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Sparse pseudospectral approximation method

机译:稀疏伪谱近似方法

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Multivariate global polynomial approximations - such as polynomial chaos or stochastic collocation methods - are now in widespread use for sensitivity analysis and uncertainty quantification. The pseudo-spectral variety of these methods uses a numerical integration rule to approximate the Fourier-type coefficients of a truncated expansion in orthogonal polynomials. For problems in more than two or three dimensions, a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation. However, when using a sparse rule to approximately integrate these coefficients, one often finds unacceptable errors in the coefficients associated with higher degree polynomials. By reexamining Smolyak's algorithm and exploiting the connections between interpolation and projection in tensor product spaces, we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients for basis functions that naturally correspond to the sparse grid integration rule. The compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation.
机译:多元全局多项式逼近(例如多项式混沌或随机搭配方法)现已广泛用于敏感性分析和不确定性量化。这些方法的伪谱变化使用数值积分规则来近似正交多项式中的截断展开的傅立叶型系数。对于超过二维或三维的问题,与张量积近似相比,稀疏网格数值积分规则可提供精度更高的节点集。但是,当使用稀疏规则近似积分这些系数时,通常会发现与较高阶多项式相关的系数存在不可接受的误差。通过重新检查Smolyak算法并利用张量积空间中插值和投影之间的联系,我们构造了一种稀疏伪谱逼近方法,该方法可以准确地再现自然对应于稀疏网格积分规则的基函数的系数。令人信服的数值结果表明,这是将稀疏网格积分规则用于伪谱近似的正确方法。

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