由于维数灾难, 高维问题的求解一般是很困难的, 张量分解是减轻维数灾难的一类有效方法. 在本文中, 我们首先研究了2011 年由Oseledets 等人提出的Tensor Train (TT) 分解. 它是一种有较大发展潜力的张量分解方法, 具有良好的算法稳定性, 既可以将存储量降低到与问题维数呈线性, 并且还可以将张量的各种基本运算的计算量降低到与问题维数呈线性, 从而减轻了维数灾难. 我们还介绍了一类常见的CP 张量及其对应的TT 张量. 接下来我们研究了2012年由Holtz 等人提出的TT 张量的交替线性法. 它依次优化TT 张量的每一个TT核, 将高维问题转化为一系列低维问题从而减轻了维数灾难. 最后, 我们利用交替线性法求解了Laplace 算子的特征值问题, 验证了算法的有效性和优越性.

Due to the curse of dimensionality, it is often difficult to solve high-dimensional problems. Tensor decomposition methods are efficient and useful to mitigate the curse of dimensionality. In this thesis, we first study a promising tensor decomposition known as the tensor train (TT) decomposition proposed by Oseledets et al. in 2011. The algorithm to compute TT decomposition is stable and robust. TT format can reduce the storage and also the complexity of various basic operations to be linear with the dimension of the problem, thus mitigating the curse of dimensionality. We also introduce a useful family of CP tensors and its corresponding TT representations. Next, we study the alternating linear scheme (ALS) in the tensor train format proposed by Holtz et al. in 2012. It optimizes each TT core of the TT tensor in turn, which transforms the high-dimensional problem into a series of low-dimensional problems, thus mitigating the curse of dimensionality. Finally, we use the ALS to solve the eigenvalue problem of the Laplace operator and verify the effectiveness and superiority of the ALS.