Parallel Iterative Methods Research Articles

An inertial parallel iterative method for solving generalized mixed equilibrium problems and common fixed point problem in reflexive Banach spaces

An inertial parallel iterative method for solving generalized mixed equilibrium problems and common fixed point problem in reflexive Banach spaces

An adaptive multi-frame parallel iterative method for accelerating real-time magnetic particle imaging reconstruction

Objective. Real-time reconstruction of magnetic particle imaging (MPI) shows promising clinical applications. However, prevalent reconstruction methods are mainly based on serial iteration, which causes large delay in real-time reconstruction. In order to achieve lower latency in real-time MPI reconstruction, we propose a parallel method for accelerating the speed of reconstruction methods. Approach. The proposed method, named adaptive multi-frame parallel iterative method (AMPIM), enables the processing of multi-frame signals to multi-frame MPI images in parallel. To facilitate parallel computing, we further propose an acceleration strategy for parallel computation to improve the computational efficiency of our AMPIM. Main results. OpenMPIData was used to evaluate our AMPIM, and the results show that our AMPIM improves the reconstruction frame rate per second of real-time MPI reconstruction by two orders of magnitude compared to prevalent iterative algorithms including the Kaczmarz algorithm, the conjugate gradient normal residual algorithm, and the alternating direction method of multipliers algorithm. The reconstructed image using AMPIM has high contrast-to-noise with reducing artifacts. Significance. The AMPIM can parallelly optimize least squares problems with multiple right-hand sides by exploiting the dimension of the right-hand side. AMPIM has great potential for application in real-time MPI imaging with high imaging frame rate.

The multiple-set split feasibility problem and its application in intensity-modulated radiation therapy

In this paper, we apply a parallel iterative method for solving the split common fixed point problem in Hilbert spaces, without prior knowledge of the transfer operator norm, to the multiple-set split feasibility problem, which plays an important role in medical image reconstruction and signal processing. By formulating a model of the intensity-modulated radiation therapy treatment in a mathematical framework based on the multiple-set split feasibility problem, we also offer an effective approach for solving the intensity-modulated radiation therapy treatment planning.

RIS-Level SIC for Uplink Cascaded-RIS Assisted NOMA System

A brand-new framework for cascaded-reconfigurable intelligent surface (RIS) empowered multi-antenna non-orthogonal multiple access (NOMA) system is conceived, wherein RISs are deployed to serve their associated users and assist data transmission for the farther RISs as well. A new concept of RIS-level successive interference cancellation (RSIC) is developed to eliminate the inter-RIS interference. With the devised framework, a transmit power minimization problem is formulated by collaboratively designing each user’s transmit power, the RISs’ phase shifts, and the BS’s equalizers. Specifically, a parallel iteration method is developed to solve the subproblem of jointly optimizing transmit power and equalizers, whereas a sequential phase-rotation algorithm is proposed to address the subproblem of phase shift optimization. It is showed by simulations that the proposed scheme outperforms the existing benchmark schemes in terms of total transmit power.

UNICADO: multidisciplinary analysis in conceptual aircraft design

In this paper, different iterative methods, so-called architectures, within the multidisciplinary analysis in conceptual aircraft design of the UNICADO software are elaborated, applied and analyzed. Possible execution sequences in the sequential method (Gauss-Seidel architecture) are derived via a graph-based algorithm in combination with expert knowledge. Sensitivities of the design disciplines are analyzed and a permitted residual for stable convergence characteristic for the aircraft design with UNICADO is derived. Prerequisites for the application of a parallel iterative method, the Jacobi architecture are conducted. Runtime and convergence characteristics of the Gauss-Seidel architecture and the Jacobi architecture are evaluated. A damping method is applied to the Jacobi architecture to enhance the convergence characteristics. The Gauss-Seidel and Jacobi architectures are used to design two different aircraft, the CSR and the CSMR, with an iteration accuracy of 2.5e–3. For these use cases studied, the Gauss-Seidel architecture converges more stably and faster than the Jacobi architecture and is, therefore, the more favorable. The aircraft design with the implemented Jacobi architecture oscillates and does not converge. Only with an implemented damping method, convergence is achieved. If the iteration time of the design loop increases, e.g., when using higher fidelity methods for aircraft design, the choice of architecture must be re-evaluated.

