【摘要】：我們首先研究基于Lanczos雙對(duì)角化的LSQR算法.LSQR算法具有天然的正則化性質(zhì),迭代次數(shù)即為正則化參數(shù).但是,至今仍然不清楚這種天然的正則化性質(zhì)能否找到最好可能的正則化解.這里最好可能的正則化解是指同TSVD方法所獲得最優(yōu)近似解,或者標(biāo)準(zhǔn)Tikhonov正則化所獲得的最優(yōu)正則化解有相同精度.我們建立了k-維Krylov子空間和k-維主右奇異空間距離的定量估計(jì),結(jié)果表明Krylov子空間對(duì)嚴(yán)重和中度不適定問(wèn)題,比對(duì)溫和不適定問(wèn)題能更好地捕獲主右奇異空間的信息.從而得出一般性結(jié)論:LSQR對(duì)前兩種問(wèn)題比對(duì)溫和不適定問(wèn)題有更好的正則化性質(zhì),并且溫和不適定問(wèn)題一般需要帶額外正則化的混合LSQR方法求解.另外,我們給出Lanczos雙對(duì)角化產(chǎn)生的秩-k逼近的精度估計(jì).數(shù)值試驗(yàn)表明,LSQR的天然正則性對(duì)于嚴(yán)重和中度不適定問(wèn)題已經(jīng)足夠獲取最好可能的近似解,而對(duì)溫和不適定問(wèn)題則需要添加額外的正則化.對(duì)于求解大規(guī)模對(duì)稱離散不適定問(wèn)題的MINRES和MR-II方法,我們首先證明MINRES的迭代近似解有過(guò)濾SVD因子的形式.之后,我們推出以下結(jié)論:(i)給定一個(gè)對(duì)稱不適定問(wèn)題,MINRES一般需要對(duì)投影問(wèn)題添加額外的正則化,才能獲取最好可能的正則化解.(ii)盡管MR-II比MINRES有更好的全局正則化特性,但是在MINRES半收斂性達(dá)到之前,k步MINRES的正則化解比(k-1)步MR-II正則化解更為精確.此外,我們同樣建立了k-維Krylov子空間和k-維主特征子空間距離估計(jì).結(jié)論表明MR-II對(duì)嚴(yán)重和中度不適定問(wèn)題比對(duì)溫和不適定問(wèn)題有更好的正則化性質(zhì),并且溫和不適定問(wèn)題一般需要混合MR-II方法來(lái)得到最好可能的正則化解.數(shù)值實(shí)驗(yàn)驗(yàn)證了我們的結(jié)論,并且實(shí)驗(yàn)表明了更強(qiáng)的結(jié)論:對(duì)于嚴(yán)重和中度不適定問(wèn)題,MR-II的天然正則化性質(zhì)已經(jīng)足夠獲取最好可能的近似解.另外,我們還驗(yàn)證了MR-II能以兩倍的效率得到與LSQR同等精度的正則化解.對(duì)于求解大規(guī)模非對(duì)稱不適定問(wèn)題的GMRES和其變型RRGMRES算法,我們從數(shù)值實(shí)驗(yàn)的角度,驗(yàn)證了k-維Krylov子空間和k-維主右奇異空間相去甚遠(yuǎn),Arnoldi過(guò)程不能獲取需要的SVD信息.從而得出結(jié)論:盡管GMRES和RRGMRES對(duì)某些不適定問(wèn)題有效,但是這種基于Arnoldi過(guò)程的迭代方法并沒(méi)有一般意義下的正則化性質(zhì).
[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.
【學(xué)位授予單位】：清華大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類(lèi)號(hào)】：O241.6

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