Biconjugate Gradient Research Articles

The MGPBiCG method for solving the generalized coupled Sylvester-conjugate matrix equations

In this paper, we extend the generalized product-type bi-conjugate gradient (GPBiCG) method for solving the generalized Sylvester-conjugate matrix equations A1XB1+C1Y¯D1=S1,A2X¯B2+C2YD2=S2 by the real representation of the complex matrix and the properties of Kronecker product and vectorization operator. Some numerical experiments demonstrate that the introduced iteration approach is efficient.

The matrix iterative methods for solving a class of generalized coupled Sylvester-conjugate linear matrix equations

The conjugate gradients-squared (CGS) method (Sonneveld, 1989) has been considered as an efficient variant of the bi-conjugate gradient (BCG) method. In Vorst (1992), a more smoothly converging variant of the BCG method which keeps the attractive convergence rate of the CGS method was investigated for the solution of certain classes of nonsymmetric linear systems, so-called bi-conjugate gradient stabilized (Bi-CGSTAB) method. In this paper, we will combine these interesting methods for solving the generalized coupled Sylvester-conjugate matrix equations A1XB1+C1Y‾D1=E,A2X‾B2+C2YD2=F after performing suitable transformation by the properties of Kronecker product and vec operator. Some numerical experiments demonstrate that the introduced iterative methods are more efficient than the existing methods.

Comparison of parallel solvers for Moving Particle Semi-Implicit method

Purpose – The purpose of this paper is to find the best solver for parallelizing particle methods based on solving Pressure Poisson Equation (PPE) by taking Moving Particle Semi-Implicit (MPS) method as an example because the solution for PPE is usually the most time-consuming part difficult to parallelize. Design/methodology/approach – To find the best solver, the authors compare six Krylov solvers, namely, Conjugate Gradient method (CG), Scaled Conjugate Gradient method (SCG), Bi-Conjugate Gradient Stabilized (BiCGStab) method, Conjugate Gradient Squared (CGS) method with Symmetric Lanczos Algorithm (SLA) method and Incomplete Cholesky Conjugate Gradient method (ICCG) in terms of convergence, time consumption, parallel efficiency and memory consumption for the semi-implicit particle method. The MPS method is parallelized by the hybrid Open Multi-Processing (OpenMP)/Message Passing Interface (MPI) model. The dam-break flow and channel flow simulations are used to evaluate the performance of different solvers. Findings – It is found that CG converges stably, runs fastest in the serial way, uses the least memory and has highest OpenMP parallel efficiency, but its MPI parallel efficiency is lower than SLA because SLA requires less synchronization than CG. Originality/value – With all these criteria considered and weighed, the recommended parallel solver for the MPS method is CG.

Iterative solvers for image denoising with diffusion models: A comparative study

In this paper we propose and compare the use of two iterative solvers using the Crank–Nicolson finite difference method, to address the task of image denoising via partial differential equations (PDEs) models such as Regularized Perona–Malik equation or C-model and Bazan model (Bilateral-filter-based model). The solvers which are considered in this paper are the Successive-over-Relaxation (SOR) and an advanced solver known as Hybrid Bi-Conjugate Gradient Stabilized (Hybrid BiCGStab) method. From numerical experiments, it is found that the Crank–Nicolson method with hybrid BiCGStab iterative solver produces better results and is more efficient than SOR and already existing, in terms of MSSIM and PSNR.

Acceleration of GPU-based Krylov solvers via data transfer reduction

Krylov subspace iterative solvers are often the method of choice when solving large sparse linear systems. At the same time, hardware accelerators such as graphics processing units continue to offer significant floating point performance gains for matrix and vector computations through easy-to-use libraries of computational kernels. However, as these libraries are usually composed of a well optimized but limited set of linear algebra operations, applications that use them often fail to reduce certain data communications, and hence fail to leverage the full potential of the accelerator. In this paper, we target the acceleration of Krylov subspace iterative methods for graphics processing units, and in particular the Biconjugate Gradient Stabilized solver that significant improvement can be achieved by reformulating the method to reduce data-communications through application-specific kernels instead of using the generic BLAS kernels, e.g. as provided by NVIDIA’s cuBLAS library, and by designing a graphics processing unit specific sparse matrix-vector product kernel that is able to more efficiently use the graphics processing unit’s computing power. Furthermore, we derive a model estimating the performance improvement, and use experimental data to validate the expected runtime savings. Considering that the derived implementation achieves significantly higher performance, we assert that similar optimizations addressing algorithm structure, as well as sparse matrix-vector, are crucial for the subsequent development of high-performance graphics processing units accelerated Krylov subspace iterative methods.

