Adaptive Leja Sparse Grid Constructions for Stochastic Collocation and High-Dimensional Approximation
Adaptive Leja Sparse Grid Constructions for Stochastic Collocation and High-Dimensional Approximation
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DOI：
10.1137/140966368
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发表时间：
2014-04
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影响因子：
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通讯作者：
A. Narayan;J. Jakeman
中科院分区：
文献类型：
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作者：
A. Narayan;J. Jakeman
我们提出了一种基于 Leja 插值序列的自适应稀疏网格随机搭配方法，用于逼近具有高维参数的参数化函数。 Leja 序列是任意粒度的（可以将任意数量的节点添加到当前序列，生成新序列），因此对于用于构造高维自适应稀疏网格的单变量复合规则来说是一个不错的选择。在进行随机搭配时，人们通常对构造加权近似感兴趣，其中权重由随机变量的概率密度确定。本文建立了一维 Leja 序列的某种加权公式，产生一个节点序列，其经验分布收敛于与权重函数相关的高斯求积节点的相应极限分布。即使对于无界域，此属性也是如此。我们将 Leja 稀疏网格方法应用于几个高维问题并且......
We propose an adaptive sparse grid stochastic collocation approach based upon Leja interpolation sequences for approximation of parameterized functions with high-dimensional parameters. Leja sequences are arbitrarily granular (any number of nodes may be added to a current sequence, producing a new sequence) and thus are a good choice for the univariate composite rule used to construct adaptive sparse grids in high dimensions. When undertaking stochastic collocation one is often interested in constructing weighted approximation where the weights are determined by the probability densities of the random variables. This paper establishes that a certain weighted formulation of one-dimensional Leja sequences produces a sequence of nodes whose empirical distribution converges to the corresponding limiting distribution of the Gauss quadrature nodes associated with the weight function. This property is true even for unbounded domains. We apply the Leja sparse grid approach to several high-dimensional problems and...