Modified parallel multisplitting iterative methods for non-Hermitian positive definite systems

Parallel multisplitting two-stage iterative methods with general weighting matrices for non-symmetric positive definite systems

Background and Objectives: Numerical iterative methods are widely used to compute reachability probabilities and expected rewards in probabilistic model checking of Markov decision processes. Several approaches have been proposed to improve the performance of these iterative methods. Reducing the total number of iterations is an important goal that is followed by several techniques in this paper.Methods: In this paper, we consider MDPs with different levels of non-determinism. We show that modified policy iteration performs better than the other standard iterative methods when the degree of non-determinism increases. We propose some novel methods to improve the performance of the modified policy iteration method. Our approach is to define several criteria to avoid useless iterations and updates of the modified policy iteration method. The first criterion considers the graphical structure of the model to define the number of iterations after each policy modification. The second criterion is a dynamic one to determine when a policy should be updated. The third proposed approach in this work is to use several priority heaps to select states for value updates.Results: Our proposed methods are implemented in the PRISM model checker and applied on several standard case studies. The results of experiments show that the running times of our approaches are less than the running time of the standard and previous methods for most case studies. The main reason for these results is that the total numbers of iterations and updates are reduced by our approaches, which results in an improvement in the performance of the method. While the running times are reduced by our approaches, the precision of computations are kept in most cases.Conclusion: The proposed techniques are able to reduce the number of iterations and accelerate the convergence to the optimal policies. The also prioritize the commutations to reduce the total number of updates.

The most common technique used to achieve primary deep biliary cannulation is the standard contrast-assisted method. To increase the success rate and reduce the risk of complications, a wire-guided cannulation strategy has been proposed. Prospective studies provided conflicting results as to whether the wire-guided cannulation technique increases the cannulation rate and reduces post-endoscopic retrograde cholangiopancreatography (post-ERCP) pancreatitis risk compared with the standard method. The objective of this study was to carry out a meta-analysis of randomized controlled trials (RCTs) that compares primary biliary cannulation and post-ERCP pancreatitis rates with the wire-guided method and the standard cannulation technique. Literature searches of electronic databases and online clinical trial registers up to March 2009 were conducted to identify RCTs comparing primary cannulation and post-ERCP pancreatitis rates with the wire-guided method and the standard cannulation technique. A meta-analysis of these clinical trials was performed. Five RCTs were included. Overall, the primary cannulation rates reported with the wire-guided cannulation technique and the standard method were 85.3 and 74.9%, respectively. The pooled analysis of all the selected studies comparing the wire-guided cannulation technique with the standard method yielded an odds ratio (OR) of 2.05 (95% confidence interval (CI): 1.27-3.31). The pooled analysis comparing the post-ERCP pancreatitis rates for the wire-guided-cannulation groups with those for the standard-method groups yielded an OR of 0.23 (95% CI: 0.13-0.41). This meta-analysis shows that the wire-guided technique increases the primary cannulation rate and reduces the risk of post-ERCP pancreatitis compared with the standard contrast-injection method. Further large, well-performed, randomized controlled studies are needed to confirm these findings.

Background: The Laryngeal Mask Airway (LMA) has traditionally become an alternative device for airway management. This clinical trial compared two standard and lateral techniques in inserting laryngeal masks in pediatric airway management.
 Methods: This single-blind clinical trial study was conducted on pediatrics aged 2-6 years who were candidates for elective inguinal herniorrhaphy with ASA I and II under general anesthesia. The pediatrics were randomly assigned to either the standard or lateral methods according to the random number table. The time required to insert LMA and airway pressure as well as the number of attempts to insert in both groups were measured. Data analysis was performed using SPSS version 25; the significance level was less than 0.05.
 Results: Chest movement and mean airway pressure had no significant difference in both groups (p>0.05). The mean LMA insertion time in the lateral and standard methods was 22.94±7.89 and 65±15.27 seconds, respectively (p=0.001). There was no case of mucosal damage in the lateral method, but ten children had mucosal damage in the standard technique (p=0.001). In the lateral method, LMA was inserted for the first time in 32 pediatrics (94.1%) and two pediatrics (5.9%) in the second time but in the standard group, LMA was inserted in the second time in 11 pediatrics (32.4%) (p=0.006).
 Conclusion: The lateral method for inserting LMA is practically easy, requires less effort, and has the least complications. Another advantage of the lateral method compared to the standard technique is that inserting a finger into the pharyngeal cavity is unnecessary.

SUMMARY A generalized skew-Hermitian triangular splitting iteration method is presented for solving non-Hermitian linear systems with strong skew-Hermitian parts. We study the convergence of the generalized skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems, as well as spectrum distribution of the preconditioned matrix with respect to the preconditioner induced from the generalized skew-Hermitian triangular splitting. Then the generalized skew-Hermitian triangular splitting iteration method is applied to non-Hermitian positive semidefinite saddle-point linear systems, and we prove its convergence under suitable restrictions on the iteration parameters. By specially choosing the values of the iteration parameters, we obtain a few of the existing iteration methods in the literature. Numerical results show that the generalized skew-Hermitian triangular splitting iteration methods are effective for solving non-Hermitian saddle-point linear systems with strong skew-Hermitian parts. Copyright © 2013 John Wiley & Sons, Ltd.

Some convergence conditions on successive over-relaxed (SOR) iterative method and symmetric SOR (SSOR) iterative method are proposed for non-Hermitian positive definite linear systems. Some examples are given to demonstrate the results obtained.

