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Journal articles on the topic 'Preconditioned conjugate gradients'

Author: Grafiati

Published: 4 June 2021

Last updated: 1 February 2022

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1

Wang, Weichung. "Final iterations in interior point methods – preconditioned conjugate gradients and modified search directions." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 41, no. 3 (2000): 312–28. http://dx.doi.org/10.1017/s0334270000011267.

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AbstractIn this article we consider modified search directions in the endgame of interior point methods for linear programming. In this stage, the normal equations determining the search directions become ill-conditioned. The modified search directions are computed by solving perturbed systems in which the systems may be solved efficiently by the preconditioned conjugate gradient solver. A variation of Cholesky factorization is presented for computing a better preconditioner when the normal equations are ill-conditioned. These ideas have been implemented successfully and the numerical results show that the algorithms enhance the performance of the preconditioned conjugate gradients-based interior point methods.

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2

Strakoš, Zdeněk, and Petr Tichý. "Error Estimation in Preconditioned Conjugate Gradients." BIT Numerical Mathematics 45, no. 4 (2005): 789–817. http://dx.doi.org/10.1007/s10543-005-0032-1.

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3

Wolkowicz, Henry. "Solving semidefinite programs using preconditioned conjugate gradients." Optimization Methods and Software 19, no. 6 (2004): 653–72. http://dx.doi.org/10.1080/1055678042000193162.

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4

Kaasschieter, E. F. "Preconditioned conjugate gradients for solving singular systems." Journal of Computational and Applied Mathematics 24, no. 1-2 (1988): 265–75. http://dx.doi.org/10.1016/0377-0427(88)90358-5.

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5

Duff, Iain S., and Gérard A. Meurant. "The effect of ordering on preconditioned conjugate gradients." BIT 29, no. 4 (1989): 635–57. http://dx.doi.org/10.1007/bf01932738.

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6

Collignon, Tijmen, and Martin Van Gijzen. "Two implementations of the preconditioned conjugate gradient method on heterogeneous computing grids." International Journal of Applied Mathematics and Computer Science 20, no. 1 (2010): 109–21. http://dx.doi.org/10.2478/v10006-010-0008-4.

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Two implementations of the preconditioned conjugate gradient method on heterogeneous computing gridsEfficient iterative solution of large linear systems on grid computers is a complex problem. The induced heterogeneity and volatile nature of the aggregated computational resources present numerous algorithmic challenges. This paper describes a case study regarding iterative solution of large sparse linear systems on grid computers within the software constraints of the grid middleware GridSolve and within the algorithmic constraints of preconditioned Conjugate Gradient (CG) type methods. We identify the various bottlenecks induced by the middleware and the iterative algorithm. We consider the standard CG algorithm of Hestenes and Stiefel, and as an alternative the Chronopoulos/Gear variant, a formulation that is potentially better suited for grid computing since it requires only one synchronisation point per iteration, instead of two for standard CG. In addition, we improve the computation-to-communication ratio by maximising the work in the preconditioner. In addition to these algorithmic improvements, we also try to minimise the communication overhead within the communication model currently used by the GridSolve middleware. We present numerical experiments on 3D bubbly flow problems using heterogeneous computing hardware that show lower computing times and better speed-up for the Chronopoulos/Gear variant of conjugate gradients. Finally, we suggest extensions to both the iterative algorithm and the middleware for improving granularity.

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7

Baxter, B. J. C. "Preconditioned conjugate gradients, radial basis functions, and Toeplitz matrices." Computers & Mathematics with Applications 43, no. 3-5 (2002): 305–18. http://dx.doi.org/10.1016/s0898-1221(01)00288-7.

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8

Manero, O. "Finite difference solution of viscoelastic flows by preconditioned conjugate gradients." Numerical Methods for Partial Differential Equations 2, no. 4 (1986): 317–26. http://dx.doi.org/10.1002/num.1690020406.

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9

Pilkington, Mark. "3-D magnetic imaging using conjugate gradients." GEOPHYSICS 62, no. 4 (1997): 1132–42. http://dx.doi.org/10.1190/1.1444214.

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A 3-D inversion approach is outlined that determines a distribution of susceptibility that produces a given magnetic anomaly. The subsurface model consists of a 3-D array of rectangular blocks, each with a constant susceptibility. The inversion incorporates a model norm that allows smoothing and depth‐weighting of the solution. Since the number of parameters can be many thousands, even for small problems, the linear system of equations is inverted using a preconditioned conjugate gradient approach. This reduces memory requirements and avoids large matrix multiplications. The method is used to determine the 3-D susceptibility distribution responsible for the Temagami magnetic anomaly in southern Ontario, Canada.

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10

Pini, G., G. Gambolati, and G. Galeati. "3-D finite element transport models by upwind preconditioned conjugate gradients." Advances in Water Resources 12, no. 2 (1989): 54–58. http://dx.doi.org/10.1016/0309-1708(89)90001-8.

