Low-rank methods for high-dimensional approximation and model order reduction
Low-rank methods for high-dimensional approximation and model order reduction
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DOI：
10.1137/1.9781611974829
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发表时间：
2015-11
期刊：
影响因子：
--
通讯作者：
A. Nouy
中科院分区：
文献类型：
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作者：
A. Nouy
张量方法是最突出的工具之一，用于高维问题的数值解，其中多个变量的函数必须近似。这些方法利用了函数空间的张量结构，并应用于计算科学中的许多问题，这些问题在张量空间中制定，如随机微积分，不确定性量化或参数分析中出现的问题。在这里，我们提出了基于低秩近似方法的复杂度降低方法。我们分析了低秩张量子集的最佳逼近问题，并讨论了它与低维约化空间中最优模型约化问题的联系。我们提出了不同的算法来计算低秩格式的函数的近似值。特别是，我们提出了建设性的算法，这是基于一个贪婪的建设的近似值（连续校正的低秩张量的子集）或贪婪的张量子空间的建设（基于子空间的低秩格式）。这些算法可以应用于张量压缩，张量完成或低秩张量格式方程的数值解。特别强调的是随机或参数依赖模型的解决方案。不同的方法提出了向量值或多元函数（张量识别）的近似，基于样本的功能（黑盒方法）或模型方程的功能满足。
Tensor methods are among the most prominent tools for the numerical solution of high-dimensional problems where functions of multiple variables have to be approximated. These methods exploit the tensor structure of function spaces and apply to many problems in computational science which are formulated in tensor spaces, such as problems arising in stochastic calculus, uncertainty quantification or parametric analyses. Here, we present complexity reduction methods based on low-rank approximation methods. We analyze the problem of best approximation in subsets of low-rank tensors and discuss its connection with the problem of optimal model reduction in low-dimensional reduced spaces. We present different algorithms for computing approximations of a function in low-rank formats. In particular, we present constructive algorithms which are based either on a greedy construction of an approximation (with successive corrections in subsets of low-rank tensors) or on the greedy construction of tensor subspaces (for subspace-based low-rank formats). These algorithms can be applied for tensor compression, tensor completion or for the numerical solution of equations in low-rank tensor formats. A special emphasis is given to the solution of stochastic or parameter-dependent models. Different approaches are presented for the approximation of vector-valued or multivariate functions (identified with tensors), based on samples of the functions (black-box approaches) or on the models equations which are satisfied by the functions.