Symmetric Successive Over-relaxation Preconditioner Research Articles

SSOR Preconditioned Gauss-Seidel Detection and Its Hardware Architecture for 5G and beyond Massive MIMO Networks

With the limitedness of the sub-6 GHz bandwidth, the world is exploring a thrilling wireless technology known as massive MIMO. This wireless access technology is swiftly becoming key for 5G, B5G, and 6G network deployment. The massive MIMO system brings together antennas at both base stations and the user terminals to provide high spectral service. Despite the fact that massive MIMO offers astronomical benefits such as low latency, high data rate, improved array gain, and far better reliability, it faces several implementation challenges due to the hundreds of antennas at the base station. The signal detection at the base station during the uplink is one of the critical issues in this technology. Detection of user signal becomes computationally complex with a multitude of antennas present in the massive MIMO systems. This paper proposes a novel preconditioned and accelerated Gauss–Siedel algorithm referred to as Symmetric Successive Over-relaxation Preconditioned Gauss-Seidel (SSORGS). The proposed algorithm will address the signal detection challenges associated with massive MIMO technology. Furthermore, we enhance the convergence rate of the proposed algorithm by introducing a novel Symmetric Successive Over-relaxation preconditioner (SSOR) scheme and an initialization scheme based on the instantaneous channel condition between the base station and the user. The simulation results show that the proposed algorithm referred to as Symmetric Successive Over-relaxation Preconditioned Gauss-Seidel (SSORGS) provides optimal BER performance. At BER =10−3, over the range of SNR, the SSORGS algorithm performs better than the traditional algorithms. Additionally, the proposed algorithm is computationally more efficient than the traditional algorithms. Furthermore, we designed a comprehensive hardware architecture for the SSORGS algorithm to find the interrelated components necessary to build the actual physical system.

Conjugate gradient method preconditioned with modified block SSOR iteration for multiplicative half-quadratic image restoration

Image restoration problem is often solved by minimizing a cost function which consists of data-fidelity terms and regularization terms. Half-quadratic regularization has the advantage that it can preserve image details well in the recovered images. In this paper, we consider solving the image restoration model which involves multiplicative half-quadratic regularization term. Newton method is employed to solve the nonlinear system of equations resulted from the optimization problem for image restoration. At each Newton iteration step, a linear system of equations with symmetric positive definite coefficient matrix arises. The preconditioned conjugate gradient method with the proposed modified block SSOR (symmetric successive over-relaxation) preconditioner is applied to solve this linear system of equations. The condition number of the preconditioned matrix is estimated and numerical experiments are also implemented for image restoration. Both theoretical and numerical results show that the modified block SSOR preconditioned PCG methods can greatly improve the computation efficiency when solving the multiplicative half-quadratic regularized image restoration problem.

A Novel and Efficient Hybrid RSILU Preconditioner for the Parallel GMRES Solution of the Coarse Mesh Finite Difference Equations for Practical Reactor Simulations

The solution of the large, nonsymmetric, sparse linear system resulting from the discretization of the Boltzmann transport equation can be computationally intensive, even when innovative coarse mesh finite difference (CMFD) acceleration methods are applied on a high-performance computer. The research reported here describes the development and implementation of an innovative hybrid Reduced Symmetric Successive Over-Relaxation (RSOR) and Incomplete Lower Upper (ILU) (RSILU) preconditioner for the preconditioned Generalized Minimal RESidual (GMRES) solution of the three-dimensional, whole-core, pin-resolved transport calculation of a nuclear reactor core. The preconditioner was designed specifically to improve parallel computing capability and minimize the computational burden for solution of the CMFD method on a high-end computing platform. The hybrid RSILU preconditioner is applied to the preconditioned GMRES method without multicolor reordering by utilizing the ILU preconditioner for internal elements on each processor and the RSOR preconditioner for boundary elements that would require interprocessor communication. The construction of the RSILU preconditioner requires only that the diagonal elements be modified, factorized, and stored, which is identical to serial ILU. The computational cost of RSILU is minimized since the factorized diagonal block requires minimal data preparation time for interprocessor communication and exchanges information only once in parallel computation. This paper reports first the serial performance of RSOR, and numerical results show that the new proposed RSOR is an effective preconditioner even for serial computing applications. The parallel performance of RSILU is then assessed and compared to conventional multicolor ILU preconditioners. The results show that RSILU provides comparable convergence rates to the Standard Incomplete Lower Upper (SILU) preconditioners, but it is easier to implement since it does not require multicolor ordering. Although the required iterations of RSILU preconditioned GMRES increase as the number of processors increases, only slightly more iterations are required than the SILU preconditioner for a practical nuclear reactor application. The number of iterations required by the RSILU preconditioner increases only slightly and significantly less than the increased number of iterations required when using conventional Block-Jacobi Incomplete Lower Upper or Symmetric Successive Over-Relaxation preconditioners. Overall, the RSILU is shown to be an efficient and practical preconditioner for the GMRES method for improving the parallel computing performance for large-scale applications.

