Low-rank tensor representations can provide highly compressed approximations of functions. These concepts, which essentially amount to generalizations of classical techniques of separation of variables, have proved to be particularly fruitful for functions of many variables. We focus here on problems where the target function is given only implicitly as the solution of a partial differential equation. A first natural question is under which conditions we should expect such solutions to be efficiently approximated in low-rank form. Due to the highly nonlinear nature of the resulting low-rank approximations, a crucial second question is at what expense such approximations can be computed in practice. This article surveys basic construction principles of numerical methods based on low-rank representations as well as the analysis of their convergence and computational complexity.

中文翻译：

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偏微分方程的低阶张量方法

低阶张量表示可以提供高度压缩的函数近似值。这些概念本质上相当于经典变量分离技术的概括，已被证明对多变量函数特别有效。我们在这里关注目标函数仅作为偏微分方程的解隐式给出的问题。第一个自然问题是在什么条件下我们应该期望这些解决方案以低秩形式有效地近似。由于由此产生的低秩近似的高度非线性特性，关键的第二个问题是在实践中可以计算这种近似的代价是什么。