Parallel Iterative Schemes Research Articles

A New High-Order Fractional Parallel Iterative Scheme for Solving Nonlinear Equations

Solving fractional-order nonlinear equations is crucial in engineering, where precision and accuracy are essential. This study introduces a novel fractional parallel technique for solving nonlinear equations. To enhance convergence, we incorporate a simple root-finding method of order 3γ + 1 as a correction term in the parallel scheme. Theoretical analysis shows that the parallel scheme achieves a convergence order of 6γ + 3. Using a dynamical system approach, we identify optimal parameter values, and the symmetry in the dynamical planes for different fractional parameters demonstrates the method’s stability and consistency in handling nonlinear problems. These parameter values are applied to the parallel scheme, yielding highly consistent results. Several engineering problems are examined to assess the method’s efficiency, stability, and consistency compared to existing methods.

Performance investigations of auxiliary‐space Maxwell solver preconditioned iterative algorithm for controlled‐source electromagnetic induction problems with electrical anisotropy

AbstractMany studies have highlighted the superior performance of iterative solvers employing the auxiliary‐space Maxwell solver preconditioner in controlled‐source electromagnetic induction problems featuring isotropic conductivity. The importance of considering the presence of electrical anisotropy in controlled‐source electromagnetic data has been well recognized. However, considering anisotropic conductivity will impose difficulty in robustly solving the final system of linear equations as the electrical anisotropy may significantly increase its condition number and degrade the performances of iterative solvers. Whether or not iterative solvers using the auxiliary‐space Maxwell solver preconditioner have similar superior performances in the case of arbitrary electrical anisotropy is still an issue to be discussed. In this study, within the framework of finite element simulation employing unstructured tetrahedral meshes, we conduct a comprehensive examination to evaluate the performance of the flexible generalized minimum residual solver with the auxiliary‐space Maxwell solver preconditioner for three‐dimensional controlled‐source electromagnetic forward modelling problems involving arbitrary anisotropic media. Tests on synthetic one‐ and three‐dimensional models show that our iterative scheme performs better than widely used iterative or direct solvers for controlled‐source electromagnetic anisotropy forward problems. Its convergence rate is nearly independent of working frequencies, anisotropy ratio and problem size. Finally, we applied the newly developed parallel iterative scheme to the Bay du Nord reservoir in a complicated real‐life offshore hydrocarbon exploration scenario characterized by anisotropic conductivity, in which our iterative scheme with an auxiliary‐space Maxwell solver preconditioner has good robustness. Furthermore, we investigated how data responses at different frequencies are sensitive to the actual hydrocarbon reservoir. Our sensitivity analysis revealed that data at large measuring offsets are considerably more sensitive to the reservoir than data at shorter measuring offsets. We also assessed the impact of neglecting anisotropy in data analysis for the realistic example and found that ignoring anisotropy can lead to noticeable changes in the data. This suggests that considering anisotropy in the interpretation of the observed data is essential to guarantee the precision of controlled‐source electromagnetic field surveys.

Viscosity approximation method for solving the multiple-set split equality common fixed-point problems for quasi-pseudocontractive mappings in Hilbert spaces

We propose a parallel iterative scheme with viscosity approximation method which converges strongly to a solution of the multiple-set split equality common fixed point problem for quasi-pseudocontractive mappings in real Hilbert spaces. We also give an application of our result to approximation of minimization problem from intensity-modulated radiation therapy. Finally, we present numerical examples to demonstrate the behaviour of our algorithm. This result improves and generalizes many existing results in literature in this direction.