Technical note: A successive over-relaxation preconditioner to solve mixed model equations for genetic evaluation.

A computationally efficient preconditioned conjugate gradient algorithm with a symmetric successive over-relaxation (SSOR) preconditioner for the iterative solution of set mixed model equations is described. The potential computational savings of this approach are examined for an example of single-step genomic evaluation of Australian sheep. Results show that the SSOR preconditioner can substantially reduce the number of iterates required for solutions to converge compared with simpler preconditioners with marked reductions in overall computing time.

Robust partitioned block preconditioners for large-scale geotechnical applications with soil-structure interactions

SUMMARY Soil–structure interaction problems are commonly encountered in geotechnical practice and remarkably characterized with significant material stiffness contrast. When solving the soil–structure interaction problems, the employed Krylov subspace iterative method may converge slowly or even fail, indicating that the adopted preconditioning method may not suit for such problems. The inexact block diagonal preconditioners proposed recently have been shown effective for the soil–structure interaction problems; however, they haven't been exploited to full capabilities. By using the same partition strategy according to the structure elements and soil elements, the partitioned block symmetric successive over-relaxation preconditioners or partitioned block constraint preconditioners are proposed. Based on two pile-group foundation problems and a tunnel problem, the proposed preconditioners are evaluated and compared with the available preconditioners for the consolidation analysis and the drained analysis, respectively. In spite of one additional solve associated with the structure block and multiplications with off-diagonal blocks in the preconditioning step, numerical results reveal that the proposed preconditioners obviously possess better performance than the recently developed inexact block preconditioners. Copyright © 2013 John Wiley & Sons, Ltd.

GPU-accelerated iterative solutions for finite element analysis of soil–structure interaction problems

Soil–structure interaction problems are commonly encountered in engineering practice, and the resulting linear systems of equations are difficult to solve due to the significant material stiffness contrast. In this study, a novel partitioned block preconditioner in conjunction with the Krylov subspace iterative method symmetric quasiminimal residual is proposed to solve such linear equations. The performance of these investigated preconditioners is evaluated and compared on both the CPU architecture and the hybrid CPU–graphics processing units (GPU) computing environment. On the hybrid CPU–GPU computing platform, the capability of GPU in parallel implementation and high-intensity floating point operations is exploited to accelerate the iterative solutions, and particular attention is paid to the matrix–vector multiplications involved in the iterative process. Based on a pile-group foundation example and a tunneling example, numerical results show that the partitioned block preconditioners investigated are very efficient for the soil–structure interaction problems. However, their comparative performances may apparently depend on the computer architecture. When the CPU computer architecture is used, the novel partitioned block symmetric successive over-relaxation preconditioner appears to be the most efficient, but when the hybrid CPU–GPU computer architecture is adopted, it is shown that the inexact block diagonal preconditioners embedded with simple diagonal approximation to the soil block outperform the others.

Parallel SSOR preconditioning implemented on dynamic SMP clusters with communication on the fly

The paper presents a comparative analysis of parallel implementation of the preconditioned conjugate gradient method with the symmetric-successive over-relaxation preconditioner. Two parallel implementations of the matrix solver are compared. The first one is a message-passing version executed on a cluster of workstations. The other one is an efficient version simulated on a novel architecture of dynamically reconfigurable shared memory clusters with a new paradigm of inter-processor communication called communication on the fly. The presented example shows high suitability of the proposed architecture for fine grain numerical computations. It can be very useful in the simulation of physical phenomena described as numerical problems suitable for fine grain parallel execution.

Preconditioning Harmonic Unsteady Potential Flow Calculations

This paper considers finite element discretizations of the Helmholtz equation and its generalization arising from harmonic acoustic perturbations to a nonuniform steady potential flow. A novel elliptic, positive-definite preconditioner with a multigrid implementation is used to accelerate the iterative convergence of Krylov subspace solvers. Both theory and numerical results show that for a model 1-D Helmholtz test problem, the preconditioner clusters the discrete system's eigenvalues and lowers its condition number to a level independent of grid resolution. For the 2-D Helmholtz equation, grid-independent convergence is achieved using a quasi-minimal residual Krylov solver, significantly outperforming the popular symmetric successive over-relaxation preconditioner. Impressive results are also presented on more complex domains, including an axisymmetric aircraft engine inlet with nonstagnant mean flow and modal boundary conditions.

