Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients
Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients
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参数椭圆偏微分方程的稀疏多项式逼近。
DOI：
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发表时间：
2015
期刊：
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通讯作者：
G. Migliorati
中科院分区：
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作者：
M. Bachmayr;A. Cohen;G. Migliorati
考虑扩散系数依赖于可数个参数的椭圆型偏微分方程。我们研究了多项式展开的函数映射参数值的PDE的解决方案，考虑泰勒和勒让德级数的可和性。我们的结果比之前已知的此类估计有了显着改进，特别是考虑到系数仿射参数化的结构特征。此外，结果结转到更一般的Jacobi多项式展开。我们证明了新的界限是尖锐的，在某些模型的情况下，我们说明了它们的数值实验。
We consider elliptic partial differential equations with diffusion coefficients that depend affinely on countably many parameters. We study the summability properties of polynomial expansions of the function mapping parameter values to solutions of the PDE, considering both Taylor and Legendre series. Our results considerably improve on previously known estimates of this type, in particular taking into account structural features of the affine parametrization of the coefficient. Moreover, the results carry over to more general Jacobi polynomial expansions. We demonstrate that the new bounds are sharp in certain model cases and we illustrate them by numerical experiments.