Biconjugate Gradient Algorithm Research Articles

A Fast Algorithm of Cross-Correlated Contrast Source Inversion in Homogeneous Background Media

Nonlinear iterative inversion methods are computationally expensive, which leads to limited applications. In this paper, a fast algorithm of the cross-correlated contrast source inversion (CC-CSI) method is proposed. By assuming a homogeneous background medium, the task of solving systems of linear equations, the most time-consuming part of the CC-CSI method, has been efficiently implemented with several FFTs and IFFTs for transverse magnetic (TM), transverse electric (TE), and 3-D inversion, respectively. Theoretical analysis and real experiments demonstrate that the inversion efficiency of CC-CSI has been improved dramatically in comparison to the strategies of LU decomposition and the Biconjugate gradient algorithm. Specifically, the inversion speed of the proposed fast CC-CSI algorithm is at least one order of magnitude higher than using the latter two approaches in 3-D inversion, while the memory requirement of the proposed algorithm is intermediary and acceptable. Furthermore, the proposed fast scheme is also applicable to all the inversion methods which are related to computing of the electric fields directly from the contrast sources in homogeneous background media.

Non-Boussinesq approach for turbulent buoyant flows in enclosure with horizontal vent and forced inlet port

The effects of forced ambient velocity on thermal plume behavior in a ceiling vented square enclosure are numerically investigated. Turbulence is modeled by unsteady Favre-averaged Navier–Stokes (UFANS) equation with Lam Bremhorst low Reynolds number k−ɛ turbulence model. A non-Boussinesq variable density approach is used to model the density variations. Simplified Marker and Cell (SMAC) algorithm is used to solve the governing equations on collocated grid with high accuracy compact finite difference schemes. The pressure Poisson equation is solved by bi-conjugate gradient algorithm and time integration is performed with four stage Runge–Kutta method (Rk-4). The results are presented for Grashof number Gr = 1011 and 1012 and Gay-Lussac number Ga = 0.2 and 2. The present model is valid when buoyancy effects are significant in comparison with forced convection effects. The heat transfer characteristics are analyzed by varying forced inlet velocity, inlet port size and inlet port location. The assisting flow enhances plume discharge rate and increases convective heat loss from cavity. The opposing flow weakens thermal buoyancy and minimizes convective heat loss from cavity. The present mathematical model and numerical method are in good agreement with the existing results available in the literature.

Matrix-Free Polynomial-Based Nonlinear Least Squares Optimized Preconditioning and Its Application to Discontinuous Galerkin Discretizations of the Euler Equations

We introduce a preconditioner that can be both constructed and applied using only the ability to apply the underlying operator. Such a preconditioner can be very attractive in scenarios where one has a highly efficient parallel code for applying the operator. Our method constructs a polynomial preconditioner using a nonlinear least squares (NLLS) algorithm. We show that this polynomial-based NLLS-optimized (PBNO) preconditioner significantly improves the performance of a discontinuous Galerkin (DG) compressible Euler equation model when run in an implicit---explicit time integration mode; note that IMEX methods are valuable only for low Mach number flows. The PBNO preconditioner achieves significant reduction in GMRES iteration counts and model wall-clock time, and significantly outperforms several existing types of generalized (linear) least squares polynomial preconditioners. Comparisons of the ability of the PBNO preconditioner to improve DG model performance when employing the Stabilized Biconjugate Gradient algorithm (BICGS) and the basic Richardson iteration are also included. In particular, we show that higher order PBNO preconditioning of the Richardson iteration (run in a dot product free mode) makes the algorithm competitive with GMRES and BICGS in a serial computing environment. Because the NLLS-based algorithm used to construct the PBNO preconditioner can handle both positive definite and complex spectra without any need for algorithm modification, we suggest that the PBNO preconditioner is, for certain types of problems, an attractive alternative to existing polynomial preconditioners based on linear least-squares methods.

Multigroup radiation hydrodynamics with flux-limited diffusion and adaptive mesh refinement

Radiative transfer plays a key role in the star formation process. Due to a high computational cost, radiation-hydrodynamics simulations performed up to now have mainly been carried out in the grey approximation. In recent years, multi-frequency radiation-hydrodynamics models have started to emerge, in an attempt to better account for the large variations of opacities as a function of frequency. We wish to develop an efficient multigroup algorithm for the adaptive mesh refinement code RAMSES which is suited to heavy proto-stellar collapse calculations. Due to prohibitive timestep constraints of an explicit radiative transfer method, we constructed a time-implicit solver based on a stabilised bi-conjugate gradient algorithm, and implemented it in RAMSES under the flux-limited diffusion approximation. We present a series of tests which demonstrate the high performance of our scheme in dealing with frequency-dependent radiation-hydrodynamic flows. We also present a preliminary simulation of a three-dimensional proto-stellar collapse using 20 frequency groups. Differences between grey and multigroup results are briefly discussed, and the large amount of information this new method brings us is also illustrated. We have implemented a multigroup flux-limited diffusion algorithm in the RAMSES code. The method performed well against standard radiation-hydrodynamics tests, and was also shown to be ripe for exploitation in the computational star formation context.

