Truncated singular value decomposition is a reduced version of the singular value decomposition in which only a few largest singular values are retained. This paper presents a novel perturbation analysis for the truncated singular value decomposition for real matrices. First, we describe perturbation expansions for the singular value truncation of order $r$. We extend perturbation results for the singular subspace decomposition to derive the first-order perturbation expansion of the truncated operator about a matrix with rank greater than or equal to $r$. Observing that the first-order expansion can be greatly simplified when the matrix has exact rank $r$, we further show that the singular value truncation admits a simple second-order perturbation expansion about a rank-$r$ matrix. Second, we introduce the first-known error bound on the linear approximation of the truncated singular value decomposition of a perturbed rank-$r$ matrix. Our bound only depends on the least singular value of the unperturbed matrix and the norm of the perturbation matrix. Intriguingly, while the singular subspaces are known to be extremely sensitive to additive noises, the newly established error bound holds universally for perturbations with arbitrary magnitude. Finally, we demonstrate an application of our results to the analysis of the mean squared error associated with the TSVD-based matrix denoising solution.
A stochastic gradient descent approach with partitioned-truncated singular value decomposition for large-scale inverse problems of magnetic modulus data
Modified Truncated Randomized Singular Value Decomposition (MTRSVD) Algorithms for Large Scale Discrete Ill-posed Problems with General-Form Regularization
A pure source transfer domain decomposition method for Helmholtz equations in unbounded domain
提出了一种求解亥姆霍兹散射问题的有界区域截断完美匹配层（PML）近似的纯源转移区域分解方法（PSTDDM）。该方法是对[Z.陈和X。Xiang，SIAM J. Numer.肛门，51（2013），pp. 2331- 2356]。STDDM将区域分解为$N$非重叠层后，由源传递和波展开两个步骤串联而成，依次求解相邻层上的$N-1$截断PML问题和$N-2$截断半空间PML问题。而PSTDDM仅由两个相反方向上的两个并行源传输步骤组成，并且在每个步骤中连续解决两个相邻层上的截断PML问题$N-1$。这样的修改的一个好处是，在两个相邻层上的截断PML问题可以通过PSTDDM沿着平行于层的方向进一步解决。因此，我们得到了一个分块PSTDDM上的分解组成的$N^2$平方，这减少了子域问题的大小，更适合于大规模的问题。在波数为常数的情况下，证明了逐层PSTDDM和逐块PSTDDM的收敛性。数值算例表明，PSTDDM对常波数的离散Helmholtz方程有很好的逼近，可以作为预处理GMRES方法中的一个有效的预处理子用于求解常波数和非均匀波数的离散Helmholtz方程.
Hot-SVD: Higher-Order t-Singular Value Decomposition for Tensors based on Tensor-Tensor Product
本文考虑将三阶张量（管矩阵）的t-SVD推广到任意阶张量（N-1阶管张量）。\color{black}这种推广不同于阶数大于3的张量的t-SVD [Martin，Shafer，Larue，SIAM J. Sci.计算：35（2013），A474-A490]。分解被称为热SVD，因为它可以被识别为HOSVD的张量-张量积版本。证明了Hot-SVD的存在性。为此，一个新的转置三阶张量。这种转置在Hot-SVD的验证中至关重要，因为它是管张量与其展开之间的桥梁。我们建立了一些性质的热SVD，类似于HOSVD，并在这样做，我们强调的观点，管张量。然后引入截断和顺序截断的Hot-SVD，其误差界为$\sqrt{N}$（对于$(N+1)$阶张量）.我们提供了数值例子来验证热SVD，截断热SVD，和顺序截断热SVD。
Split Bregman Isotropic and Anisotropic Image Deblurring with Kronecker Product Sum Approximations using Single Precision Enlarged-GKB or RSVD Algorithms to provide low rank truncated SVDs
