在 Genz 测试函数上数值验证了该算法的收敛性，并在一个实际的化学动力学问题上展示了自适应方法的准确性和效率。
英文摘要：
Polynomial approximations of computationally intensive models are central to uncertainty quantification. This paper describes an adaptive method for non-intrusive pseudospectral approximation, based on Smolyak's algorithm with generalized sparse grids. We rigorously analyze and extend the non-adaptive method proposed in [6], and compare it to a common alternative approach for using sparse grids to construct polynomial approximations, direct quadrature. Analysis of direct quadrature shows that O(1) errors are an intrinsic property of some configurations of the method, as a consequence of internal aliasing. We provide precise conditions, based on the chosen polynomial basis and quadrature rules, under which this aliasing error occurs. We then establish theoretical results on the accuracy of Smolyak pseudospectral approximation, and show that the Smolyak approximation avoids internal aliasing and makes far more effective use of sparse function evaluations. These results are applicable to broad choices of quadrature rule and generalized sparse grids. Exploiting this flexibility, we introduce a greedy heuristic for adaptive refinement of the pseudospectral approximation. We numerically demonstrate convergence of the algorithm on the Genz test functions, and illustrate the accuracy and efficiency of the adaptive approach on a realistic chemical kinetics problem.
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