鞍点问题的若干有效算法及其在图像复原中的应用

发布时间：2018-04-22 01:21

本文选题：鞍点问题 + 图像复原问题　；参考：《兰州大学》2017年博士论文

【摘要】：鞍点线性系统源于许多科学计算与工程应用领域,如计算流体力学、椭圆偏微分方程的有限元和有限差分离散、加权等式约束最小二乘估计、图像处理等.鞍点系统的求解不仅对整个问题的解决起着至关重要的作用,而且具有十分重要的理论意义和实际应用价值.如何根据具体物理背景和鞍点结构矩阵性质设计出一类高效、稳健、实用的数值解法既是现代科学与工程计算的核心,又是当前数值计算工作者和工程技术人员的研究热点.本文主要研究了离散化偏微分方程中一类鞍点问题的数值解法,并将所得解法应用于图像复原问题中出现的一类结构化线性系统的求解.第一章给出了鞍点线性系统研究的背景意义、研究现状,并概述了本文的主要研究内容、特色和创新之处.第二章基于松弛预处理思想和松弛正定反Hermitian分裂方法,为大型稀疏非Hermitian鞍点问题提出了一类有效的广义松弛正定反Hermitian分裂(GRPSS)预处理方法.理论研究了GRPSS预处理矩阵的特征值分布和收敛性,并且发现GRPSS预处理子在某些范数意义下比RPSS预处理子更加接近初始系数矩阵.最后通过数值实验验证了此方法的有效性,并且发现理论与实验结果完全吻合.第三章对不可压缩Navier-Stokes方程中广义鞍点问题提出了一类修正松弛分裂(MRS)预处理解法.详细地研究了此预处理方法所对应预处理矩阵最小多项式次数及其预处理矩阵的特征值分布.与GRS方法相比,在保持计算量不变的前提下,MRS预处理子更加接近原始矩阵.实验证明了MRS方法的可行性和有效性.然而在求解MDS和MRS方法所对应的预处理子系统时,每步都需要求解两个子矩阵的逆.为此我们提出了一类新的块上下三角分裂(BULT)迭代法,理论分析发现当结合Krylov子空间方法求解时,可以很好地避免上述子系统求逆这一困难,从而大大提高了Krylov子空间方法的求解效率.第四章针对稳态不可压缩Navier-Stokes方程中的一类鞍点问题,提出了一类修正的SIMPLE(MS)预处理方法.通过对MS预处理矩阵的谱分析发现,在适当的条件下,预处理矩阵的所有特征值将会紧紧地聚集在(1,0)点附近.从而克服了松弛的HSS方法其余特征值分布很广的这一缺点.最后,从理论和实验上得到MS预处理子比已有的一些较好的预处理子更为有效.第五章研究了两类特殊鞍点系统的数值解法,即复线性系统和奇异鞍点线性系统.对复线性系统提出了一类广义的PMHSS(GPMHSS)方法,理论分析表明在选取适当的参数下,GPMHSS方法的谱半径比PMHSS方法和ADPMHSS方法的谱半径都要小.此外,对奇异鞍点系统提出了一类增广块三角分裂(ABTS)预处理方法.此方法对应产生鞍点线性系统的一个恰当分裂且理论分析证明,ABTS预处理迭代方法会收敛到奇异鞍点问题的广义逆解.同时发现,在结合ABTS预处理方法和GMRES方法进行求解时,也会收敛到预处理奇异鞍点系统的广义逆解.最后给出了ABTS方法的最优参数以及最佳收敛因子表达式.第六章研究了图像复原中得到的鞍点结构线性系统的上下三角(ULT)分裂迭代解法,给出了某些特定条件下的最优参数和最优收敛因子.实验结果显示,与已有的SHSS和RGHSS方法相比,ULT方法更具竞争性和有效性,且可以有效地应用于图像复原问题.第七章首先将广义的反Hermitian三角分裂(GSTS)迭代方法进行推广并得到一类修正的广义反Hermitian三角分裂(MGSTS)迭代解法.理论上给出了MGSTS方法求解图像复原问题时的收敛性和拟最优参数.最后通过数值比较验证了此方法在在求解图像复原问题时的高效性和精确性.第八章对全文进行总结,并在该章给出了以后工作的方向和展望.
[Abstract]:The saddle point linear system is derived from many fields of scientific computing and engineering applications, such as computational fluid mechanics, finite element and finite difference separation, weighted equality constrained least squares estimation, image processing and so on. The solution of saddle point system not only plays a vital role in solving the whole problem, but also is very important. How to design a class of efficient, robust and practical numerical solutions based on the specific physical background and the saddle point structure matrix is not only the core of modern science and engineering calculation, but also the research hotspot of current numerical computing workers and engineering technicians. This paper mainly studies the discrete partial differential square. The numerical solution of a kind of saddle point problem is used in the process, and the solution method is applied to the solution of a class of structured linear systems which appear in the image restoration problem. Chapter 1 gives the background significance of the research on the saddle point linear system, the research status, and summarizes the main research contents, features and innovations in this paper. The second chapter is based on the relaxation preprocessing. A class of effective generalized relaxation positive definite inverse Hermitian splitting (GRPSS) preprocessing method for large sparse non Hermitian saddle point problem is proposed by the thought and relaxation positive definite inverse Hermitian splitting method. The eigenvalue distribution and convergence of the GRPSS preconditioning matrix are studied in theory, and it is found that GRPSS preconditioner is pretreated in some norm sense than RPSS. The processing sub is closer to the initial coefficient matrix. Finally, the validity of the method is verified by numerical experiments, and the theory is in perfect agreement with the experimental results. In the third chapter, a class of modified