PBR Clutter Suppression Algorithm Based on Dilation Morphology of Non-Uniform Grid

Many new challenges are faced by the PBR (passive bi-static radar) employing non-cooperative radar illuminators. After the CFAR (constant false alarm) processor, the appearance of the amount of false alarm clutter points impacts the following tracing performance. To enhance the PBR tracing performance, we consider to reduce these clutter points before target tracing as soon as possible. In this paper, we propose a PBR clutter suppression algorithm based on dilation morphology of non-uniform grid. Firstly, we construct the non-uniform polar grid based on the acquisition geometry of PBR. Then, with the help of the grid platform, we separate the false alarm clutter points based on the dilation morphology. To efficiently operate the algorithm, we build up its parallel iteration scheme. To verify the performance of the proposed algorithm, we utilize both simulated data and field data to do the experiment. Experimental results show that the algorithm can effectively suppress most of the clutter points. Besides, we respectively combine the proposed suppression algorithm with two typical tracking algorithms to test the performance. Experimental results reveal that the compound tracing algorithm outperforms the traditional one. It can enhance the PBR tracing performance, reduce the occurrence probability of false tracks and meanwhile save time.

GRIDHPC: A decentralized environment for high performance computing

SummaryThis paper presents GRIDHPC, a decentralized environment dedicated to high performance computing. It relies on the reconfigurable multi network protocol RMNP to support data exchange between computing nodes on multi network systems with Ethernet, Infiniband, Myrinet, and on OpenMP for the exploitation of computing resources of multicore CPU. We report on scalability of several parallel iterative schemes of computation combined with GRIDHPC. In particular, the experimental results show that GRIDHPC scales up when combined with asynchronous iterative schemes of computation.

A Conservative Parallel Iteration Scheme for Nonlinear Diffusion Equations on Unstructured Meshes

AbstractIn this paper, a conservative parallel iteration scheme is constructed to solve nonlinear diffusion equations on unstructured polygonal meshes. The design is based on two main ingredients: the first is that the parallelized domain decomposition is embedded into the nonlinear iteration; the second is that prediction and correction steps are applied at subdomain interfaces in the parallelized domain decomposition method. A new prediction approach is proposed to obtain an efficient conservative parallel finite volume scheme. The numerical experiments show that our parallel scheme is second-order accurate, unconditionally stable, conservative and has linear parallel speed-up.

A Low-Complexity Algorithm of Joint Multiple Frequency Offsets and Channels Estimation in Cooperative Relay Systems

A low complexity joint estimator of multiple CFOs and Channels is presented in cooperative relay systems. The new algorithm first utilizes correlation-based frequency estimator to get CFOs initial estimation, and then serial interference cancellation based on correlation properties of training sequence in different relays is done to obtain the initial channel estimation. Moreover, a parallel iteration scheme with interference cancellation is proposed to reduce time complexity of the traditional serial iteration. In the overall process, matrix inversion is avoided. Thus, the complexity of the proposed algorithm in both time and computation is much less than the existing algorithms. In last, simulation results verify the iterative algorithm achieves a good performance in Decode and Forward (DF) relay system.

On a multiscale strategy and its optimization for the simulation of combined delamination and buckling

SUMMARYThis paper investigates a computational strategy for studying the interactions between multiple through‐the‐width delaminations and global or local buckling in composite laminates taking into account possible contact between the delaminated surfaces. To achieve an accurate prediction of the quasi‐static response, a very refined discretization of the structure is required, leading to the resolution of very large and highly nonlinear numerical problems. In this paper, a nonlinear finite element formulation along with a parallel iterative scheme based on a multiscale domain decomposition is used for the computation of three‐dimensional mesoscale models. Previous works by the authors already dealt with the simulation of multiscale delamination assuming small perturbations. This paper presents the formulation used to include geometric nonlinearities into this existing multiscale framework and discusses the adaptations that need to be made to the iterative process to ensure the rapid convergence and the scalability of the method in the presence of buckling and delamination. These various adaptations are illustrated by simulations involving large numbers of DOFs. Copyright © 2012 John Wiley & Sons, Ltd.

Numerical performance of parallel group explicit solvers for the solution of fourth order elliptic equations

Many applications in applied mathematics and engineering involve numerical solutions of partial differential equations (PDEs). Various discretisation procedures such as the finite difference method result in a problem of solving large, sparse systems of linear equations. In this paper, a group iterative numerical scheme based on the rotated (skewed) five-point finite difference discretisation is proposed for the solution of a fourth order elliptic PDE which represents physical situations in fluid mechanics and elasticity. The rotated approximation formulas lead to schemes with lower computational complexities compared to the centred approximation formulas since the iterative procedure need only involve nodes on half of the total grid points in the solution domain. We describe the development of the parallel group iterative scheme on a cluster of distributed memory parallel computer using Message-Passing Interface (MPI) programming environment. A comparative study with another group iterative scheme derived from the centred difference formula is also presented. A detailed performance analysis of the parallel implementations of both group methods will be reported and discussed.