A Stable and Accurate Preconditioner for Bidirectional Beam Propagation Method

A stable and accurate preconditioner based on the Denman-Beavers iteration (DBI) for the square-root operator to construct an efficient bidirectional beam propagation method (Bi-BPM) is proposed. Utilizing a branch-cut rotation technique, numerical instability from the evanescent wave in classical DBI is nearly removed. The biconjugate gradient stabilized method is applied to solve the preconditioned formulations due to its fast convergence rate. Moreover, the present preconditioner is shown to be highly efficient for dealing with the optical devices with multiple longitudinal dielectric interfaces, especially for those with aperiodic structures since it can be simultaneously obtained during the calculation of the square root of the characteristic matrix. To validate the present approach, a typical single facet waveguide structure and two kinds of multilayer coating structures are considered as the numerical examples. Results show that the present Bi-BPM has the advantages of excellent convergence, and high accuracy and stability, superior to the widely used Pade approximation.

Finite element method to improve the computational performance in optical devices analysis

This work presents a robust bidimensional finite element method (FEM) to disassembly and assembly only modified elements in global matrix. It integrates biconjugate gradient stabilized iterative method and incomplete LU threshold preconditioner to solve linear systems for nondiagonal sparse matrixes, decreasing FEM runtime around 70% in some applications. © 2015 Wiley Periodicals, Inc. Microwave Opt Technol Lett 57:1423–1426, 2015

Partitioned solution strategies for coupled BEM–FEM acoustic fluid–structure interaction problems

This paper investigates two FEM–BEM coupling formulations for acoustic fluid–structure interaction (FSI) problems, using the Finite Element Method (FEM) to model the structure and the Boundary Element Method (BEM) to represent a linear acoustic fluid. The coupling methods described interconnect fluid and structure using classical or localized Lagrange multipliers, allowing the connection of non-matching interfaces. First coupling technique is the well known mortar method, that uses classical multipliers and is compared with a new formulation of the method of localized Lagrange multipliers (LLM) for FSI applications with non-matching interfaces. The proposed non-overlapping domain decomposition technique uses a classical non-symmetrical acoustic BEM formulation for the fluid, although a symmetric Galerkin BEM formulation could be used as well. A comparison between the localized methodology and the mortar method in highly non conforming interface meshes is presented. Furthermore, the methodology proposes an iterative preconditioned and projected bi-conjugate gradient solver which presents very good scalability properties in the solution of this kind of problems.

A new quasi-minimal residual method based on a biconjugate A-orthonormalization procedure and coupled two-term recurrences

Recently, some novel variants of Lanczos-type methods were explored that are based on the biconjugate A-orthonormalization process. Numerical experiments coming from some physical problems indicate that these new methods are competitive with or superior to other popular Krylov subspace methods. Among them, the biconjugate A-orthogonal residual (BiCOR) method is the archetype method. However, like the biconjugate gradient (BiCG) method, the BiCOR method often shows irregular convergence behavior which can lead to numerical instability. To overcome this drawback, motivated by the effectiveness and robustness of the quasi-minimal residual (QMR) method, we derive a new QMR-like approach based on the coupled two-term biconjugate A-orthonormalization process and simple recurrences instead of the QR decomposition. Some convergence properties are given, and the special case for solving complex symmetric linear system is considered. Finally, numerical experiments are reported to illustrate the performances of our methods and their preconditioners on linear systems discretizing the Helmholtz equation or taken from Florida Sparse Matrix Collection.

Three-Dimensional Simulation of Pulsatile Flow Through a Porous Bulge

The present research deals with simulation of pulsatile flow through a tubular bulge filled with homogeneous, isotropic, saturated porous medium. For comparison, flow fields are also computed for the flow through clear medium in the same geometry. Both cases involve three-dimensional, unsteady, laminar, incompressible flow of Newtonian fluid. Flow through the porous medium is modeled via Forchheimer–Brinkman-extended Darcy model. While semi-analytical solutions are presented for the asymptotic cases, numerical techniques are used for the general solutions of the conservation equations. The governing equations, in both porous and clear media, are discretized by an implicit finite volume technique over an unstructured tetrahedral mesh. The resulting algebraic equations are solved using stabilized biconjugate gradient technique. Simulations include Darcy numbers of $$10^{-4}$$ and $$10^{-2}$$ at Reynolds numbers of 500 and 2000 based on the peak incoming average velocity and main tube diameter. The Womersley number is chosen to be 11.3 to mimic biomedical application. Results show that the porous medium naturalizes the recirculations and secondary flows while experiencing higher-pressure drop and wall shear compared to that of the clear medium.

