Tensor Products of Sobolev Spaces and Applications
Tensor Products of Sobolev Spaces and Applications
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Sobolev空间的张量积及其应用
DOI：
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发表时间：
2007
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通讯作者：
Gerhard ZUMBUSCHxAbstract
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作者：
A. Hochmuth;Stephan KNAPEKx;Gerhard ZUMBUSCHxAbstract
在许多情况下，涉及各向同性Sobolev空间的变分问题的近似解具有指数依赖于维数的复杂性。然而，如果解具有控制混合导数，则可以以较低的复杂性{有时甚至与维数无关}将其离散化到相应的变分问题。为了分析这些效应，我们将Sobolev空间的张量积与具有控制混合导数的空间联系起来。基于这些考虑，我们构造了nite维各向异性近似空间的家庭，特别是稀疏网格的推广。得到的估计表明，在这种情况下，可以预期的复杂性独立或几乎独立的尺寸。最后，数值实验表明建议的近似空间的有效性。
In many cases the approximation of solutions to variational problems involving isotropic Sobolev spaces has a complexity which depends exponentially on the dimension. However, if the solutions possess dominating mixed derivatives one can nd discretizations to the corresponding variational problems with a lower complexity { sometimes even independent of the dimension. In order to analyse these eeects, we relate tensor products of Sobolev spaces with spaces with dominating mixed derivatives. Based on these considerations we construct families of nite dimensional anisotropic approximation spaces which generalize in particular sparse grids. The obtained estimates demonstrate, in which cases a complexity independent or nearly independent of the dimension can be expected. Finally numerical experiments demonstrate the usefulness of the suggested approximation spaces.