Stabilizing large‐scale generalized systems on parallel computers using multithreading and message‐passing

AbstractWe discuss the parallelization of an efficient algorithm for the partial stabilization of large‐scale linear control systems in generalized state‐space form. The algorithm is composed of highly parallel iterative schemes that appear in the computation of certain matrix functions. Here we evaluate different approaches to exploit parallelism at two levels, based on threads and processes. Our experimental results on a cluster of symmetric multiprocessors and a CC‐NUMA platform show that the efficiency of the matrix operations underlying the iterative schemes carry over to the parallel implementation of the stabilization algorithm. Copyright © 2006 John Wiley & Sons, Ltd.

Parallel iterative difference schemes based on prediction techniques for transport method

Particle transport calculations are inherently expensive because of the large variable space that must be discretized, and usually the fully implicit schemes in time discretization are used. There exist some bottleneck problems in implementing the Sn transport method on massively parallel computers, such as strong correlation of data and algorithmic consecutiveness. In this paper the spatial domain decomposition procedures are studied for numerically solving the discrete ordinates Sn transport equations. We introduce a parallel solution strategy for transport calculations,which combines the source iterative methods with a technique based on the prediction-correction on the interface of sub-domains. Moreover the two choices of initial values of the source iterative methods are proposed and compared. Numerical experiments show the parallel iterative schemes with interface prediction-correction are unconditionally stable, and have the same accuracy as the original implicit schemes. Moreover, they have fairly high parallelism, and high efficiency due to the construction of our parallel scheme and the good choice of the initial guess of source iteration.

Parallel Iterative Schemes of Linear Algebra with Application to the Stability Analysis of Solutions of Systems of Linear Differential Equations

Parallel modifications of linear iteration schemes are proposed that are used to solve systems of liner algebraic equations and to achieve the time complexity equal to T = log2 k · O(log2 n), where k is the number of iterations of an original scheme and n is the dimension of a system. Such schemes are extended to the case of approximate solution of systems of linear differential equations with constant coefficients. Based on them and using a program, the stability of solutions in the Lyapunov sense is analyzed.

Parallel iterative solvers involving fast wavelet transforms for the solution of BEM systems

This paper describes the parallelization of a strategy to speed up the convergence of iterative methods applied to boundary element method (BEM) systems arising from problems with non-smooth boundaries and mixed boundary conditions. The aim of the work is the application of fast wavelet transforms as a black box transformation in existing boundary element codes. A new strategy was proposed, applying wavelet transforms on the interval, so it could be used in case of non-smooth coefficient matrices. Here, we describe the parallel iterative scheme and we present some of the results we have obtained.

On the convergence of the parallel multisplitting PSD algorithm

O'Leary and White have suggested a parallel multisplitting iteration scheme of solving a nonsingular linear system Ax = b. We present a class of relaxed parallel multisplitting algorithms called the parallel multisplitting PSD (MPSD) algorithm of solving same system. Based on the new algorithm model, we establish another algorithm called the extrapolated MPSD algorithm. If A is an H-matrix, these methods converge under suitable conditions.

Parallel algorithm for solving coupled algebraic Lyapunov equations of discrete-time jump linear systems

A parallel iterative scheme for solving coupled algebraic Lyapunov equations of discrete-time jump linear systems with Markovian transitions is introduced. The algorithm is computationally efficient since it operates on reduced-order decoupled algebraic discrete Lyapunov equations. Furthermore, the solutions at every iteration are computed by elementary matrix operations. Hence, the number of operations is minimal. Monotonicity of convergence is established under the existence conditions of unique positive solutions.