The SCBiCG class of algorithms for complex symmetric linear systems with applications in several electromagnetic model problems

Frequency domain formulations of computational electromagnetic problems often require the solutions of complex-valued non-Hermitian systems of equations, which are still symmetric. For this kind of problems a whole class of sub-variant solver methods derived from the complex-valued Bi-Conjugate Gradient method is available. This class of methods contains established iterative methods as the Conjugate Orthogonal Conjugate Gradient (COCG) method, Bi-Conjugate Gradient Conjugate Residual (BiCGCR) method and Conjugate A-Orthogonal Conjugate Residual (COCR) method. The mathematical equivalence of the BiCGCR method and COCR method is shown and preconditioned variants of the various solvers are derived. An efficient kind of two-step preconditioning technique is also proposed. Numerical experiments involving e.g. electro-quasistatic frequency domain simulation are employed to show the difference in the convergence behaviors of these iterative methods and effectiveness of the two-step preconditioning techniques.

Matrix GPBiCG algorithms for solving the general coupled matrix equations

Linear matrix equations have important applications in control and system theory. In the study, we apply Kronecker product and vectorisation operator to extend the generalised product bi-conjugate gradient (GPBiCG) algorithms for solving the general coupled matrix equations ∑ l j=1(A) i,1, jX 1 Bi ,1,j +Ai ,2,j X 2 Bi, 2,j +…+Ai,l,jXi,l,j ) = Di for i = 1,2,…,l (including the (coupled) Sylvester, the second-order Sylvester and coupled Markovian jump Lyapunov matrix equations). We propose four effective matrix algorithms for finding solutions of the matrix equations. Numerical examples and comparison with other well-known algorithms demonstrate the effectiveness of the proposed matrix algorithms.

Formulation of a Preconditioned Algorithm for the Conjugate Gradient Squared Method in Accordance with Its Logical Structure

In this paper, we propose an improved preconditioned algorithm for the conjugate gradient squared method (improved PCGS) for the solution of linear equations. Further, the logical structures underlying the formation of this preconditioned algorithm are demonstrated via a number of theorems. This improved PCGS algorithm retains some mathematical properties that are associated with the CGS derivation from the bi-conjugate gradient method under a non-preconditioned system. A series of numerical comparisons with the conventional PCGS illustrate the enhanced effectiveness of our improved scheme with a variety of preconditioners. This logical structure underlying the formation of the improved PCGS brings a spillover effect from various bi-Lanczos-type algorithms with minimal residual operations, because these algorithms were constructed by adopting the idea behind the derivation of CGS. These bi-Lanczos-type algorithms are very important because they are often adopted to solve the systems of linear equations that arise from large-scale numerical simulations.

Three-Dimensional Scattering and Inverse Scattering from Objects With Simultaneous Permittivity and Permeability Contrasts

There has been increasing efforts in enhanced biomedical and geophysical imaging in the recent years exploring the magnetic contrast agent. The handling of both dielectric and magnetic contrasts adds on difficulties to the forward and inverse scattering problems. In this paper, the 3-D scattering and the inverse scattering from objects having simultaneous electric and magnetic contrasts are presented, where the permittivity, conductivity, and permeability of the objects can all be different from the background. To cope with the challenging high computation cost in the 3-D scattering problem, we formulate the combined field volume integral equations and extend the stabilized biconjugate gradient method and fast Fourier transform (BCGS-FFT) method to compute the electromagnetic field incorporating both the electric and magnetic contrasts. The variational Born iterative method for electrical contrast inversion in axisymmetric media is generalized to 3-D and to the simultaneous reconstruction of objects with electric and magnetic contrasts. The BCGS-FFT method provides the predicted scattered field from the 3-D heterogeneous objects and the Frechet derivatives in the inverse scattering problem. The efficient forward solver also dramatically reduces the computation time of the inverse problem. Numerical results are presented to validate the forward solver and to demonstrate the effectiveness of the inverse scattering method.