Application of a two-step preconditioning strategy to the finite element analysis for electromagnetic problems

A two-step preconditioning strategy is presented for the conjugate gradient (CG) iterative method to solve a large system of linear equations resulting from the use of edge-based finite-element discretizations of Helmholtz equations. The key idea is to combine both the factorized sparse approximate inverse (FSAI) and the symmetric successive overrelaxation (SSOR) preconditioning techniques in two successive steps in order to obtain a better preconditioner for the original matrix equations. The newly constructed preconditioner combines the advantages of both the FSAI and SSOR preconditioners with less computational complexity without the breakdowns of incomplete factorization technique. Numerical simulations show that this jointly preconditioned CG iterative method has a faster convergence speed than both FSAI and SSOR preconditioned CG methods. © 2006 Wiley Periodicals, Inc. Microwave Opt Technol Lett 48: 1623–1627, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.21683

Performance of preconditioners for the distributed vector finite-element time-domain algorithm

This paper deals with some aspects of performance of the symmetric successive over-relaxation preconditioner in a distributed environment. The details of distributed formulation of the preconditioner are presented. Some performance metrics are compared and discussed for the message passing interface implementation of the algorithm. The properties of the solver are estimated for concurrent three-dimensional formulation of the finite-element time-domain method. The analyzed benchmark models are approximated by tetrahedral first order Whitney elements.

Three-dimensional DC resistivity forward modelling using finite elements in comparison with finite-difference solutions

SUMMARY A 3-D finite-element scheme for direct current resistivity modelling is presented. The singularity is removed by formulating the problem in terms of the secondary potential, which improves the accuracy considerably. The resulting system of linear equations is solved using the conjugate gradient method. The incomplete Cholesky preconditioner with a scaled matrix has been proved to be faster than the symmetric successive overrelaxation preconditioner. A compact storage scheme fully utilizes the sparsity and symmetry of the system matrix. The finite-element (FE) and a previously developed finite-difference (FD) scheme are compared in detail. Generally, both schemes show good agreement, the relative error in apparent resistivity for a vertical dike model presented in this paper is less than 0.5 per cent overall. The FD scheme produces larger errors near the conductivity contrast, whereas the FE scheme requires approximately 3.4 times as much storage as the FD scheme and is less robust with respect to coarse grids. As an improvement to the forward modelling scheme, a modified singularity removal technique is presented. A horizontally layered earth or a vertical contact is regarded as the normal structure, the solution of which is the primary potential. The effect of this technique is demonstrated by two examples: a cube in two-layered earth and a cube near a vertical contact.

Calculating Flux Perturbations with a Davidson Algorithm

In perturbation calculations, obtaining an accurate flux shape of a perturbed core is more difficult than the multiplication factor. Generalized Davidson algorithms using a symmetric successive overrelaxation preconditioner are developed to solve the unperturbed eigenvalue problem and the related perturbed eigenvalue problem of large sparse matrices. The bases of the subspace obtained from the sequence of solving the unperturbed problem through the algorithm can be used in the perturbed problem to save computational time. One- and two-dimensional test problems indicate that by incorporating symmetric successive overrelaxation iteration, the optimized relaxation factor, and the newly developed shifted form-function vector method for a large perturbation, a considerable amount of computational time can be saved in the perturbed calculations with accuracy comparable to the existing CITATION code. This method also provides an efficient means for survey calculations where the requirement of accuracy is not stringent.

Iterative Solution of the Helmholtz Equation by a Second-Order Method

The numerical solution of the Helmholtz equation subject to nonlocal radiation boundary conditions is studied. The specific problem is the propagation of hydroacoustic waves in a two-dimensional curvilinear duct. The problem is discretized with a second-order accurate finite difference method, resulting in a linear system of equations. To solve the system of equations, a preconditioned Krylov subspace method is employed. We construct a preconditioner that is based on fast transforms and yields a direct fast Helmholtz solver for rectangular domains. Numerical experiments for curved ducts demonstrate that the rate of convergence is high. The fast transform preconditioner is compared with a symmetric successive over-relaxation (SSOR) preconditioner, and also with band Gaussian elimination. For the preconditioned iterative methods, the gains in storage requirement are large compared with band Gaussian elimination. Regarding the arithmetic complexity, the fast transform preconditioner yields a significant gain, whereas the SSOR preconditioner performs worse than band Gaussian elimination.

