大规模离散不适定问题迭代正则化方法的研究
发布时间：2018-11-15 19:02
【摘要】：我们首先研究基于Lanczos双对角化的LSQR算法.LSQR算法具有天然的正则化性质,迭代次数即为正则化参数.但是,至今仍然不清楚这种天然的正则化性质能否找到最好可能的正则化解.这里最好可能的正则化解是指同TSVD方法所获得最优近似解,或者标准Tikhonov正则化所获得的最优正则化解有相同精度.我们建立了k-维Krylov子空间和k-维主右奇异空间距离的定量估计,结果表明Krylov子空间对严重和中度不适定问题,比对温和不适定问题能更好地捕获主右奇异空间的信息.从而得出一般性结论:LSQR对前两种问题比对温和不适定问题有更好的正则化性质,并且温和不适定问题一般需要带额外正则化的混合LSQR方法求解.另外,我们给出Lanczos双对角化产生的秩-k逼近的精度估计.数值试验表明,LSQR的天然正则性对于严重和中度不适定问题已经足够获取最好可能的近似解,而对温和不适定问题则需要添加额外的正则化.对于求解大规模对称离散不适定问题的MINRES和MR-II方法,我们首先证明MINRES的迭代近似解有过滤SVD因子的形式.之后,我们推出以下结论:(i)给定一个对称不适定问题,MINRES一般需要对投影问题添加额外的正则化,才能获取最好可能的正则化解.(ii)尽管MR-II比MINRES有更好的全局正则化特性,但是在MINRES半收敛性达到之前,k步MINRES的正则化解比(k-1)步MR-II正则化解更为精确.此外,我们同样建立了k-维Krylov子空间和k-维主特征子空间距离估计.结论表明MR-II对严重和中度不适定问题比对温和不适定问题有更好的正则化性质,并且温和不适定问题一般需要混合MR-II方法来得到最好可能的正则化解.数值实验验证了我们的结论,并且实验表明了更强的结论:对于严重和中度不适定问题,MR-II的天然正则化性质已经足够获取最好可能的近似解.另外,我们还验证了MR-II能以两倍的效率得到与LSQR同等精度的正则化解.对于求解大规模非对称不适定问题的GMRES和其变型RRGMRES算法,我们从数值实验的角度,验证了k-维Krylov子空间和k-维主右奇异空间相去甚远,Arnoldi过程不能获取需要的SVD信息.从而得出结论:尽管GMRES和RRGMRES对某些不适定问题有效,但是这种基于Arnoldi过程的迭代方法并没有一般意义下的正则化性质.
[Abstract]:We first study the LSQR algorithm based on Lanczos double diagonalization. The LSQR algorithm has the natural regularization property and the iteration number is the regularization parameter. However, it is still unclear whether this natural regularization property can find the best possible regularization solution. Here the best possible regularization solution is the same precision as the optimal approximate solution obtained by the TSVD method or the optimal regularization solution obtained by the standard Tikhonov regularization. We establish the quantitative estimation of the distance between k- dimensional Krylov subspaces and k- dimensional principal right singular spaces. The results show that Krylov subspaces can capture the information of principal-right singular spaces better than mild ill-posed problems. It is concluded that LSQR has better regularization properties for the first two kinds of problems than the mild ill-posed problems, and the mild ill-posed problems generally need to be solved by mixed LSQR method with extra regularization. In addition, we estimate the accuracy of rank-k approximation generated by Lanczos bidiagonalization. Numerical experiments show that the natural regularity of LSQR is sufficient to obtain the best possible approximate solution for severe and moderate ill-posed problems, while additional regularization is needed for mild ill-posed problems. For the MINRES and MR-II methods for solving large-scale symmetric discrete ill-posed problems, we first prove that the iterative approximate solutions of MINRES have the form of filtered SVD factors. Then we draw the following conclusion: (i) is given a symmetric ill-posed problem, and MINRES generally needs to add additional regularization to the projection problem. In order to obtain the best possible regularization. (ii), although MR-II has better global regularization than MINRES, the regularization solution of k step MINRES is more accurate than (k-1) step MR-II regularization solution before MINRES semi-convergence is achieved. In addition, we also establish the estimation of distance between k- dimensional Krylov subspaces and k- dimensional principal feature subspaces. The results show that MR-II has better regularization properties for severe and moderate ill-posed problems than mild ill-posed problems, and that the mixed MR-II method is generally required to obtain the best possible regularization solutions for mild ill-posed problems. Numerical experiments verify our conclusion and show a stronger conclusion: for severe and moderate ill-posed problems, the natural regularization properties of MR-II are sufficient to obtain the best possible approximate solutions. In addition, we also verify that MR-II can obtain the regularization solution with the same accuracy as LSQR with twice the efficiency. For the GMRES and its modified RRGMRES algorithm for solving large-scale asymmetric ill-posed problems, we verify that the k-dimensional Krylov subspace is very different from the k-dimensional principal right singular space from the point of view of numerical experiments, and the Arnoldi process cannot obtain the required SVD information. It is concluded that although GMRES and RRGMRES are effective for some ill-posed problems, this iterative method based on Arnoldi process has no regularization property in general sense.