Butcher–Kuntzmann methods for nonstiff problems on parallel computers

From a theoretical point of view, the Butcher–Kuntzmann Runge–Kutta methods belong to the best step- by-step methods for nonstiff problems. These methods integrate first-order initial-value problems by means of formulas based on Gauss–Legendre quadrature, and combine excellent stability features with the property of superconvergence at the step points. Like the IVP itself, they only need the given initial value without requiring additional starting values, and therefore are a natural discretization of the initial-value problem. On the other hand, from a practical point of view, these methods have the drawback of requiring in each step an approximation to the solution of a system of equations of dimension sd, s and d being the number of stages and the dimension of the initial-value problem, respectively. However, parallel computers have changed the scene and enable us to design parallel iteration methods for approximating the solution of the implicit systems such that the Butcher–Kuntzmann methods become efficient step-by-step methods for integrating initial-value problems. In this contribution, we address nonstiff initial-value problems and we investigate the possibility of introducing preconditioners into the iteration method. In particular, the iteration error will be analysed. By a number of numerical experiments it will be shown that the Butcher–Kuntzmann method, in combination with the preconditioned, parallel iteration scheme, performs much more efficiently than the best sequential methods.

A method for finite element parallel viscous compressible flow calculations

AbstractThis paper presents the parallelization aspects of a solution method for the fully coupled 3D compressible Navier‐Stokes equations. The algorithmic thrust of the approach, embedded in a finite element code NS3D, is the linearization of the governing equations through Newton methods, followed by a fully coupled solution of velocities and pressure at each non‐linear iteration by preconditioned conjugate gradient‐like iterative algorithms. For the matrix assembly, as well as for the linear equation solver, efficient coarse‐grain parallel schemes have been developed for shared memory machines, as well as for networks of workstations, with a moderate number of processors. The parallel iterative schemes, in particular, circumvent some of the difficulties associated with domain decomposition methods, such as geometry bookkeeping and the sometimes drastic convergence slow‐down of partitioned non‐linear problems.

A HIGHLY EFFICIENT ITERATIVE PARALLEL COMPUTATIONAL METHOD FOR FINITE ELEMENT SYSTEMS

A fully parallel algorithm for the solution of a finite element system using a MIMD (multiple‐instruction multiple‐data architecture) parallel computer is presented. The formulation includes a simple domain decomposer that automatically divides a finite element mesh into a list of subdomains to guarantee the load balancing. Furthermore, each subdomain is assigned to a processor of a parallel computer and treated as a sub‐finite element system with information exchanged through the interface between two adjacent subdomains. With this new algorithm, these sub‐finite element systems are solved fully parallelly as independent finite element systems, not only the computations of the interior nodes but also the computations of the interface nodes can be executed parallelly. Also, the inherently sequential Gauss‐Seidel and SOR schemes are altered into fully parallel iterative schemes. An implementation of this new scheme on an iPSC/2 D5 Hypercube Concurrent Computer reached an efficiency of more than 100% when c...

Parallel simulation dynamics for elastic multibody chains

A solution procedure for simulation dynamics of elastic multibody systems specifically designed for parallel processing is presented. The method is applicable to open chains with general (rotational and/or translational) interbody constraints. It is based on obtaining an explicit solution for the joint constraint forces by means of iterative techniques. Numerical results for a three-link anthropomorphic flexible-link manipulator are presented. The simulated comparison, on a serial computer, of different parallel iterative schemes indicates that the preconditioned conjugate-gradient methods are computationally most efficient. Moreover, their parallel implementation yields computational complexity that, based on theoretical estimates, is approximately constant with the number of bodies in the chain. >

On the convergence of parallel asynchronous block-iterative computations

This paper considers the convergence problem of parallel asynchronous block-iterative computation schemes. A new mathematical state-space model for a class of nonlinear time-varying parallel iterative schemes is proposed. Using this model, which generalizes several models of the Chazan-Miranker type, together with large-scale systems and Liapunov techniques, it is shown that the well-known quasidominance condition on a certain aggregated matrix guarantees exponential convergence of this class of methods.