USE OF QMR ALGORITHM TO CALCULATE VERY SLOW-MOTIONAL SPECTRA

使用 QMR 算法计算非常慢运动的光谱

• 批准号: 7602665
• 负责人: YUN-WEI CHIANG
• 金额: $ 0.12万
• 依托单位: CORNELL UNIVERSITY
• 依托单位国家: 美国
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://reporter.nih.gov/project-details/7602665
• 关键词: Algorithms; Complex; Computer Retrieval of Information on Scientific Projects Database; Coupled; Equation; Funding; Grant; Institution; Methods; Monitor; Numbers; Research; Research Personnel; Residual state; Resources; Solutions; Source; Systemic Lupus Erythematosus; United States National Institutes of Health; programs; vector

This subproject is one of many research subprojects utilizing the resources provided by a Center grant funded by NIH/NCRR. The subproject and investigator (PI) may have received primary funding from another NIH source, and thus could be represented in other CRISP entries. The institution listed is for the Center, which is not necessarily the institution for the investigator. The diagonalization of the stochastic Liouville (SL) matrix using the Lanczos algorithm (LA) is optimized with the aid of the conjugate (CG) method for calculating 2D-ELDOR spectra in the slow-motional regime. In each step of the LA recursion, the convergence is monitored according to the residual norm calculated in the CG iterations. Thus the methods of the CG and LA can be coupled together to tri-diagonalize a large symmetric and complex sparse matrix efficiently. The LA-CG has been very successfully used to tri-diagonalize the SLE matrix in the slow-motional regime. However, due to the loss of orthogonality in the LA vectors, we found that the LA-CG method can break down in calculating spectra in the very slow-motional regime. This is mainly due to the fact that CG requires the residual norm to calculate the solution direction in each iteration. This indicates that the whole algorithm would be spoiled due to the loss of orthogonality after a certain number of LA projections. This would be particularly true when one calculates a very slow-motional spectrum that requires a large number of LA projections in order to adequately obtain good eigenvalues. The quasi-minimal residual (QMR), which was originally developed to be a linear equation solver, has been adapted for determining the number of LA projections in our program. In the QMR method, the solution vector is obtained by minimizing the quasi-residual norm, using QR factorization. It is of great advantage to us to replace CG with QMR to determine the number of LA projections, because the residual norm is NOT used iteratively as in the CG method to determine the solution direction in each iteration. Therefore, this LA-QMR method that we utilize provides an advantage of avoiding the breakdown noted above.

该子项目是利用该技术的众多研究子项目之一 资源由 NIH/NCRR 资助的中心拨款提供。子项目和 研究者 (PI) 可能已从 NIH 的另一个来源获得主要资金， 因此可以在其他 CRISP 条目中表示。列出的机构是 对于中心来说，它不一定是研究者的机构。 使用 Lanczos 算法 (LA) 的随机刘维尔 (SL) 矩阵的对角化借助共轭 (CG) 方法进行了优化，用于计算慢运动状态下的 2D-ELDOR 谱。在 LA 递归的每一步中，根据 CG 迭代中计算的残差范数来监控收敛性。因此，CG 和 LA 的方法可以耦合在一起，以有效地对大型对称且复杂的稀疏矩阵进行三对角化。 LA-CG 已非常成功地用于慢动作状态下的 SLE 矩阵的三对角化。 然而，由于 LA 向量中正交性的损失，我们发现 LA-CG 方法在非常慢运动的情况下计算谱时可能会失败。这主要是由于CG在每次迭代中都需要残差范数来计算解方向。这表明，在一定数量的 LA 投影之后，由于正交性的丧失，整个算法将被破坏。当计算一个非常慢的运动谱（需要大量 LA 投影才能充分获得良好的特征值）时，尤其如此。 准最小残差 (QMR) 最初是作为线性方程求解器开发的，现已适用于确定我们程序中 LA 投影的数量。在 QMR 方法中，通过使用 QR 分解最小化准残差范数来获得解向量。用 QMR 代替 CG 来确定 LA 投影的数量对我们来说有很大的优势，因为残差范数不像 CG 方法那样迭代地使用来确定每次迭代的解方向。因此，我们使用的这种 LA-QMR 方法具有避免上述故障的优点。