relaxation splitting (MRS) preview method is proposed for the generalized saddle point problem in incompressible Navier-Stokes equations. Compared with the GRS method, the MRS preconditioner is more close to the original matrix compared with the GRS method. The experiment proves the feasibility and effectiveness of the MRS method. However, every step in the solution of the pre processing subsystem corresponding to the MDS and MRS methods We need to solve the inverse of the two submatrices. Therefore, we propose a new class of block upper and lower triangulation (BULT) iterative method. The theoretical analysis shows that when the Krylov subspace method is used to solve the problem, the difficulty of the above subsystem inversion can be avoided, and the solution efficiency of the Krylov subspace method is greatly improved. The fourth chapter is aimed at the steady state. A class of saddle point problems in incompressible Navier-Stokes equations is proposed. A modified SIMPLE (MS) preprocessing method is proposed. By spectral analysis of the MS preconditioning matrix, it is found that under appropriate conditions, all the eigenvalues of the pretreated matrix will be closely clustered in the (1,0) point close. Thus the remaining eigenvalue distribution of the relaxed HSS method is overcome. In the fifth chapter, the numerical solution of the two special saddle point systems, the complex linear system and the singular saddle point linear system, is studied in the fifth chapter. A generalized PMHSS (GPMHSS) method is proposed for the complex linear system, and the theoretical points are divided into two chapters. The analysis shows that the spectral radius of the GPMHSS method is smaller than that of the PMHSS method and the ADPMHSS method under the appropriate parameters. In addition, a class of augmented block triangulation (ABTS) preprocessing method for the singular saddle point system is proposed. This method corresponds to a proper splitting of the saddle point linear system and the theoretical analysis shows that the ABTS preprocessing is superposition. The generation method converges to the generalized inverse solution of the singular saddle point problem. It is found that when the ABTS preprocessing method and the GMRES method are solved, the generalized inverse solution of the pretreated singular saddle point system will also converge. Finally, the optimal parameters of the ABTS method and the best convergent factor table are given. The sixth chapter studies the image restoration. The optimal parameter and optimal convergence factor under certain conditions are given by the upper and lower trigonometric (ULT) splitting iterative method for the linear system of saddle point structure. The experimental results show that compared with the existing SHSS and RGHSS methods, the ULT method is more competitive and effective, and it can be effectively used for image restoration. The seventh chapter first will be generalized. The inverse Hermitian triangulation (GSTS) iterative method is generalized and a class of modified generalized inverse Hermitian triangulation (MGSTS) iterative solution is obtained. The convergence and the quasi optimal parameters of the MGSTS method in solving the image restoration problem are theoretically given. Finally, the efficiency of the method is verified by numerical comparison. Sex and accuracy. The eighth chapter summarizes the full text, and gives the direction and Prospect of future work in this chapter.

【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.6

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