AI 中文总结
本文提出采用分层塔克分解的高阶秩自适应隐式算法，用于求解高维扩散方程，通过数值实验验证其高阶精度及解秩捕获能力。
AI 中文摘要
本文提出一种用于求解高维扩散方程张量解的高阶秩自适应隐式积分器。由于塔克分解的存储复杂度随维度数d>3呈指数增长，我们将该方法的三维版本从塔克分解扩展到更高维度，采用分层塔克（HT）分解。HT格式通过根据二叉树分解解来避免该问题，该二叉树由各维度的基和连接这些基的核心张量组成。空间离散化采用谱方法，时间离散化采用对角隐式龙格-库塔方法。在龙格-库塔方法的每个阶段，会扩充前序阶段计算得到的基，以预测即将到来的基并构建投影子空间。通过投影到这些扩充的子空间，可从叶节点到根节点沿树的上行方向依次更新基和核心张量。与仅具有单个核心张量的三维塔克方法不同，HT方法还会更新中间核心张量。数值实验表明，该方法具有高阶精度，并测试了积分器对不同时变扩散系数集合的解秩捕获能力。
英文摘要
This paper presents a high-order rank-adaptive implicit integrator for the tensor solution of high-dimensional diffusion equations. We extend the 3D version of this method from the Tucker decomposition to higher dimensions using the hierarchical Tucker (HT) decomposition, since the storage complexity for the Tucker decomposition increases exponentially with the number of dimensions $d>3$. The HT format avoids this issue by decomposing the solution according to a binary tree consisting of bases for each dimension and core tensors which connect the bases. Spectral methods are considered for spatial discretization, and diagonally implicit Runge-Kutta methods are considered for time discretization. At each stage of the Runge-Kutta method, the bases computed at the previous stages are augmented to predict the upcoming basis and construct projection subspaces. By projecting onto these enriched subspaces, the bases and cores can be updated in a sequential manner going up the tree from leaf-to-root. Unlike the 3D Tucker method which has a single core tensor, the HT method also updates the intermediate core tensors. Numerical experiments demonstrate that the method observes high-order accuracy, and test how well the integrator captures the solution rank for various sets of time-dependent diffusion coefficients.
CommentsThis work was done while PBP was an undergraduate at Swarthmore College. Project Advisor: Dr. Joseph Nakao (Swarthmore College)
