Adaptive Low-Rank Methods: Problems on Sobolev Spaces
Adaptive Low-Rank Methods: Problems on Sobolev Spaces
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自适应低秩方法：Sobolev 空间上的问题
DOI：
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发表时间：
2014
影响因子：
2.9
通讯作者：
W. Dahmen
中科院分区：
文献类型：
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作者：
M. Bachmayr;W. Dahmen
本文研究了基于子空间的低秩张量格式的高维二阶椭圆问题的迭代求解器。产生低秩近似的子空间和相应的低维张量分量的稀疏近似自适应地确定。一个主要的障碍，同时控制的秩增长和准确性原来是这样一个事实，即基本的椭圆算子是一个同构的空间之间没有赋予交叉规范。因此，作为该方案的核心部分，我们设计了一种预处理低秩张量表示的方法。在标准的数据假设下，我们建立了收敛到连续问题的解决方案，保证减少误差。此外，对于该解决方案具有一定的低秩结构和表示稀疏的情况下，我们推导出的计算复杂性，包括，在partic…
This paper is concerned with the development and analysis of an iterative solver for high-dimensional second-order elliptic problems based on subspace-based low-rank tensor formats. Both the subspaces giving rise to low-rank approximations and the corresponding sparse approximations of lower-dimensional tensor components are determined adaptively. A principal obstruction to a simultaneous control of rank growth and accuracy turns out to be the fact that the underlying elliptic operator is an isomorphism only between spaces that are not endowed with cross norms. Therefore, as the central part of this scheme, we devise a method for preconditioning low-rank tensor representations of operators. Under standard assumptions on the data, we establish convergence to the solution of the continuous problem with a guaranteed error reduction. Moreover, for the case that the solution exhibits a certain low-rank structure and representation sparsity, we derive bounds on the computational complexity, including, in partic...