论文标题
一个维度的奇异扰动部分微分方程的低等级张量近似
Low rank tensor approximation of singularly perturbed partial differential equations in one dimension
论文作者
论文摘要
我们在量化反应扩散偏微分方程（PDES）的量级张量序列（QTT）上得出了等级边界。具体而言，我们表明，与单数扰动参数的比例无关，具有精度$ 0 <ε<1 $的数值解决方案可以以QTT格式表示，并且具有许多参数，这些参数仅取决于$ε$。换句话说，就参数数的根而言，QTT压缩解决方案将指数收敛于精确解决方案。我们还通过数值验证了秩界的估计，并通过调整预处理策略以在所有尺度上获得稳定的方案来克服基于QTT的PDE解决方案的已知稳定性问题。因此，我们发现基于QTT的策略是一种奇异扰动PDE的解决方案的快速收敛算法，该算法不需要在单数扰动的规模和边界层的形状上进行先验知识。
We derive rank bounds on the quantized tensor train (QTT) compressed approximation of singularly perturbed reaction diffusion partial differential equations (PDEs) in one dimension. Specifically, we show that, independently of the scale of the singular perturbation parameter, a numerical solution with accuracy $0<ε<1$ can be represented in QTT format with a number of parameters that depends only polylogarithmically on $ε$. In other words, QTT compressed solutions converge exponentially to the exact solution, with respect to a root of the number of parameters. We also verify the rank bound estimates numerically, and overcome known stability issues of the QTT based solution of PDEs by adapting a preconditioning strategy to obtain stable schemes at all scales. We find, therefore, that the QTT based strategy is a rapidly converging algorithm for the solution of singularly perturbed PDEs, which does not require prior knowledge on the scale of the singular perturbation and on the shape of the boundary layers.