A two-step parallel iteration method for large sparse horizontal linear complementarity problems

In this work, a two-step modulus-based synchronous multisplitting iteration method is constructed for solving large sparse horizontal linear complementarity problems. Some convergence theorems of the proposed method are presented, which can generalize the convergence results of some existing methods. Numerical tests on parallel computers by OpenACC show that the proposed method is efficient.

Convergence of restricted additive Schwarz with impedance transmission conditions for discretised Helmholtz problems

The Restricted Additive Schwarz method with impedance transmission conditions, also known as the Optimised Restricted Additive Schwarz (ORAS) method, is a simple overlapping one-level parallel domain decomposition method, which has been successfully used as an iterative solver and as a preconditioner for discretised Helmholtz boundary-value problems. In this paper, we give, for the first time, a convergence analysis for ORAS as an iterative solver—and also as a preconditioner—for nodal finite element Helmholtz systems of any polynomial order. The analysis starts by showing (for general domain decompositions) that ORAS is an unconventional finite element approximation of a classical parallel iterative Schwarz method, formulated at the PDE (non-discrete) level. This non-discrete Schwarz method was recently analysed in [Gong, Gander, Graham, Lafontaine, and Spence, Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation], and the present paper gives a corresponding discrete version of this analysis. In particular, for domain decompositions in strips in 2-d, we show that, when the mesh size is small enough, ORAS inherits the convergence properties of the Schwarz method, independent of polynomial order. The proof relies on characterising the ORAS iteration in terms of discrete ‘impedance-to-impedance maps’, which we prove (via a novel weighted finite-element error analysis) converge as h → 0 h\rightarrow 0 in the operator norm to their non-discrete counterparts.

Dynamic regional splitting planning of remote sensing satellite swarm using parallel genetic PSO algorithm

The strips from region splitting method are critical premise of meta-task generation and task planning for satellite swarm. To achieve the regional observation with high coverage and low overlap ratio and satellites usage, a novel region splitting algorithm is designed based on Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) algorithm. In the hybrid algorithm of GA and PSO, these two algorithms are executed paralleled, and then exchange the best genes and individuals periodically, to overcome the disadvantage of GA's poor local searching ability and PSO's low convergence speed. The encoding method and parallel iterative optimal method are given in detail. Finally, a simulation scenario is designed and executed to verify proposed algorithm, and the effects of algorithm parameters are analyzed by simulation results.

On Intelligent Reflecting Surface-Assisted NOMA Uplinks With Imperfect SIC

As the interplay between intelligent reflecting surface (IRS) and non-orthogonal multiple access (NOMA) can boost both the spectrum- and the power-efficiency, this letter investigates an IRS-NOMA system, taking into account the multi-group detection and imperfect successive interference cancellation (SIC). Specifically, a group-level SIC (GSIC) is conceived to remove the decoded groups’ signals, whilst the joint design of power control and equalizers is leveraged to mitigate the aggregated interference caused by NOMA, imperfect SIC, and concurrent transmissions per group. According to the newly formulated multi-group NOMA framework, the transmit power and the phase shifts are optimized alternately to reduce the total energy consumption together with the minimum mean square error equalizer. Thereof, the power control and the phase shift subproblems are respectively addressed by a group-level parallel iteration method and an element-specific sequential rotation method. Compared with various existing methods, the superiority of the proposed IRS-NOMA scheme is demonstrated through simulations.