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.

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.

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.

Comparative study of conjugate gradient algorithms performance on the example of steady-state axisymmetric heat transfer problem

Abstract The finite element method (FEM) is one of the most frequently used numerical methods for finding the approximate discrete point solution of partial differential equations (PDE). In this method, linear or nonlinear systems of equations, comprised after numerical discretization, are solved to obtain the numerical solution of PDE. The conjugate gradient algorithms are efficient iterative solvers for the large sparse linear systems. In this paper the performance of different conjugate gradient algorithms: conjugate gradient algorithm (CG), biconjugate gradient algorithm (BICG), biconjugate gradient stabilized algorithm (BICGSTAB), conjugate gradient squared algorithm (CGS) and biconjugate gradient stabilized algorithm with l GMRES restarts (BICGSTAB(l)) is compared when solving the steady-state axisymmetric heat conduction problem. Different values of l parameter are studied. The engineering problem for which this comparison is made is the two-dimensional, axisymmetric heat conduction in a finned circular tube.

Calculation of normal mode spectra in laterally heterogeneous earth models using an iterative direct solution method

Normal mode observations play an important role in studying broad-scale lateral variations in the Earth. Such studies require the calculation of accurate synthetic spectra in realistic earth models, and this remains a computationally challenging problem. Here, we describe a new implementation of the direct solution method for calculating normal mode spectra in laterally heterogeneous earth models. In this iterative direct solution method, the mode-coupling equations are solved in the frequency-domain using the preconditioned biconjugate gradient algorithm, and the time-domain solution is recovered using a numerical inverse Fourier transform. A number of example calculations are presented to demonstrate the accuracy and efficiency of the method for performing large ‘full coupling’ calculations as compared to methods based on matrix diagonalization and the traditional direct solution method.

Efficient Numerical Method in Second-Order Inelastic Analysis of Space Trusses

In this paper, the Newton-Raphson method is combined with three different algorithms. These algorithms are the generalized minimum residual (GMRES), the least squares (LSQR), and the biconjugate gradient (BCG). Of these algorithms, the most effective at reducing the number of iterations and the time required is identified. A common characteristic of these algorithms is that they replace the inversion of the tangent stiffness matrix with an iterative procedure to solve the linearized system of equations. A computer program based on three algorithms is developed to numerically solve a system of nonlinear equations. The procedure can be applied to analysis of structures with complex behaviors, including unloading, snap-through buckling, and inelastic postbuckling analyses. To demonstrate the efficiency and accuracy of the method developed here, some well-known trusses are investigated and analyzed using the various aforementioned algorithms. Results show that the biconjugate gradient algorithm is a more effective scheme to be coupled with the Newton-Raphson method when nonlinear structural problems are to be solved, having considerable savings in computational cost and time. Furthermore, the BCG algorithm is improved for solving a system of nonlinear equations. Finally, results reveal that the improved BCG method drastically reduces the computational time and the number of iterations while the accuracy of the results is maintained.

The analysis of gradient algorithm effectiveness - two dimensional heat transfer problem

The analysis of gradient algorithm effectiveness - two dimensional heat transfer problemThe analysis of effectiveness of the gradient algorithm for the two-dimension steady state heat transfer problems is being performed. The three gradient algorithms - the BCG (biconjugate gradient algorithm), the BICGSTAB (biconjugate gradient stabilized algorithm), and the CGS (conjugate gradient squared algorithm) are implemented in a computer code. Because the first type boundary conditions are imposed, it is possible to compare the results with the analytical solution. Computations are carried out for different numerical grid densities. Therefore it is possible to investigate how the grid density influences the efficiency of the gradient algorithms. The total computational time, residual drop and the iteration time for the gradient algorithms are additionally compared with the performance of the SOR (successive over-relaxation) method.

An efficient impedance method for induced field evaluation based on a stabilized Bi-conjugate gradient algorithm

This paper presents a stabilized Bi-conjugate gradient algorithm (BiCGstab) that can significantly improve the performance of the impedance method, which has been widely applied to model low-frequency field induction phenomena in voxel phantoms. The improved impedance method offers remarkable computational advantages in terms of convergence performance and memory consumption over the conventional, successive over-relaxation (SOR)-based algorithm. The scheme has been validated against other numerical/analytical solutions on a lossy, multilayered sphere phantom excited by an ideal coil loop. To demonstrate the computational performance and application capability of the developed algorithm, the induced fields inside a human phantom due to a low-frequency hyperthermia device is evaluated. The simulation results show the numerical accuracy and superior performance of the method.

