A composite step bi-conjugate gradient algorithm for nonsymmetric linear systems

The Bi-Conjugate Gradient (BCG) algorithm is the simplest and most natural generalization of the classical conjugate gradient method for solving nonsymmetric linear systems. It is well-known that the method suffers from two kinds of breakdowns. The first is due to the breakdown of the underlying Lanczos process and the second is due to the fact that some iterates are not well-defined by the Galerkin condition on the associated Krylov subspaces. In this paper, we derive a simple modification of the BCG algorithm, the Composite Step BCG (CSBCG) algorithm, which is able to compute all the well-defined BCG iterates stably, assuming that the underlying Lanczos process does not break down. The main idea is to skip over a step for which the BCG iterate is not defined.The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440.The work of this author was supported by the Office of Naval Research under contracts N00014-90J-1695 and N00014-92J-1890, the Department of Energy under contract DE-FG03-87ER25307, the National Science Foundation under contracts ASC 90-03002 and 92-01266, and the Army Research Office under contract DAAL03-91-G-0150. Part of this work was completed during a visit to the Computer Science Dept., The Chinese University of Hong Kong.     

The internal magnetic field distribution, and single exponential magnetic resonance free induction decay, in rocks

When fluid saturated porous media are subjected to an applied uniform magnetic field, an internal magnetic field, inside the pore space, is induced due to magnetic susceptibility differences between the pore-filling fluid and the solid matrix. The microscopic distribution of the internal magnetic field, and its gradients, was simulated based on the thin-section pore structure of a sedimentary rock. The simulation results were verified experimentally. We show that the 'decay due to diffusion in internal field' magnetic resonance technique may be applied to measure the pore size distribution in partially saturated porous media. For the first time, we have observed that the internal magnetic field and its gradients in porous rocks have a Lorentzian distribution, with an average gradient value of zero. The Lorentzian distribution of internal magnetic field arises from the large susceptibility contrast and an intrinsic disordered pore structure in these porous media. We confirm that the single exponential magnetic resonance free induction decay commonly observed in fluid saturated porous media arises from a Lorentzian internal field distribution. A linear relationship between the magnetic resonance linewidth, and the product of the susceptibility difference in the porous media and the applied magnetic field, is observed through simulation and experiment.     

X光激光在等离子体中的传播

由于等离子体柱中有垂直于轴向的电子密度梯度,致使X光激光在等离子体柱内传播时发生折射。折射效应限制了增益区的长度,采用“双靶对接”的实验方法,能有效地抑制折射的影响。本文从理论上阐述了折射效应对激光增益的影响相当于在小讯号增益系数上加了一个负项;并给出X光激光出靶端的强度角分布及偏离轴向的折射角;为“双靶对接”实验提供靶面的夹角值。理论计算结果与实验结果符合得很好。     

Treatment of near-breakdown in the CGS algorithm

Lanczos' method for solving the system of linear equationsAx=b consists in constructing a sequence of vectors (x _k ) such thatr _k =b–Ax _k =P _k (A)r ₀ wherer ₀=b–Ax ₀.P _k is an orthogonal polynomial which is computed recursively. The conjugate gradient squared algorithm (CGS) consists in takingr _k =P _k ² (A)r₀. In the recurrence relation forP _k , the coefficients are given as ratios of scalar products. When a scalar product in a denominator is zero, then a breakdown occurs in the algorithm. When such a scalar product is close to zero, then rounding errors can seriously affect the algorithm, a situation known as near-breakdown. In this paper it is shown how to avoid near-breakdown in the CGS algorithm in order to obtain a more stable method.     

Parallel algorithms for panel methods

A parallel algorithm for the solution of potential flow problems using the panel method of Hess and Smith and conjugate and bi-conjugate gradient techniques is presented. Analysis of the parallelism for the matrix. solvers shows the algorithms to have scalable properties as the problem size grows indefinitely large. Speed-up and efficiency values are presented along with experimental and theoretical values for the optimum number of processors for maximum speed-up. It is envisaged that the parallel techniques presented here have applications using other boundary integral methods for solving engineering problems of a more complex nature.     

