This paper is concerned with the construction, analysis, and realization of a numerical method to approximate the solution of high-dimensional elliptic partial differential equations. We propose a new combination of an adaptive wavelet Galerkin method (AWGM) and the well-known hierarchical tensor (HT) format. The arising HT-AWGM is adaptive both in the wavelet representation of the low-dimensional factors and in the tensor rank of the HT representation. The point of departure is an adaptive wavelet method for the HT format using approximate Richardson iterations and an AWGM for elliptic problems. HT-AWGM performs a sequence of Galerkin solves based upon a truncated preconditioned conjugate gradient (PCG) algorithm in combination with a tensor-based preconditioner. Our analysis starts by showing convergence of the truncated conjugate gradient method. The next step is to add routines realizing the adaptive refinement. The resulting HT-AWGM is analyzed concerning convergence and complexity. We show that the performance of the scheme asymptotically depends only on the desired tolerance with convergence rates depending on the Besov regularity of low-dimensional quantities and the low-rank tensor structure of the solution. The complexity in the ranks is algebraic with powers of four stemming from the complexity of the tensor truncation. Numerical experiments show the quantitative performance.

中文翻译：

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HT-AWGM：高维椭圆问题的分层塔克自适应小波Galerkin方法

本文涉及一种数值方法的构造，分析和实现，该方法可以近似高维椭圆偏微分方程的解。我们提出了一种自适应小波伽勒金方法（AWGM）和众所周知的分层张量（HT）格式的新组合。产生的HT-AWGM在低维因子的小波表示和HT表示的张量秩中都是自适应的。出发点是针对HT格式的自适应小波方法，使用近似Richardson迭代和针对椭圆问题的AWGM。HT-AWGM基于截断的预处理共轭梯度（PCG）算法结合基于张量的预处理器执行一系列Galerkin解。我们的分析开始于截断共轭梯度法的收敛性。下一步是添加实现自适应细化的例程。分析所得的HT-AWGM关于收敛性和复杂性。我们表明，该方案的性能渐近仅取决于收敛速度的期望容差，收敛速度取决于低维数量的Besov规律性和解决方案的低秩张量结构。由于张量截断的复杂性，行列中的复杂度是四次方的代数。数值实验表明了定量性能。我们表明，该方案的性能渐近仅取决于收敛速度的期望容差，收敛速度取决于低维数量的Besov规律性和解决方案的低秩张量结构。由于张量截断的复杂性，行列中的复杂度是四次方的代数。数值实验表明了定量性能。我们表明，该方案的性能渐近仅取决于收敛速度的期望容差，收敛速度取决于低维数量的Besov规律性和解决方案的低秩张量结构。由于张量截断的复杂性，行列中的复杂度是四次方的代数。数值实验表明了定量性能。