Unconditionally stable Crank–Nicolson finite-different time-domain method for simulation of three-dimensional microwave circuits

An unconditionally stable Crank–Nicolson finite-difference time-domain (CN-FDTD) algorithm is presented for three-dimensional microwave circuit analysis. First, Mur's first–order absorbing boundary condition is applied this CN-FDTD algorithm. A symmetric successive over relaxation–preconditioned biconjugate-gradient algorithm is also proposed to solve the large sparse matrix equation obtained in the CN-FDTD method. Resonant cavity and several planar microstrip circuits are presented to illustrate the versatility of this technique. Numerical results indicate that with a time-step size excessively larger than the Courant–Friedrich–Levy limit, the accuracy of CN-FDTD is still much higher than that of the alternating-direction implicit FDTD.

Improved iterative algorithm for 3D edge FEM analysis of electromagnetic field boundary value problems

In this article, an improved iterative arithmetic of the symmetric successive over-relaxation preconditioning biconjugate-gradient algorithm (ISSOR-PBCG) is utilized to solve the 3D edge FEM equations derived from the time-harmonic electromagnetic-field boundary value problems. Several typical structures have been analyzed, and the computation time is compared with that of other algorithms such as biconjugate-gradient (BCG) algorithm and the conventional symmetric successive over-relaxation preconditioning biconjugate- gradient algorithm (SSOR-PBCG). The CPU time saved using the ISSOR-PBCG algorithm is nearly 27% and 65.5%, as compared with that using the SSOR-PBCG and the BCG algorithm. It can be seen that the ISSOR-PBCG algorithm is efficient for edge FEM equation sets derived from large-scale time-harmonic electromagnetic-field problems. © 2006 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2006.

Analysis of the finite precision bi-conjugate gradient algorithm for nonsymmetric linear systems

In this paper we analyze the bi-conjugate gradient algorithm in finite precision arithmetic, and suggest reasons for its often observed robustness. By using a tridiagonal structure, which is preserved by the finite precision bi-conjugate gradient iteration, we are able to bound its residual norm by a minimum polynomial of a perturbed matrix (i.e. the residual norm of the exact GMRES applied to a perturbed matrix) multiplied by an amplification factor. This shows that occurrence of near-breakdowns or loss of biorthogonality does not necessarily deter convergence of the residuals provided that the amplification factor remains bounded. Numerical examples are given to gain insights into these bounds.

Multiparameter descent methods

A descent method for solving a system of linear equations Ax= b consists of the iterations x k+1 = x k + λ k z k , where z k is a vector and λ k a parameter chosen to minimize some functional. In this paper, we will consider multiparameter generalizations of such descent methods, namely iterations of the form x k+1 = x k + Z k Λ k , where Z k is a matrix and Λ k a vector chosen to minimize some functional. Multiparameter generalizations of the conjugate direction method and the Lanczos method will be obtained and their algebraic properties discussed. Multiparameter conjugate and biconjugate gradient algorithms and other Lanczos-type algorithms for implementing this Lanczos method will also be given.

Efficient Pseudospectral Flow Simulations in Moderately Complex Geometries

A computationally efficient pseudospectral method is developed for incompressible flow simulations in two-dimensional geometries involving periodicity in one direction and significant surface deformations. A pseudoconformal mapping is used to map the flow domain into a rectangle, thereby establishing an orthogonal curvilinear coordinate system within which the governing equations are formulated. The time integration of the spectrally discretized, two-dimensional momentum equations is performed by a second-order mixed explicit/implicit time integration scheme. The satisfaction of the continuity equation is obtained through the solution of a Poisson equation for the pressure and the use of the influence matrix technique. A highly efficient iterative solver has been developed for the solution of a generalized Stokes problem at each time step based on a spectrally preconditioned biconjugate gradient algorithm, which exhibits almost linear scalability, requiring an orderNlog2Nnumber of operations, whereNis the number of unknowns. Numerical results are presented for two-dimensional steady, oscillatory, and peristaltic flows within an undulating channel, which agree well with previous results that have appeared in the literature.