QTBiCGSTAB Algorithm for Large Linear Systemand Parallelized

According to the traditional stabilized biconjugate gradient algorithm (BiCGSTAB) deficiency in data locality, this paper proposed a QTBiCGSTAB algorithm whose core idea is that recursively divides sparse matrix with quarter tree into sub-matrix and reorders them, to improve the hit ratio of cache and enhance the algorithm’s efficiency. And the idea is good for algorithm being parallized, that is proved by the numerical experiments later. It mainly shows, firstly, QTBiCGSTAB algorithm is more efficiency than BiCGSTAB, and the speedup would reach 1:330. The target division length would be influnced on the algorithm’s performance; Secondly, for large linear system, parallized QTBiCGSTAB is more efficiency than serial’s.

RETRACTED ARTICLE: Formal derivation of biconjugate gradient method with its modification for distributed parallel computing

RETRACTED ARTICLE: Formal derivation of biconjugate gradient method with its modification for distributed parallel computing

Extending the GPBiCG algorithm for solving the generalized Sylvester-transpose matrix equation

By applying Kronecker product and vectorization operator, we extend the generalized product bi-conjugate gradient (GPBiCG) algorithm for solving the generalized Sylvester-transpose matrix equation $\sum\nolimits_{i = 1}^r {(A_i XB_i + C_i X^T D_i ) = E} $ . By using numerical results, we compare the new method with other popular iterative solvers in use today.

Performance of Preconditioned Nonstationary Methods for Electromagnetic Scattering From One Dimensional Dielectric Rough Surfaces

For the analysis of electromagnetic scattering from one dimensional dielectric randomly rough surfaces, we apply the right-preconditioning technique to three nonstationary iterative algorithms, namely, the generalized minimal residual (GMRES), conjugate gradient squared (CGS), and biconjugate gradient stabilized (BiCGSTAB). The spectral acceleration technique is also utilized to expedite computation of matrix-vector product. Among the nonstationary methods considered, with the right-preconditioning (RP) applied, GMRES-RP is the most robust, followed by BiCGSTAB-RP, with CGS-RP falling behind. In terms of number of iterations to achieve convergence, for cases when they all converge, GMRES-RP requires the most whereas BiCGSTAB-RP the least, with CGS-RP slightly more demanding than BiCGSTAB-RP. However, in terms of run time, GMRES-RP is the most efficient, followed by BiCGSTAB-RP, and CGS-RP is the least efficient. This relative performance is in contrast to the unpreconditioned case as reported in the literature, which speaks of the impact of the adopted right-preconditioning on the spectral distribution of these nonstationary algorithms.

Numerical Modeling of Distributed Inhomogeneities and Their Effect on Rolling-Contact Fatigue Life

The present work proposes a new efficient numerical solution method based on Eshelby's equivalent inclusion method (EIM) to study the influence of distributed inhomogeneities on the contact of inhomogeneous materials. Benchmark comparisons with the results obtained with an existing numerical method and the finite element method (FEM) demonstrate the accuracy and efficiency of the proposed solution method. An effective influence radius is defined to quantify the scope of influence for inhomogeneities, and the biconjugate gradient stabilized method (Bi-CGSTAB) is introduced to determine the eigenstrains of a large number of inclusions efficiently. Integrated with a rolling-contact fatigue (RCF) life prediction model, the proposed numerical solution is applied to investigate the RCF life of (TiB + TiC)/Ti-6Al-4V composites, and the results are compared with those of a group of RCF tests, revealing that the presence of the reinforcements causes reduction in the RCF lives of the composites. The comparison illustrates the capability of the proposed solution method on RCF life prediction for inhomogeneous materials.

Axisymmetric fully spectral code for hyperbolic equations

We present a fully pseudo-spectral scheme to solve axisymmetric hyperbolic equations of second order. With the Chebyshev polynomials as basis functions, the numerical grid is based on the Lobbato (for two spatial directions) and Radau (for the time direction) collocation points. The method solves two issues of previous algorithms which were restricted to one spatial dimension, namely, (i) the inversion of a dense matrix and (ii) the acquisition of a sufficiently good initial-guess for non-linear systems of equations. For the first issue, we use the iterative bi-conjugate gradient stabilized method, which we equip with a pre-conditioner based on a singly diagonally implicit Runge–Kutta (“SDIRK”-) method. In this paper, the SDIRK-method is also used to solve issue (ii). The numerical solutions are correct up to machine precision and we do not observe any restriction concerning the time step in comparison with the spatial resolution. As an application, we solve general-relativistic wave equations on a black-hole space–time in so-called hyperboloidal slices and reproduce some recent results available in the literature.