Performance and robustness of block constraint preconditioners in finite element coupled consolidation problems

Block constraint preconditioners are a most recent development for the iterative solution to large‐scale, often ill‐conditioned, coupled consolidation problems. A major limitation to their practical use, however, is the somewhat difficult selection of a number of user‐defined parameters (at least 4) in a more or less optimal way. The present paper investigates the robustness of three variants of the block constraint preconditioning in relation to the above parameters. A theoretical analysis of the eigenspectrum of the preconditioned matrix provides relatively simple bounds of the eigenvalues as a function of these parameters. A number of test problems used to validate the theoretical results show that both the mixed constraint preconditioner (MCP) combined with the symmetric quasi‐minimal residual (SQMR) solver and the MCP triangular variant (T‐MCP) combined with the bi‐conjugate gradient stabilized (Bi‐CGSTAB) are efficient and robust tools for the solution to difficult Finite Element‐discretized coupled consolidation problems. Moreover, the practical selection of the user‐defined parameters is relatively easy as a stable behavior is observed for a wide range of fill‐in degree values. The theoretical bounds on the eigenspectrum of the preconditioned matrix may help to suggest the most appropriate parameter combination. Copyright © 2009 John Wiley & Sons, Ltd.     

Comparison of Two-Level Preconditioners Derived from Deflation,Domain Decomposition and Multigrid Methods

For various applications, it is well-known that a multi-level, in particular two-level, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors.     

Toward Non Commutative Numerical Analysis: High Order Integration in Time

We review some methods for high precision time integration: it is not easy to ensure stability, precision and numerical efficiency at the same time. Operator splitting—when it works—can be a good way to satisfy all these constraints; in some cases, the order of the splitting schemes can be enhanced by extrapolation; nevertheless, the applicability of splitting is limited due to non commutativity. As an alternative to splitting, we introduce preconditioned Runge–Kutta (PRK) schemes: the preconditioning is included in the scheme, instead of being put aside for implementation. Examples of PRK schemes are given including the extrapolation of the residual smoothing scheme, and sufficient conditions for stability are described.     

A review of sensory preconditioning.

Research employing the sensory preconditioning (SPC) paradigm is examined in the light of 3 questions: (a) can SPC be dealt with by learning theory, (b) if so, what principles does it follow, (c) what problems does the theorist face in integrating the SPC data into his system? The author concludes that SPC, tentatively, can be considered as a phenomenon in learning and that there is a need for further research in determining the principles SPC follows. A suggested paradigm for such research is presented. Finally, one problem the theorist must face is that reinforcement, as classically defined, is an unnecessary condition for SPC to be effective. At the moment, a more tenable approach would be a Hebbian-type view or an S-S analysis. 28 refs. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Application of polynomial preconditioners to conservation laws

Polynomial preconditioners which are suitable in implicit time-stepping methods for conservation laws are reviewed and analyzed. The preconditioners considered are either based on a truncation of a Neumann series or on Chebyshev polynomials for the inverse of the system-matrix. The latter class of preconditioner is optimal in a space of polynomials of certain degree if the matrix has only real eigenvalues and a non-singular system of eigenvectors. The preconditioning can be applied to any convergent splitting of the system matrix, i.e. to any classical implicit time-stepping method for conservation laws that is based on a quasi-Newton iteration. An efficient implementation based on SSOR is presented and the approach is applied to simulations of the viscous unsteady Burgers equation and to inviscid steady flow around an airfoil in two spatial dimensions to illustrate the method in large-scale computations. For viscous flows the efficiency increase due to preconditioning is considerable.     

弹性波全波形反演中的四种优化方法对比

弹性波全波形反演(EFWI)是一种高精度成像方法.由于EFWI本质是一个强非线性问题,因此常采用局部优化算法进行求解,不同优化算法的反演结果差异很大.在较为常用的共轭梯度法(CG)、L-BFGS法(Limited-memory Broyden-Fletcher-Goldfarb-Shanno Algorithm)的基础...     

基于预条件共轭梯度法的直流电阻率三维有限元正演研究

对直流电阻率法勘探而言,对其反演结果的精度和速度的要求越来越高,这就需要提出一套优化有限元数值正演速度和精度的计算方案。设计了系数矩阵的一维非零元素压缩存储模式,设置了索引数组以便按照行号和列号对元素进行索引,与变带宽存储模式相比,其内存占用量明显减小。为提高正演计算速度,利用预条件共轭梯度法（PCG）求解有限元中的大型稀疏线性方程组。在PCG法中,将雅可比迭代中的对角阵作为预处理矩阵,与其它预处理矩阵相比,其具有求逆方便、无需存储空间的特点,使得大型线性方程组的求解速度大大提高。另外,在直流电阻率三维正演中,采用了异常电位法,提高了电源点附近的解的精度。以二层地层的电阻率勘探为例,初步验证了计算方案的实用性。利用上述方案,重点对隧道含水断层的电阻率法超前探测进行了有限元数值正演,并进行了相应的物理模型试验。对比显示,数值正演结果与试验数据基本一致,且数值正演的速度和精度均显著提高。     

Mesh independence of galerkin approach by preconditioning

Preconditioning the discrete Galerkin system for a bounded elliptic linear problem, by the Choleski factors of the Gramian, we obtain a mesh independent condition number. Moreover, the rate of convergence of the multigrid method is also mesh independent. Our example give a justification to the oscillatory movement of the discrete solution of a well known problem.     

Shifted FSAI preconditioners for the efficient parallel solution of non‐linear groundwater flow models

The present paper investigates the performance of a shifted factorized sparse approximate inverse as a parallel preconditioner for the iterative solution to the linear systems arising in the finite element discretization of non‐linear groundwater flow models. The shift strategy is based on an inexpensive preconditioner update exploiting the structure of the coefficient matrix. The proposed algorithm is experimented with in the parallel simulation of a large‐scale real multi‐aquifer system characterized by a stochastic distribution of the hydraulic conductivity. The numerical results show that the shifted factorized sparse approximate inverse algorithm may yield an overall computational gain up to 300% with respect to the non‐shifted scheme with an excellent parallel efficiency. Copyright © 2011 John Wiley & Sons, Ltd.     

The Effect of Treadmill Training Pre-Exercise on Glutamate Receptor Expression in Rats after Cerebral Ischemia

Physical exercise has been demonstrated to be neuroprotective in both clinical and laboratory settings. However, the exact mechanism underlying this effect is unclear. Our study aimed to investigate whether pre-ischemic treadmill training could serve as a form of ischemic preconditioning in a rat model undergoing middle cerebral artery occlusion (MCAO). Thirty-six rats were divided into three groups: a sham control group, a non-exercise with operation group and an exercise with operation group. After treadmill training, ischemia was induced by occluding the MCA for 2 h, followed by reperfusion. Half of the rats in each group were sacrificed for mRNA detection of mGluR5 and NR2B 80 min after occlusion. The remaining animals were evaluated for neurological deficits by behavioral scoring and then decapitated to assess the infarct volume. The mRNA expression of mGluR5 and NR2B was detected by real-time PCR. The results suggest that pre-ischemic treadmill training may induce brain ischemic tolerance by reducing the mRNA levels of mGluR5 and NR2B, and thus, the results indicate that physical exercise might be an effective method to establish ischemic preconditioning.