Globalization of Nonlinear FETI-DP Domain Decomposition Methods Using an SQP Approach

The globalization of Nonlinear FETI-DP (Dual Primal Finite Element Tearing and Interconnecting) methods is considered using a Sequential Quadratic Programming (SQP) approach. Nonlinear FETI-DP methods are parallel iterative solution methods for nonlinear finite element problems, based on divide and conquer, using Lagrange multipliers. In these methods, nonlinear elimination is an important ingredient to increase the convergence radius of Newton’s method. We prove standard globalization results for SQP-based globalization of Nonlinear FETI-DP, first for the case that the elimination set is empty. We then show how to combine nonlinear elimination and SQP-based globalization. The globalization preserves the block structure of the FETI-DP operator, which is the basis of the computational parallelism.Supporting numerical experiments using homogenous and heterogeneous model problems from nonlinear structural mechanics are provided. In the numerical experiments, we consider four standard choices of different elimination sets and different problem setups including stiff or almost incompressible inclusions in every subdomain. The numerical experiments illustrate that a good elimination set is important. However, the use of the SQP-based globalization approach presented here can improve the convergence of Nonlinear FETI-DP methods further, especially, if combined with a good choice of the elimination set.

Power minimization for GSIC-based uplink cell-free massive MIMO-NOMA systems

Non-orthogonal multiple access (NOMA) and multiple-input multiple-output (MIMO) are considered promising techniques to satisfy the demands for high spectrum efficiency and massive connectivity in future wireless communication. In this paper, a novel framework to realize transmission in cell-free massive MIMO-NOMA system with deep integration between MIMO and NOMA is proposed. A new method is developed to divide users into different groups according to their equivalent path loss, and then “group-level successive interference cancellation” (GSIC) is used to cancel the inter-group interference when demodulating the users. Based on the new framework, a rigorous closed-form expression of the achievable sum rate in uplink cell-free massive MIMO-NOMA system is derived. In addition, a parallel iterative method is used to obtain the best power control scheme. Simulations show that the proposed scheme can effectively reduce the total power consumption and outperforms the orthogonal multiple access (OMA) transmission and traditional SIC-NOMA schemes based on user clustering.

Parallel Direct and Iterative Methods for Solving the Time-Fractional Diffusion Equation on Multicore Processors

The work is devoted to developing the parallel algorithms for solving the initial boundary problem for the time-fractional diffusion equation. After applying the finite-difference scheme to approximate the basis equation, the problem is reduced to solving a system of linear algebraic equations for each subsequent time level. The developed parallel algorithms are based on the Thomas algorithm, parallel sweep algorithm, and accelerated over-relaxation method for solving this system. Stability of the approximation scheme is established. The parallel implementations are developed for the multicore CPU using the OpenMP technology. The numerical experiments are performed to compare these methods and to study the performance of parallel implementations. The parallel sweep method shows the lowest computing time.

Scaled parallel iterative method for finding real roots of nonlinear equations

Given a nonconvex minimization problem where the objective function is nonlinear and twice differentiable. To gain more information about the objective function, it is essential to obtain all its stationary points and study the behaviour of these points. Since many nonlinear functions are expressible as polynomials via interpolation, there is a need to devise fast and accurate algorithms in finding root(s) of the interpolating polynomial. Through interval computation, the Weierstrass-like parallel iterative methods are known for their efficiency in finding polynomial zeros. However, these schemes are highly dependent on the midpoints of each interval in generating successive intervals. In this study, we propose a scaling function on some Weierstrass-like parallel iterative methods such that the procedures are less dependent on the generated midpoints, hence allowing a more efficient search for the zeros while reducing the width of the intervals. The proposed procedures with the shifted centres of the enclosing intervals are tested on 120 problems and we compare their efficiency with the existing Weierstrass-like methods in terms of the number of iterations and largest final interval width. The results indicate that the proposed procedures outperform the original procedures, giving more reduction on the final interval width with a lesser number of iterations.

Target Fast Reconstruction of Real Aperture Radar Using Data Extrapolation-Based Parallel Iterative Adaptive Approach

Real aperture radar (RAR) usually sweeps a wide sector to continuously observe the scenario of interest. Because its angular resolution is limited by the size of the antenna aperture, target reconstruction methods are widely applied to obtain super-resolution radar images. However, the wide-sector processing mode suffers from high operational complexity because of the high-dimensional matrix inversion. Even worse, for the targets located at the scene edge, its echo data are received less than half beamwidth. The incomplete echo data will lead to the deformation of the reconstructed targets by existing reconstruction methods. To overcome the two problems, a data extrapolation-based parallel iterative adaptive approach is proposed to fast reconstruct the targets in the whole sector without the distortion at the scene edge. First, the echo model of RAR is repaired to remedy the model error. Then, based on the correlation of the echo data within one beamwidth, an autoregressive model is adopted to extrapolate the data of the missing half beamwidth. Finally, a parallel iterative adaptive approach method is proposed to efficiently recover the targets by exploiting the regular characteristics of the repaired steering matrix. Simulations and experimental data are applied to verify the proposed method.

