标题： 分裂Bregman各向同性和各向异性图像去模糊，使用单精度扩展-GKB或RSVD算法进行Kronecker乘积和近似，以提供低秩截断SVDs 显示英文标题

标题： 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

摘要： 我们考虑使用各向同性和各向异性正则化方法，通过分裂Bregman算法求解$\ell_1$正则化的图像去模糊问题。 对于大规模问题，我们使用通过重新排列矩阵$\mathcal{R}(A)$的近似截断奇异值分解得到的Kronecker乘积近似来替换系统矩阵$A$。 为了获得矩阵$\mathcal{R}(A)$的近似分解，我们提出了扩展的Golub Kahan双对角化算法，该算法通过将Krylov子空间扩展到给定的秩以实现所需的近似，或者使用自动停止测试来提供适合近似的秩。 所得的展开与使用相同项数的截断和随机奇异值分解进行了对比。 为了进一步扩大可以考虑的问题规模，我们使用单精度进行近似确定，而在正则化的所有步骤中使用标准双精度。 报告的数值测试表明，将近似单精度Kronecker乘积展开应用于$A$，结合使用分裂Bregman算法实现的各向同性或各向异性正则化，在求解图像去模糊问题方面是有效的。 随着问题规模的增大，我们的结果表明，主要成本与确定Kronecker乘积近似有关，而不是正则化算法的成本。 此外，扩展的Golub Kahan双对角化算法在估计近似奇异值分解方面与随机奇异值分解相比具有竞争力。 显示英文摘要

摘要： We consider the solution of the $\ell_1$ regularized image deblurring problem using isotropic and anisotropic regularization implemented with the split Bregman algorithm. For large scale problems, we replace the system matrix $A$ using a Kronecker product approximation obtained via an approximate truncated singular value decomposition for the reordered matrix $\mathcal{R}(A)$. To obtain the approximate decomposition for $\mathcal{R}(A)$ we propose the enlarged Golub Kahan Bidiagonalization algorithm that proceeds by enlarging the Krylov subspace beyond either a given rank for the desired approximation, or uses an automatic stopping test that provides a suitable rank for the approximation. The resultant expansion is contrasted with the use of the truncated and the randomized singular value decompositions with the same number of terms. To further extend the scale of problem that can be considered we implement the determination of the approximation using single precision, while performing all steps for the regularization in standard double precision. The reported numerical tests demonstrate the effectiveness of applying the approximate single precision Kronecker product expansion for $A$, combined with either isotropic or anisotropic regularization implemented using the split Bregman algorithm, for the solution of image deblurring problems. As the size of the problem increases, our results demonstrate that the major costs are associated with determining the Kronecker product approximation, rather than with the cost of the regularization algorithm. Moreover, the enlarged Golub Kahan Bidiagonalization algorithm competes favorably with the randomized singular value decomposition for estimating the approximate singular value decomposition.