Multipoint iterative parallel methods for solving equations

The pre-computation of data cubes is critical to improving the response time of on-line analytical processing (OLAP) systems and can be instrumental in accelerating data mining tasks in large data warehouses. However, as the size of data warehouses grows, the time it takes to perform this pre-computation becomes a significant performance bottleneck. This work presents a fast parallel method for generating ROLAP data cubes on a shared-nothing multiprocessor based on a novel optimized data partitioning technique. Since no shared disk is required, this method can be applied on highly scalable processor clusters consisting of standard PCs with local disks, connected via a data switch. The approach taken, which uses a ROLAP representation of the data cube, is well suited to large data warehouses on high dimensional data, and supports the generation of both fully materialized and partially materialized cubes. In comparison with previous approaches, our new method does significantly improve the scalability with respect to both, the number of processors and the I/O bandwidth (number of parallel disks). We have implemented our new parallel shared-nothing data cube generation method and evaluated it on a PC cluster, exploring relative speedup, scaleup, sizeup, output sizes and data skew. For a fact table with 16 million rows and 8 attributes, our parallel data cube generation method achieves close to optimal speedup for as many as 32 processors, generating a full data cube in under 7 minutes. For a fact table with 256 million rows and 8 attributes, our parallel method achieves optimal speedup for 32 processors, generating a full data cube consisting of /spl ap/7 billion rows (200 Gigabytes) in under 88 minutes.

Parallel diagonally implicit Runge-Kutta-Nyström methods

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.

Parallel iterative regularization methods for solving systems of ill-posed equations

Short-range molecular dynamics simulations of molecular systems are commonly parallelized by replicated-data methods, where each processor stores a copy of all atom positions. This enables computation of bonded 2-, 3-, and 4-body forces within the molecular topology to be partitioned among processors straightforwardly. A drawback to such methods is that the inter-processor communication scales as N, the number of atoms, independent of P, the number of processors. Thus, their parallel efficiency falls off rapidly when large numbers of processors are used. In this paper a new parallel method called force-decomposition for simulating macromolecular or small-molecule systems is presented. Its memory and communication costs scale as N/{radical}P, allowing larger problems to be run faster on greater numbers of processors. Like replicated-data techniques, and in contrast to spatial-decomposition approaches, the new method can be simply load-balanced and performs well even for irregular simulation geometries. The implementation of the algorithm in a prototypical macromolecular simulation code ParBond is also discussed. On a 1024-processor Intel Paragon, ParBond runs a standard benchmark simulation of solvated myoglobin with a parallel efficiency of 61% and at 40 times the speed of a vectorized version of CHARMM running on a single Cray Y-MP processor.

Abstract. A novel iterative method based on Picard iterations and Berstein polynomials is proposed for solving weakly singular and fractional Fredholm integral equations. On a uniform mesh, at each iterative step a Bernstein type spline is constructed by using the values computed at the previous step. The error estimates are obtained in terms of the Lipschitz constants and the convergence of the method is proved. Some numerical examples are presented in order to illustrate the accuracy of this iterative method. Mathematics Subject Classification (2010): 65R20. Keywords: Weakly singular and fractional Fredholm integral equations, iterative numerical method, piecewise Bernstein polynomials spline, order of convergence.

Newton trust-region methods with primary variable switching for simulating high temperature multiphase porous media flow

It is widely accepted that the asynchronous parallel methods are more suitable than the synchronous ones on a grid architecture. Indeed, they outperform the synchronous methods, because they overlap the communications of the synchronous methods with computations. However, they also usually execute more iterations than the synchronous ones and thus consume more energy. To reduce the energy consumption of the CPUs executing such methods, the Dynamic voltage and frequency scaling technique can be used. It lowers the frequency of a CPU to reduce its energy consumption, but it also decreases its computing power. Therefore, the frequency that gives the best trade-off between energy consumption and performance must be selected. This paper presents a new online frequency selecting algorithm for parallel iterative asynchronous methods running over grids. It selects a vector of frequencies that gives the best trade-off between energy consumption and performance. New energy and performance models were used in this algorithm to predict the execution time and the energy consumption of synchronous, asynchronous, or hybrid iterative applications running over grids. The proposed algorithm was evaluated on the SimGrid simulator. The experiments showed that synchronously applying the proposed algorithm to the asynchronous version of the application reduces on average its energy consumption by 22% and speeds it up by 5.72%. Finally, the proposed algorithm was also compared to a method that uses the well-known energy and delay product and the comparison results showed that it outperforms this method in terms of energy consumption and performance.