Parallel iterative methods for variational integration applied to navigation problems

Abstract Discrete variational methods have shown an excellent performance in numerical simulations of different mechanical systems. In this paper, we introduce an iterative method for discrete variational methods appropriate for boundary value problems. More concretely, we explore a parallelization strategy that leverages the power of multicore CPUs and GPUs (graphics cards). We study this parallel method for first-order and second-order Lagrangians and we illustrate its excellent behavior in some interesting applications, namely Zermelo’s navigation problem, a fuel-optimal navigation problem, and an interpolation problem.

Serial and Parallel Iterative Splitting Methods: Algorithms and Applications to Fractional Convection-Diffusion Equations

The benefits and properties of iterative splitting methods, which are based on serial versions, have been studied in recent years, this work, we extend the iterative splitting methods to novel classes of parallel versions to solve nonlinear fractional convection-diffusion equations. For such interesting partial differential examples with higher dimensional, fractional, and nonlinear terms, we could apply the parallel iterative splitting methods, which allow for accelerating the solver methods and reduce the computational time. Here, we could apply the benefits of the higher accuracy of the iterative splitting methods. We present a novel parallel iterative splitting method, which is based on the multi-splitting methods, The flexibilisation with multisplitting methods allows for decomposing large scale operator equations. In combination with iterative splitting methods, which use characteristics of waveform-relaxation (WR) methods, we could embed the relaxation behavior and deal better with the nonlinearities of the operators. We consider the convergence results of the parallel iterative splitting methods, reformulating the underlying methods with a summation of the individual convergence results of the WR methods. We discuss the numerical convergence of the serial and parallel iterative splitting methods with respect to the synchronous and asynchronous treatments. Furthermore, we present different numerical applications of fluid and phase field problems in order to validate the benefit of the parallel versions.

A Parallel Iterative Finite Element Method for the Linear Elliptic Equations

By combining the two-grid discretization with the partition of unity method, a parallel iterative finite element method for the linear elliptic equations is proposed and investigated. Since the construction of the partition of unity is based on the coarse mesh triangulation, the computational domain of each subproblem can be divided automatically and the number of subproblems can be arbitrarily huge as the coarse mesh parameter H tends to zero. That means our method can be easily implemented in high performance supercomputers or cluster of workstations. Theoretical results based on a priori error estimation of the scheme are obtained, which indicate that our method can reach the optimal convergence orders within a few two-grid iterations. Numerical results are reported to assess the theoretical results.

Parallel iterative methods for solving the generalized split common null point problem in Hilbert spaces

We study the recently introduced generalized split common null point problem in Hilbert spaces. In order to solve this problem, we propose two new parallel algorithms and establish strong convergence theorems for both of them. Our schemes combine the hybrid and shrinking projection methods with the proximal point algorithm.

A parallel iterative method for solving a class of variational inequalities in Hilbert spaces

A parallel iterative method for solving a class of variational inequalities in Hilbert spaces

Parallel Iterative Methods for Solving the Split Common Fixed Point Problem in Hilbert Spaces

We introduce three new parallel algorithms for solving the split common fixed point problem in Hilbert spaces. Those iterative methods for solving split common fixed point problems which involve step sizes that depend on the norm of a given bounded linear operator are often not easy to implement because one has to compute the norm of this operator. Therefore, in addition to two such algorithms, we also propose a new iterative method involving step sizes which are selected in such a way that the implementation of the method does not require the computation or estimation of the norm of the given operator. Several corollaries of our main results regarding the solution of multiple-set split feasibility, split common null point, split variational inequality and split mixed equilibrium problems are also presented.