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Tensor-Sparsity of Solutions to High-Dimensional Elliptic Partial Differential Equations

机译：高维椭圆型偏微分方程解的张量稀​​疏性

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A recurring theme in attempts to break the curse of dimensionality in the numerical approximation of solutions to high-dimensional partial differential equations (PDEs) is to employ some form of sparse tensor approximation. Unfortunately, there are only a few results that quantify the possible advantages of such an approach. This paper introduces a class of functions, which can be written as a sum of rank-one tensors using a total of at most parameters, and then uses this notion of sparsity to prove a regularity theorem for certain high-dimensional elliptic PDEs. It is shown, among other results, that whenever the right-hand side of the elliptic PDE can be approximated with a certain rate in the norm of by elements of , then the solution can be approximated in from to accuracy for any . Since these results require knowledge of the eigenbasis of the elliptic operator considered, we propose a second "basis-free" model of tensor-sparsity and prove a regularity theorem for this second sparsity model as well. We then proceed to address the important question of the extent to which such regularity theorems translate into results on computational complexity. It is shown how this second model can be used to derive computational algorithms with performance that breaks the curse of dimensionality on certain model high-dimensional elliptic PDEs with tensor-sparse data.
机译：试图打破高维偏微分方程（PDE）解的数值逼近的维数诅咒的一个反复出现的主题是采用某种形式的稀疏张量逼近。不幸的是，只有少数结果量化了这种方法的可能优势。本文介绍了一类函数，该函数可以使用最多总共参数来表示为一阶张量之和，然后使用这种稀疏性的概念证明某些高维椭圆PDE的正则定理。除其他结果外，结果表明，只要椭圆PDE的右手边可以按的元素的范数以一定比率近似，则对于任何一个，解都可以近似于到。由于这些结果需要了解所考虑的椭圆算子的本征基础，因此我们提出了张量稀疏性的第二个“无基础”模型，并证明了该第二个稀疏模型的正则定理。然后，我们着手解决这个正则定理在多大程度上转化为计算复杂性结果的重要问题。展示了如何使用第二个模型来推导具有破坏具有张量稀疏数据的某些模型高维椭圆PDE上的维数诅咒的性能的计算算法。

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