本文關(guān)鍵詞：全局Krylov子空間方法研究及其應(yīng)用　出處：《電子科技大學(xué)》2015年碩士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 重啟的加權(quán)全局廣義最小殘量方法 全局Krylov子空間方法 多右端向量線性方程組

【摘要】：本文主要研究求解大規(guī)模稀疏多右端向量線性方程組的全局Krylov子空間方法。由于許多應(yīng)用領(lǐng)域,如求解偏微分方程問題,流體動力學(xué),電路仿真,電磁場計(jì)算,線性控制論等,均需要求解多右端向量線性系統(tǒng)。因此建立多右端向量線性方程組的高效穩(wěn)健的數(shù)值方法具有十分重要的意義。本文主要由下面兩部分構(gòu)成:首先,全面系統(tǒng)地闡述了基于Arnoldi過程的Krylov子空間方法,包括完全正交化方法(FOM)和廣義最小殘量方法(GMRES),數(shù)值實(shí)驗(yàn)表明GMRES方法要優(yōu)于FOM方法。其次,按照這兩種方法推導(dǎo)思路,分析了基于全局Arnoldi過程的Krylov子空間方法,即全局完全正交化方法(GL-FOM)和全局廣義最小殘量方法(GL-GMRES)。在此基礎(chǔ)上,運(yùn)用Givens旋轉(zhuǎn)變換,給出了迭代解與相應(yīng)的殘量矩陣范數(shù)的表達(dá)式,并且使它們的表達(dá)式分別與FOM和GMRES的迭代解與相應(yīng)的殘量矩陣范數(shù)的表達(dá)式結(jié)構(gòu)上保持類似。數(shù)值實(shí)驗(yàn)表明GL-GMRES方法要優(yōu)于GL-FOM方法。基于全局Arnoldi過程,結(jié)合加權(quán)技術(shù),提出了一種重啟的加權(quán)全局廣義最小殘量方法用于求解多右端向量線性方程組。給出了兩個(gè)定理和兩個(gè)命題,目的在于:1.保證了D內(nèi)積和D范數(shù)定義的合理性,并用拉直技術(shù)和克羅內(nèi)克積表達(dá)D內(nèi)積;2.使得所提的重啟的加權(quán)全局廣義最小殘量方法的迭代解和相應(yīng)的殘量矩陣范數(shù)能夠進(jìn)一步計(jì)算;3.確保所提的重啟的加權(quán)全局廣義最小殘量方法求出的迭代解和相應(yīng)的殘量矩陣范數(shù)的表達(dá)式與全局廣義最小殘量方法和廣義最小殘量方法求出的迭代解和相應(yīng)的殘量矩陣范數(shù)的表達(dá)式結(jié)構(gòu)上保持一致;4.保證通過加權(quán)全局Arnoldi過程構(gòu)造了塊Krylov子空間的一組D正交基;5.旨在說明重啟的加權(quán)全局廣義最小殘量算法具有尺度不變性。最后通過數(shù)值實(shí)驗(yàn)對收斂曲線、迭代次數(shù)、CPU消耗總時(shí)間方面進(jìn)行比較,驗(yàn)證了所提重啟的加權(quán)全局廣義最小殘量算法的有效性。
[Abstract]:This paper mainly studies how the right vectors for solving large sparse linear equations of global Krylov subspace methods. Because of many application areas, such as the problem of solving partial differential equations of fluid dynamics, circuit simulation, electromagnetic field calculation, linear control theory, requires solving multi right vector linear system. Therefore the establishment of a robust and efficient numerical method the right end of the vector of linear equations is very important. This paper consists of two parts following components: first, comprehensively and systematically elaborated the Krylov subspace method based on Arnoldi process, including complete orthogonal method (FOM) and generalized minimal residual method (GMRES), the numerical experiments show that the GMRES method is better than FOM according to this method. Secondly, the two methods are ideas, analysis of the Krylov subspace method based on global Arnoldi process, namely the global full orthogonalization method (GL-FOM) and the global generalized minimal Residual method (GL-GMRES). On this basis, using Givens transformation, an iteration solution and the corresponding residual matrix norm, and make them the expression of FOM and GMRES respectively and the iterative solution structure and residual matrix norm on the corresponding remain similar. Numerical experiments show the method to GL-GMRES better than the GL-FOM method. The global Arnoldi process based on combination weighting technique, a restart of the weighted global generalized minimal residual method for solving multi right vector linear equations is proposed. Two theorems are given and two propositions: 1., aims to ensure the rationality of D product and D norm is defined, and straighten Kronecker product technology and the expression of D 2. makes the inner product; iterative weighted global generalized minimal residual method for the restart of the solution and the corresponding residual matrix norm can be further calculated; 3. to ensure the resumption of the Consistent expression structure iteration expression and global GMRES iterative weighted global generalized minimal residual method to find solutions and the corresponding residual matrix norm and generalized minimal residual method to find solutions and the corresponding residual matrix norm; 4. is guaranteed by the weighted global Arnoldi process to construct a group D orthogonal basis block Krylov subspace; 5. to illustrate the weighted global generalized minimal residual algorithm restart with scale invariance. Finally, numerical experiments on the convergence curve, the number of iterations, the total time consumption of CPU were compared. The test validity of proposed weighted global generalized minimal residual algorithm restart.

【學(xué)位授予單位】：電子科技大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2015
【分類號】：O177

【引證文獻(xiàn)】

相關(guān)期刊論文 前1條

1 陳磊;陳業(yè)慧;金建;;應(yīng)用自適應(yīng)交叉近似算法快速計(jì)算導(dǎo)體RCS[J];新鄉(xiāng)學(xué)院學(xué)報(bào);2016年06期

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本文編號：1387462