Computation of electromagnetic fields for high-frequency magnetic resonance imaging applications

A numerical method is presented to compute electromagnetic fields inside a 2 mm high resolution, anatomically detailed model of a human head for high-frequency magnetic resonance imaging (MRI) applications. The method uses the biconjugate gradient algorithm in combination with the fast Fourier transform to solve a matrix equation resulting from the discretization of an integrodifferential equation representing the original physical problem. Given the current distribution in an MRI coil, the method can compute both the electric field (thus the specific energy absorption rate (SAR)) and the magnetic field, also known as the field. Results for the SAR and field distribution, excited by a linear and a quadrature birdcage coil, are calculated and presented at 64 MHz, 128 MHz and 256 MHz, corresponding to the operating frequencies of the 1.5 T, 3 T and 6 T MRI systems. It is shown that compared with that at 64 MHz, the SAR at 128 MHz is increased by a factor over 5 and the SAR at 256 MHz is increased by a factor over 10, assuming the same current strength in the coil. Furthermore, compared with the linear excitation, the average SAR for the quadrature excitation is reduced by a factor over 2 and the maximum SAR is reduced by a factor over 3. It is also shown that the field at high frequencies exhibits a strong inhomogeneity, which is attributed to dielectric resonance.

Composite Step Product Methods for Solving Nonsymmetric Linear Systems

First proposed in [Numer. Math., 66 (1993), pp. 295–319, Numer. Algorithms, 7 (1994), pp. 1–16] by Bank and Chan, the composite step (CS) method is a technique for curing the pivot breakdown (one of two possible breakdowns) in the biconjugate gradient (BCG) algorithm by skipping over steps in which the iterate is not defined. We show how to extend this method to cure the same breakdown inherited in product methods such as conjugate gradients squared (CGS) [SIAM J. Sci. Statist. Comput., 10 (1989), pp. 36–52], Bi-CGSTAB [SIAM J. Sci. Statist. Comput., 13 (1992), pp. 631–644], Bi-CGSTAB2 [Variants of BiCGSTAB for Matrices with Complex Spectrum, IPS Research Report No. 91-14, ETH Zurich], which are derived from a product of the BCG polynomial with another polynomial of the same degree. New methods introduced in this paper are CS-CGSTAB (composite step applied to Bi-CGSTAB) and a more stable variant of this, CS-CGSTAB2, which handles a possible additional breakdown problem due to the Bi-CGSTAB process in the case where A is skew symffietric. CS-CGSTAB2 can be viewed as an improvement over the Bi-CGSTAB$(l)$ algorithm (Sleijpen and Fokkema [ETNA, 1 (1993), pp. 11–32]) with $l = 2$ in that although Bi-CGSTAB(2) is designed to cure the skew-symmetric breakdown, it does not handle pivot breakdown as CS-CGSTAB2 does. The new methods require only a minor modification to the existing product methods. Moreover, the sizes of the steps taken in our methods can vary as opposed to fixed step methods like Bi-CGSTAB$(l)$ and Bi-CGSTAB2.Our strategy for deciding whether to skip a step does not involve any machine dependent parameters and is designed to skip near breakdowns as well as produce smoother iterates. Numerical experiments show that the new methods do produce improved performance over those without composite step on practical problems. Furthermore, we extend the “best approximation” result in [Numer. Algorithms, 7 (1994), pp. 1–16] to obtain convergence proofs for CGS and Bi-CGSTAB and their composite step stabilized versions.

Relations between Galerkin and Norm-Minimizing Iterative Methods for Solving Linear Systems

Several iterative methods for solving linear systems $Ax = b$ first construct a basis for a Krylov subspace and then use the basis vectors, together with the Hessenberg (or tridiagonal) matrix generated during that construction, to obtain an approximate solution to the linear system. To determine the approximate solution, it is necessary to solve either a linear system with the Hessenberg matrix as coefficient matrix or an extended Hessenberg least squares problem. In the first case, referred to as a Galerkin method, the residual is orthogonal to the Krylov subspace, whereas in the second case, referred to as a norm-minimizing method, the residual (or a related quantity) is minimized over the Krylov subspace. Examples of such pairs include the full orthogonalization method (FOM) (Arnoldi) and generalized minimal residual (GMRES) algorithms, the biconjugate gradient (BCG) and quasi-minimal residual (QMR) algorithms, and their symmetric equivalents, the Lanczos and minimal residual (MINRES) algorithms. A relationship between the solution of the linear system and that of the least squares problem is used to relate the residual norms in Galerkin processes to the norms of the quantities minimized in the corresponding norm-minimizing processes. It is shown that when the norm-minimizing process is converging rapidly, the residual norms in the corresponding Galerkin process exhibit similar behavior, whereas when the norm-minimizing process is converging very slowly, the residual norms in the corresponding Galerkin process are significantly larger. This is a generalization of the relationship established between Arnoldi and GMRES residual norms in P. N. Brown, A theoretical comparison of the Arnoldi and GMRES algorithms, SIAM J. Sci. Statist. Comput., 12, 1991, pp. 58–78. For MINRES and Lanczos, and for two nonsymmetric bidiagonalization procedures, we extend the arguments to incorporate the effects of finite precision arithmetic.