【学位授予单位】：清华大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.6
[Abstract]:We first study the LSQR algorithm based on Lanczos double diagonalization. The LSQR algorithm has the natural regularization property and the iteration number is the regularization parameter. However, it is still unclear whether this natural regularization property can find the best possible regularization solution. Here the best possible regularization solution is the same precision as the optimal approximate solution obtained by the TSVD method or the optimal regularization solution obtained by the standard Tikhonov regularization. We establish the quantitative estimation of the distance between k- dimensional Krylov subspaces and k- dimensional principal right singular spaces. The results show that Krylov subspaces can capture the information of principal-right singular spaces better than mild ill-posed problems. It is concluded that LSQR has better regularization properties for the first two kinds of problems than the mild ill-posed problems, and the mild ill-posed problems generally need to be solved by mixed LSQR method with extra regularization. In addition, we estimate the accuracy of rank-k approximation generated by Lanczos bidiagonalization. Numerical experiments show that the natural regularity of LSQR is sufficient to obtain the best possible approximate solution for severe and moderate ill-posed problems, while additional regularization is needed for mild ill-posed problems. For the MINRES and MR-II methods for solving large-scale symmetric discrete ill-posed problems, we first prove that the iterative approximate solutions of MINRES have the form of filtered SVD factors. Then we draw the following conclusion: (i) is given a symmetric ill-posed problem, and MINRES generally needs to add additional regularization to the projection problem. In order to obtain the best possible regularization. (ii), although MR-II has better global regularization than MINRES, the regularization solution of k step MINRES is more accurate than (k-1) step MR-II regularization solution before MINRES semi-convergence is achieved. In addition, we also establish the estimation of distance between k- dimensional Krylov subspaces and k- dimensional principal feature subspaces. The results show that MR-II has better regularization properties for severe and moderate ill-posed problems than mild ill-posed problems, and that the mixed MR-II method is generally required to obtain the best possible regularization solutions for mild ill-posed problems. Numerical experiments verify our conclusion and show a stronger conclusion: for severe and moderate ill-posed problems, the natural regularization properties of MR-II are sufficient to obtain the best possible approximate solutions. In addition, we also verify that MR-II can obtain the regularization solution with the same accuracy as LSQR with twice the efficiency. For the GMRES and its modified RRGMRES algorithm for solving large-scale asymmetric ill-posed problems, we verify that the k-dimensional Krylov subspace is very different from the k-dimensional principal right singular space from the point of view of numerical experiments, and the Arnoldi process cannot obtain the required SVD information. It is concluded that although GMRES and RRGMRES are effective for some ill-posed problems, this iterative method based on Arnoldi process has no regularization property in general sense.
【学位授予单位】：清华大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.6
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