The main goal of this article is to introduce a numerical procedure for the calculation of solidification and melting using the PISO algorithm. Since coupling among velocity, pressure, temperature, and liquid fraction is important in phase-change problems, an improved PISO algorithm is required to couple these variables properly. To achieve this goal, the present study proposes the use of the enthalpy method for the inner iterations of the matrix solver, in order to predict more accurate temperature and liquid fraction that satisfy the energy balance criterion simultaneously. Another difference of the present method from other conventional coupling methods is in updating the permeability term of the momentum equation with liquid fractions predicted in the first correction step. This may couple the velocities with the liquid fractions more closely. A comparison of the proposed method with a standard iterative method is made for three different cases, showing that improved predictions are obtained in terms of computing speed and solution robustness.

The solution of large sparse linear systems arises in many applications, such as computational fluid dynamics and oil reservoir simulation. In realistic cases the matrices are often so large that they require large scale distributed parallel computing to obtain the solution of interest in a reasonable time. In this paper we discuss the design and implementation of the AmgX library, which provides drop-in GPU acceleration of distributed algebraic multigrid (AMG) and preconditioned iterative methods. The AmgX library implements both classical and aggregation-based AMG methods with different selector and interpolation strategies, along with a variety of smoothers and preconditioners, including block-Jacobi, Gauss--Seidel, and incomplete-LU factorization. The library contains many of the standard and flexible preconditioned Krylov subspace iterative methods, which can be combined with any of the available multigrid methods or simpler preconditioners. The parallelism in the aggregation scheme exploits parallel graph matching techniques, while the smoothers and preconditioners often rely on parallel graph coloring algorithms. The AMG algorithm implemented in the AmgX library achieves $2$--$5\times$ speedup on a single GPU against a competitive implementation on the CPU. As will be shown in the numerical experiments section, both setup and solve phases scale well across multiple nodes, sustaining this performance advantage.

A numerical solution of general frictional contact problems by the direct boundary element and mathematical programming approach

Modified HSS iteration methods for a class of non-Hermitian positive-definite linear systems

In this paper we show the results of some research carried out on parallel iterative methods to solve equations. In particular we study general classes of one point parallel methods and multipoint ones without memory, and we point out the convergence order of these methods and the conditions which are both necessary and sufficient for them to be optimal. In addition we prove that the convergence order for multipoint parallel procedures without memory cannot be more thenr(r+)m−1, wherer indicates the number of the parallel processor used andm the number of the functions and eventual derivatives, calculated not simultaneously in every iteration.

This paper proposes a complete orthogonal decomposition (COD) algorithm for solving weighted least-squares problems. In applications, the weight matrix can be highly ill conditioned, and this can cause standard methods like QR factorization to return inaccurate answers in floating-point arithmetic. Stewart and Todd independently established a norm bound for the weighted least-squares problem that is independent of the weight matrix. Vavasis proposed a definition of a "stable" solution of weighted least squares based on this norm bound: The solution computed by a stable algorithm must satisfy an accuracy bound that is not affected by ill conditioning in the weight matrix. A forward error analysis shows that the COD algorithm is stable in this sense, but it is simpler and more efficient than the algorithm proposed by Vavasis. Our forward error bound is contrasted to the backward error analysis of other previous works on weighted least squares.

Central screw positioning in the scaphoid provides biomechanical advantages. A prospective randomized study of six fluoroscopically guided and six electromagnetically navigated screw (ENS) placements was performed on human cadavers. Accuracy of screw position was determined. Intraoperative fluoroscopy exposure times, readjustments of drilling directions, complete restarts and complications were documented. The ENS method provided a mean time benefit of 7.34 min compared with the standard method and the mean screw length ratio (SLR coronar: ENS 0.96 ± 0.04 mm, SFF: 0.92 ± 0.04 mm, P = 0.065; SLR sagittal: ENS 0.98 ± 0.02 mm, SFF: 0.91 ± 0.04 mm, P = 0.009) and the screw axis deviation angle (AD coronar: ENS 3.33 ± 2.34°, SFF: 10.33 ± 2.58°, P = 0.002; AD sagittal: ENS 2.83 ± 0.98°, SFF: 11.00 ± 6.16°, P = 0.002) were lower. Using the electromagnetic navigation procedure no drilling readjustments or restarts were required, no cortical breach occurred. Compared with the standard fluoroscopic technique, the ENS method used in this study showed higher accuracy, less complications, required less operation and radiation exposure time.

A simple procedure is proposed for the determination of the water supplies and demands which maximize the total net benefit from water use in a region. The technique is in some ways a standard iterative market method, and, in fact, the convergence to the optimal solution is achieved when the marginal benefit at each demand point is equal to the marginal cost of delivering water to that point. The possibility of applying this technique to a given problem can immediately be ascertained from the pattern of the links between supply and demand units. An example is presented in which the method is applied to analyze the possible interbasin water transfer in the Northwest Water Plan in Mexico.

Recently, acoustic analysis techniques are used in the design of acoustic environments both inside and outside of rooms containing a sound source. However, when the analysis area is expanded or the frequency is increased, unknowns of a solving liner equation increase, and large-scale analyses becomes necessary. Therefore, in this study, a large-scale acoustic analysis method has been developed based on a parallel finite element method. In the proposed large-scale analysis method, an iterative method is applied to the interface problem as the iterative domain decomposition method, which is known to be an effective parallelization method. To improve the convergence of the iterative method, balancing domain decomposition has been applied as a preconditioner step, and its effectiveness in large-scale acoustic analysis is demonstrated. The acoustic analysis code is developed in this study that is shown to be capable of the large-scale analysis of finite element models on the order of tens of millions of elements with high accuracy.