论文标题
通过指数总和在分数运算符中保存低量张量的结构
Low-rank tensor structure preservation in fractional operators by means of exponential sums
论文作者
论文摘要
分数微分方程的使用是建模非本地现象的关键工具。通常，求解涉及分数运算符离散化的线性系统的有效方案是评估矩阵函数$ x = \ Mathcal a^{ - α} c $，其中$ \ nathcal a $是经典的laplacian和$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ 0 $ 0 $ 0 $ 1 $ $ 1 $ 1 $ 1 $ 1 $ 1 $ 1。在这项工作中，我们得出了$ f（z）= z^{ - α} $的指数和近似值，该$与$ [1，\ infty）$准确，并允许有效地近似于以各种低级格式（cp，tt，tt，fucker，fucker，fucker）存储在有限和无界操作器上。从理论的角度来看，结果也与众不同，因为它们预测了这些线性系统以低量张量格式的溶液的低级别近似性。
The use of fractional differential equations is a key tool in modeling non-local phenomena. Often, an efficient scheme for solving a linear system involving the discretization of a fractional operator is evaluating the matrix function $x = \mathcal A^{-α} c$, where $\mathcal A$ is a discretization of the classical Laplacian, and $α$ a fractional exponent between $0$ and $1$. In this work, we derive an exponential sum approximation for $f(z) =z^{-α}$ that is accurate over $[1, \infty)$ and allows to efficiently approximate the action of bounded and unbounded operators of this kind on tensors stored in a variety of low-rank formats (CP, TT, Tucker). The results are relevant from a theoretical perspective as well, as they predict the low-rank approximability of the solutions of these linear systems in low-rank tensor formats.
