论文标题
关于基于竞争性的算法的内部迭代，用于解决大规模不足问题
On inner iterations of the joint bidiagonalization based algorithms for solving large scale ill-posed problems
论文作者
论文摘要
矩阵对$ \ {a，l \} $的联合竞争性过程可用于开发通用tikhonov正规化中的大规模不足问题的迭代正则化算法$ \ min_x \ left \ {\ | ax-b \ | _ {2}^2+λ^2 \ | lx \ | _ {2}^2}^2 \ right \} $或基本上等效的一个$ \ min \ min \ | | lx \ | lx \ | _ {2} \ | _ {2} \ \ \ \ \ \ \ \ \ mbox} x \ in \ Mathcal {s} = \ {x | \ \ | ax-b \ | _ {2} \ leq leqη\ | e \ | _ {2} \} $，其中$ e $是高斯白噪声，$ l $是正则化矩阵，$ l $是正则化矩阵，$η> 1 $ $。一系列算法是，$（a^{t}，l^{t}）^{t} $的大规模内部最小二乘问题作为系数矩阵必须在每个外部迭代中解决，尤其是当这些问题的解决方案准确性时，这可能是昂贵的。在本文中，我们对内部最小二乘问题的解决方案准确性要求进行了详细的研究，并提出了内部迭代的实际停止标准。结果表明，对于噪声水平不小的问题，可以大大放松内部最小二乘问题的解决方案的准确性，同时无法降低正则化解决方案的准确性，因此可以大大提高算法的总体效率。进行数值实验以说明我们的结果并显示算法的一些数值性能。
The joint bidiagonalization process of a matrix pair $\{A,L\}$ can be used to develop iterative regularization algorithms for large scale ill-posed problems in general-form Tikhonov regularization $\min_x\left\{\|Ax-b\|_{2}^2+λ^2\|Lx\|_{2}^2\right\}$ or the essentially equivalent one $\min\|Lx\|_{2} \ \mbox{\rm s.t.} \ x\in\mathcal{S} = \{x|\ \|Ax-b\|_{2}\leq η\|e\|_{2}\}$, where $e$ is a Gaussian white noise, $L$ is a regularization matrix and $η>1$ slightly. A bottleneck of the algorithms is that a large scale inner least squares problem with $(A^{T}, L^{T})^{T}$ as the coefficient matrix must be solved at each outer iteration, which may be costly, especially when the solution accuracy of these problems is high. In this paper, we give a detailed investigation on the solution accuracy requirement on the inner least squares problems and propose a practical stopping criterion for inner iterations. The results show that for ill-posed problems with not too small noise levels, the solution accuracy of the inner least squares problems can be relaxed considerably while it will not reduce the accuracy of regularized solutions, thus the overall efficiency of the algorithms can be improved substantially. Numerical experiments are made to illustrate our results and show some numerical performance of the algorithms.