Efficient Solution of Spacecraft Thermal Models Using Preconditioned Conjugate Gradient Methods

The thermal mathematical models ofspacecraft areusually constructed using a lumped parameternetwork formulation. Thegoverning equations under the steady-state condition are a system of nonlinearalgebraicequations, often large and sparse. Generally, the quadratic convergent Newton’ s method is used to solve nonlinear systems, in which a linear system with a Jacobian coefe cient matrix is solved at each iteration. It is necessary to solve this linear system in an efe cient and economical way, exploiting the sparsity of the coefe cient matrix in storage as well as in computation. The conjugate gradient method with appropriate preconditioners is a very successful tool to solve large, sparse, linear systems. Numerical experiments were conducted to investigate the effectiveness of preconditioned conjugate gradient methods for large spacecraft thermal problems. The resultsare compared with thoseofsparseelimination methodsand thesuccessive overrelaxation method. Ithasbeen found thattheconjugate gradient method with the symmetric successive overrelaxation preconditioner is a simple and efe cient method to solve large-order problems. Nomenclature A = area,m 2 Fa = albedo load factor Fe = Earth-shine load factor I = solar constant, W/m 2 K = conduction exchange factor, W/K L = Cholesky factor for a positive dee nite symmetric matrix M = preconditioning matrix n = total number of nodes; number of equations; order of the matrix P = internal power dissipation, W R = radiation exchange factor, m 2 T = temperature, K ® = absorptance = emittance µ = solar incidence angle ·.A/ = spectral condition number of the matrix A Ω = Earth’ s albedo factor or ree ectance Ω.A/ = spectral radius of the matrix A ae = Stefan‐Boltzmann constant, 5.67E-8 W/m 2 -K 4 ! = relaxation parameter Subscripts i = node under consideration j = node thermally coupled to node i opt = optimum s = space node Superscripts T = transpose i1 = inverse * = exact solution

Advances in adaptive parallel processing for field applications

Techniques for the adaptive solution of two-dimensional vector systems for hyperbolic and elliptic partial differential equations on shared-memory parallel computers are described. Hyperbolic systems are approximated by an explicit finite volume technique and solved by a recursive local mesh refinement procedure. Several computational procedures have been developed, and results comparing a variety of heuristic processor load-balancing techniques and refinement strategies are presented. For elliptic problems, the spatial domain is discretized using a finite quadtree mesh-generation procedure and the differential system is discretized by a finite-element Galerkin technique with a hierarchical piecewise polynomial basis. Resulting linear algebraic systems are solved in parallel on noncontiguous quadrants by a conjugate gradient technique with element-by-element and symmetric successive over-relaxation preconditioners. Noncontiguous regions are determined by using a linear-time complexity coloring procedure that requires a maximum of six colors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Parallel computation with adaptive methods for elliptic and hyperbolic systems

We consider the solution of two-dimensional vector systems of elliptic and hyperbolic partial differential equations on a shared memory parallel computer. For elliptic problems, the spatial domain is discretized using a finite quadtree mesh generation procedure and the differential system is discretized by a finite element-Galerkin technique with a piecewise linear polynomial basis. Resulting linear algebraic systems are solved using the conjugate gradient technique with element-by-element and symmetric successive over-relaxation preconditioners. Stiffness matrix assembly and linear system solutions are processed in parallel with computations schedules on noncontiguous quadrants of the tree in order to minimize process synchronization. Determining noncontiguous regions by coloring the regular finite quadtree structure is far simpler than coloring elements of the unstructured mesh that the finite quadtree procedure generates. We describe linear-time complexity coloring procedures that use six and eight colors. For hyperbolic problems, the rectangular spatial domain is discretized into a grid of rectangular cells, and the differential system is discretized by an explicit finite difference technique. Recursive local refinement of the time steps and spatial cells of a coarse base mesh is performed in regions where a refinement indicator exceeds a prescribed tolerance. Data management involves the use of a tree of grids with finer grids regarded as offspring of coarser ones. Computational procedures that sequentially traverse the tree structure while processing solutions on each grid in parallel and that process solutions at the same tree level in parallel have been developed. Computational results using the sequential tree traversal scheme are presented and compared with results using a non-adaptive strategy. Heuristic processor load balancing techniques are suggested for the parallel tree traversal procedure.

Two-Color Fourier Analysis of Iterative Algorithms for Elliptic Problems with Red/Black Ordering

The red/black ordering scheme is often used to increase the parallelism of iterative methods for solving elliptic partial differential equations (PDEs). However, the convergence rates are also affected, often adversely. This paper provides a unified approach, called the two-color Fourier analysis, to study the convergence rates of iterative algorithms for elliptic problems with the red/black ordering. This Fourier tool is used to analyze different types of iterative algorithms, including the successive over-relaxation (SOR) method, symmetric successive over-relaxation (SSOR) method, preconditioned iterative methods with SSOR, ILU, and MILU preconditioners, and multigrid (MG) methods. By comparing the convergence rates of algorithms with the natural and red/black orderings, it is shown that although the red/black ordering does not affect the rate of convergence in the context of SOR and MG methods, it slows down the convergence significantly in the context of SSOR and preconditioned iterative methods.