PARALLEL PREDICTOR-CORRECTOR METHODS FOR THE SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS II

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

In this paper we present three modified parallel multisplitting iterative methods for solving non-Hermitian positive definite systems Ax?=?b. The first is a direct generalization of the standard parallel multisplitting iterative method for solving this class of systems. The other two are the iterative methods obtained by optimizing the weighting matrices based on the sparsity of the coefficient matrix A. In our multisplitting there is only one that is required to be convergent (in a standard method all the splittings must be convergent), which not only decreases the difficulty of constructing the multisplitting of the coefficient matrix A, but also releases the constraints to the weighting matrices (unlike the standard methods, they are not necessarily be known or given in advance). We then prove the convergence and derive the convergent rates of the algorithms by making use of the standard quadratic optimization technique. Finally, our numerical computations indicate that the methods derived are feasible and efficient.

Parallel iterative methods for fast solution of large-scale acoustic radiation and scattering problems are developed using exact Dirichlet-to-Neumann (DtN), nonreflecting boundaries. A separable elliptic nonreflecting boundary is used to efficiently model unbounded regions surrounding elongated structures. We exploit the special structure of the non-local DtN map as a low-rank update of the system matrix to efficiently compute the matrix-by-vector products found in Krylov subspace based iterative methods. For the complex non-hermitian matrices resulting from the Helmholtz equation, we use a distributed-memory parallel BICG-STAB iterative method in conjunction with a parallel Jacobi preconditioner. Domain decomposition with interface minimization was performed to ensure optimal interprocessor communication. For the architectures tested, and using the MPICH version of MPI, we show that when implemented as a low-rank update, the non-local character of the DtN map does not signicantly decrease the scale up and parallel eciency versus a purely approximate local boundary condition.

A pair of parallel sequences $\{ u_k \} \to u,\{ v_k \} \to 0$ (1.3) is generated to solve the $n \times n$ linear system $Au = (I_n - B)u = f$. Convergence depends only on the geometry or shape of $\sigma (B)$, the set of eigenvalues of B. The parallel method is applied to the singularly perturbed convection-diffusion equation (6.1), when the Reynolds number in the direction of flow is large. Numerical comparisons with known results are given. Our theory also applies to the class of possibly nonsymmetric A with real spectrum (cf. Theorem 5.1) and to several other classes of systems as well. Computations to generate the sequences are relatively straightforward, as is indicated in our main result, Theorem 4.1. In fact, the parameters of the embracing ellipse for $\sigma (B)^2 $ (4.6) completely determine (i) the coefficients for the parallel sequences $\{ u_k \} \to u$ and $\{ v_k \} \to 0$ and (ii) the spectral radius (4.4), which characterizes their asymptotic convergence rate (2.4). Figure 5.1 illustrates some geometries for $\sigma (B)$ that are accommodated by our theory and Figure 7.1 shows the eigenvalue bowtie region arising from the convection-diffusion equation with large Reynolds number.

The present article deals with the effect of convexity in the study of the well-known Whittaker iterative method, because an iterative method converges to a unique solution \(t^*\) of the nonlinear equation \(\psi(t)=0\) faster when the function's convexity is smaller. Indeed, fractional iterative methods are a simple way to learn more about the dynamic properties of iterative methods, i.e., for an initial guess, the sequence generated by the iterative method converges to a fixed point or diverges. Often, for a complex root search of nonlinear equations, the selective real initial guess fails to converge, which can be overcome by the fractional iterative methods. So, we have studied a Caputo fractional double convex acceleration Whittaker's method (CFDCAWM) of order at least (\(1+2\zeta\)) and its global convergence in broad ways. Also, the faster convergent CFDCAWM method provides better results than the existing Caputo fractional Newton method (CFNM), which has (\(1+\zeta\)) order of convergence. Moreover, we have applied both fractional methods to solve the nonlinear equations that arise from different real-life problems.

This paper presents two partition methods that speed up iterative search methods applied to vehicle routing problems including a large number of vehicles. Indeed, using a simple implementation of taboo search as an iterative search method, every best‐known solution to classical problems was found. The first partition method (based on a partition into polar regions) is appropriate for Euclidean problems whose cities are regularly distributed around a central depot. The second partition method is suitable for any problem and is based on the arborescence built from the shortest paths from any city to the depot. Finally, solutions that are believed to be optimum are given for problems generated on a grid. © 1993 by John Wiley & Sons, Inc.