Donor-hydrogen complexes in crystalline silicon

Summary Experimental results are presented on the study of Sb-H complexes in crystalline silicon, employing¹¹⁹Sb→¹¹⁹Sn source M?ssbauer spectroscopy and a low-energy H implantation technique. In addition to a visible component, we observe a large decrease of the M?ssbauer intensity associated with the trapping of hydrogen, even at low temperatures. This is interpreted as the formation of a component with a negligible recoilless fraction. The different M?ssbauer components were studied as a function of H dose, H-implantation temperature and annealing temperature. The data show that the visible component is associated with the well-known SbH complex, whereas the invisible component is associated with the formation of SbH _n (n≥2) complexes. We show that these complexes are in thermal equilibrium with a larger hydrogen reservoir (H ₂ ^* ), which governs their thermal stability. No Sb-H complexes are observed inp-type Si after H-implantation, in agreement with the current belief that hydrogen has a deep donor level in the gap. The microscopic structure of the various Sb-H and Sn-H complexes was studied with first-principles calculations using the pseudopotentialdensity-functional approach. The structure of the Sb-H complex is found to be similar to the P-H complex, with the H in an antibonding site of a Si atom neighbouring the Sb impurity. For SbH₂ three configurations are found with energies differing by less than ≈ 0.1 eV. We find that the reaction SbH+H≠SbH₂ is exothermic. We argue that the SbH₂ complexes are shallow donors, irrespective of the structure. Therefore, the formation of SbH₂ may depassivate the sample. Paper presented at ICAME-95, Rimini, 10–16 September 1995.     

Jacobi condition for elliptic forms in Hilbert spaces

In this paper, we give the definitions of conjugate and semi-conjugate points for a quadratic elliptic form in a Hilbert space, and we state the corresponding necessary and sufficient conditions (the Jacobi conditions) for the form to be positive (nonnegative). We apply the abstract results to a one-dimensional problem in the calculus of variations where both endpoints are allowed to vary. The conditions that we obtain complete and generalize previously known results.This research was partly supported by MURST Research Grant, Teoria del Controllo dei Sistemi Dinamici.     

Object‐oriented parallel algorithms for computing three‐dimensional isopycnal flow

In this paper, we derive an object‐oriented parallel algorithm for three‐dimensional isopycnal flow simulations. The matrix formulation is central to the algorithm. It enables us to apply an efficient preconditioned conjugate gradient linear solver for the global system of equations, and leads naturally to an object‐oriented data structure design and parallel implementation. We discuss as well, in less detail, a similar algorithm based on the reduced system, suitable also for parallel computation. Favorable performances are observed on test problems. Copyright © 2002 John Wiley & Sons, Ltd.     

Al quadrupole interaction in zeolites loaded with probe molecules—a quantum-chemical study of trends in electric field gradients and chemical bonds in clusters

The electric field gradient (EFG) has been calculated in zeolite clusters at the aluminium site surrounded by four SiO₄ tetrahedra. Density functional theory (DFT) with the 6-31G^{* *} basis set has been employed. Formation of a Brønsted acid site by protonation of one oxygen atom of the A1O₄ tetrahedron perturbs the coordination of aluminium, i.e., the corresponding Al-O bond is considerably weaker than in the unprotonated case. This leads to a large EFG, and the calculated quadrupole coupling constant (QCC) for ²⁷Al is 18.2 MHz. Different probe molecules were adsorbed on the Brønsted site. The hydrogen bond formed between the acid proton and the probe molecule weakens the zeolitic O---H bond. For conservation of the overall bond order of the oxygen atom, its bonds to the neighboring tetrahedral atoms (Si, Al) become stronger. As a consequence, the perturbation of the AlO₄ tetrahedron and the EFG at the aluminium position decrease depending on the strength of the hydrogen bond. Perturbation of an oxygen atom of the AlO₄ tetrahedron by accepting a hydrogen bond from the base molecule also affects the corresponding AlO---O bond order. A linear correlation is found between the calculated QCC constants for ²⁷Al and the AlO---O bond orders of the oxygen atoms which are perturbed by protonation or by hydrogen bonds. A geometrical shear strain parameter and a simple electrostatic point charge model are less successful at predicting the trends in EFG which clearly shows the importance of the chemical bonds. Published by Elsevier Science B.V.     

工序能力Bayes推断

本文从Ｂａｙｅｓ观点研究工序能力，对无信息先验和共轭先验，给出了Ｃｐ的后验分布、条件期望估计和最大后验估计、Ｂａｙｅｓ置信下限和判断工序是否有能力的临界值，适于对相似工序作统计推断。