本文選題：鞍點(diǎn)問題 + 圖像復(fù)原問題��； 參考：《蘭州大學(xué)》2017年博士論文

【摘要】：鞍點(diǎn)線性系統(tǒng)源于許多科學(xué)計(jì)算與工程應(yīng)用領(lǐng)域,如計(jì)算流體力學(xué)、橢圓偏微分方程的有限元和有限差分離散、加權(quán)等式約束最小二乘估計(jì)、圖像處理等.鞍點(diǎn)系統(tǒng)的求解不僅對整個(gè)問題的解決起著至關(guān)重要的作用,而且具有十分重要的理論意義和實(shí)際應(yīng)用價(jià)值.如何根據(jù)具體物理背景和鞍點(diǎn)結(jié)構(gòu)矩陣性質(zhì)設(shè)計(jì)出一類高效、穩(wěn)健、實(shí)用的數(shù)值解法既是現(xiàn)代科學(xué)與工程計(jì)算的核心,又是當(dāng)前數(shù)值計(jì)算工作者和工程技術(shù)人員的研究熱點(diǎn).本文主要研究了離散化偏微分方程中一類鞍點(diǎn)問題的數(shù)值解法,并將所得解法應(yīng)用于圖像復(fù)原問題中出現(xiàn)的一類結(jié)構(gòu)化線性系統(tǒng)的求解.第一章給出了鞍點(diǎn)線性系統(tǒng)研究的背景意義、研究現(xiàn)狀,并概述了本文的主要研究內(nèi)容、特色和創(chuàng)新之處.第二章基于松弛預(yù)處理思想和松弛正定反Hermitian分裂方法,為大型稀疏非Hermitian鞍點(diǎn)問題提出了一類有效的廣義松弛正定反Hermitian分裂(GRPSS)預(yù)處理方法.理論研究了GRPSS預(yù)處理矩陣的特征值分布和收斂性,并且發(fā)現(xiàn)GRPSS預(yù)處理子在某些范數(shù)意義下比RPSS預(yù)處理子更加接近初始系數(shù)矩陣.最后通過數(shù)值實(shí)驗(yàn)驗(yàn)證了此方法的有效性,并且發(fā)現(xiàn)理論與實(shí)驗(yàn)結(jié)果完全吻合.第三章對不可壓縮Navier-Stokes方程中廣義鞍點(diǎn)問題提出了一類修正松弛分裂(MRS)預(yù)處理解法.詳細(xì)地研究了此預(yù)處理方法所對應(yīng)預(yù)處理矩陣最小多項(xiàng)式次數(shù)及其預(yù)處理矩陣的特征值分布.與GRS方法相比,在保持計(jì)算量不變的前提下,MRS預(yù)處理子更加接近原始矩陣.實(shí)驗(yàn)證明了MRS方法的可行性和有效性.然而在求解MDS和MRS方法所對應(yīng)的預(yù)處理子系統(tǒng)時(shí),每步都需要求解兩個(gè)子矩陣的逆.為此我們提出了一類新的塊上下三角分裂(BULT)迭代法,理論分析發(fā)現(xiàn)當(dāng)結(jié)合Krylov子空間方法求解時(shí),可以很好地避免上述子系統(tǒng)求逆這一困難,從而大大提高了Krylov子空間方法的求解效率.第四章針對穩(wěn)態(tài)不可壓縮Navier-Stokes方程中的一類鞍點(diǎn)問題,提出了一類修正的SIMPLE(MS)預(yù)處理方法.通過對MS預(yù)處理矩陣的譜分析發(fā)現(xiàn),在適當(dāng)?shù)臈l件下,預(yù)處理矩陣的所有特征值將會(huì)緊緊地聚集在(1,0)點(diǎn)附近.從而克服了松弛的HSS方法其余特征值分布很廣的這一缺點(diǎn).最后,從理論和實(shí)驗(yàn)上得到MS預(yù)處理子比已有的一些較好的預(yù)處理子更為有效.第五章研究了兩類特殊鞍點(diǎn)系統(tǒng)的數(shù)值解法,即復(fù)線性系統(tǒng)和奇異鞍點(diǎn)線性系統(tǒng).對復(fù)線性系統(tǒng)提出了一類廣義的PMHSS(GPMHSS)方法,理論分析表明在選取適當(dāng)?shù)膮?shù)下,GPMHSS方法的譜半徑比PMHSS方法和ADPMHSS方法的譜半徑都要小.此外,對奇異鞍點(diǎn)系統(tǒng)提出了一類增廣塊三角分裂(ABTS)預(yù)處理方法.此方法對應(yīng)產(chǎn)生鞍點(diǎn)線性系統(tǒng)的一個(gè)恰當(dāng)分裂且理論分析證明,ABTS預(yù)處理迭代方法會(huì)收斂到奇異鞍點(diǎn)問題的廣義逆解.同時(shí)發(fā)現(xiàn),在結(jié)合ABTS預(yù)處理方法和GMRES方法進(jìn)行求解時(shí),也會(huì)收斂到預(yù)處理奇異鞍點(diǎn)系統(tǒng)的廣義逆解.最后給出了ABTS方法的最優(yōu)參數(shù)以及最佳收斂因子表達(dá)式.第六章研究了圖像復(fù)原中得到的鞍點(diǎn)結(jié)構(gòu)線性系統(tǒng)的上下三角(ULT)分裂迭代解法,給出了某些特定條件下的最優(yōu)參數(shù)和最優(yōu)收斂因子.實(shí)驗(yàn)結(jié)果顯示,與已有的SHSS和RGHSS方法相比,ULT方法更具競爭性和有效性,且可以有效地應(yīng)用于圖像復(fù)原問題.第七章首先將廣義的反Hermitian三角分裂(GSTS)迭代方法進(jìn)行推廣并得到一類修正的廣義反Hermitian三角分裂(MGSTS)迭代解法.理論上給出了MGSTS方法求解圖像復(fù)原問題時(shí)的收斂性和擬最優(yōu)參數(shù).最后通過數(shù)值比較驗(yàn)證了此方法在在求解圖像復(fù)原問題時(shí)的高效性和精確性.第八章對全文進(jìn)行總結(jié),并在該章給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.

【學(xué)位授予單位】：蘭州大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.6

【參考文獻(xiàn)】

相關(guān)期刊論文 前2條

1 Michael K.Ng;;BLOCK-TRIANGULAR PRECONDITIONERS FOR SYSTEMS ARISING FROM EDGE-PRESERVING IMAGE RESTORATION[J];Journal of Computational Mathematics;2010年06期

2 ;ITERATIVE METHODS WITH PRECONDITIONERS FOR INDEFINITE SYSTEMS[J];Journal of Computational Mathematics;1999年01期

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本文編號(hào)：1784976