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11

Kredler, Christian, Christian Zillober, Frank Johannes, and Georg Sigl. "An application of preconditioned conjugate gradients to relative placement in chip design." International Journal for Numerical Methods in Engineering 36, no. 2 (1993): 255–71. http://dx.doi.org/10.1002/nme.1620360206.

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12

Klein, Ole, Olaf A. Cirpka, Peter Bastian, and Olaf Ippisch. "Efficient geostatistical inversion of transient groundwater flow using preconditioned nonlinear conjugate gradients." Advances in Water Resources 102 (April 2017): 161–77. http://dx.doi.org/10.1016/j.advwatres.2016.12.006.

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13

Gohberg, I., and I. Koltracht. "A fast realization of preconditioned Conjugate Gradients for Wiener-Hopf integral equations." Applied Mathematics Letters 8, no. 6 (1995): 65–72. http://dx.doi.org/10.1016/0893-9659(95)00087-7.

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14

van der Vorst, Henk A. "The performance of FORTRAN implementations for preconditioned conjugate gradients on vector computers." Parallel Computing 3, no. 1 (1986): 49–58. http://dx.doi.org/10.1016/0167-8191(86)90006-2.

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15

Dupont, S., and J. M. Marchal. "Preconditioned conjugate gradients for solving the transient Boussinesq equations in three-dimensional geometries." International Journal for Numerical Methods in Fluids 8, no. 3 (1988): 283–303. http://dx.doi.org/10.1002/fld.1650080303.

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16

Bielawski, Stanislav S., Stepan G. Mulyarchik, and Andrew V. Popov. "The construction of an algebraically reduced system for the acceleration of preconditioned conjugate gradients." Journal of Computational and Applied Mathematics 70, no. 2 (1996): 189–200. http://dx.doi.org/10.1016/0377-0427(95)00204-9.

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17

Kraft, Rosilene A., and Alvaro L. G. A. Coutinho. "Deflated preconditioned conjugate gradients applied to a Petrov-Galerkin generalized least squares finite element formulation for incompressible flows with heat transfer." International Journal of Numerical Methods for Heat & Fluid Flow 25, no. 2 (2015): 272–98. http://dx.doi.org/10.1108/hff-12-2012-0272.

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Purpose – The purpose of this paper is to show benefits of deflated preconditioned conjugate gradients (CG) in the solution of transient, incompressible, viscous flows coupled with heat transfer. Design/methodology/approach – This paper presents the implementation of deflated preconditioned CG as the iterative driver for the system of linearized equations for viscous, incompressible flows and heat transfer simulations. The De Sampaio-Coutinho particular form of the Petrov-Galerkin Generalized Least Squares finite element formulation is used in the discretization of the governing equations, leading to symmetric positive definite matrices, allowing the use of the CG solver. Findings – The use of deflation techniques improves the spectral condition number. The authors show in a number of problems of coupled viscous flow and heat transfer that convergence is achieved with a lower number of iterations and smaller time. Originality/value – This work addressed for the first time the use of deflated CG for the solution of transient analysis of free/forced convection in viscous flows coupled with heat transfer.

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18

KANTZAS, APOSTOLOS, and IOANNIS CHATZIS. "APPLICATION OF THE PRECONDITIONED CONJUGATE GRADIENTS METHOD IN THE SIMULATION OF RELATIVE PERMEABILITY PROPERTIES OF POROUS MEDIA." Chemical Engineering Communications 69, no. 1 (1988): 169–89. http://dx.doi.org/10.1080/00986448808940611.

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19

Xu, Linan, and Mauricio D. Sacchi. "Preconditioned acoustic least-squares two-way wave-equation migration with exact adjoint operator." GEOPHYSICS 83, no. 1 (2018): S1—S13. http://dx.doi.org/10.1190/geo2017-0167.1.

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We have investigated the problem of designing the forward operator and its exact adjoint for two-way wave-equation least-squares migration. We study the problem in the time domain and pay particular attention to the individual operators that are required by the algorithm. We derive our algorithm using the language of linear algebra and establish a simple path to design forward and adjoint operators that pass the dot-product test. We also found that the exact adjoint operator is not equal to the classic reverse time migration algorithm. For instance, one must pay particular attention to boundary conditions to compute the exact adjoint that accurately passes the dot-product test. Forward and adjoint operators are adopted to solve the so-called least-squares reverse time migration problem via the method of conjugate gradients. We also examine a preconditioning strategy to invert extended images.

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20

Ek, David, and Anders Forsgren. "Exact linesearch limited-memory quasi-Newton methods for minimizing a quadratic function." Computational Optimization and Applications 79, no. 3 (2021): 789–816. http://dx.doi.org/10.1007/s10589-021-00277-4.

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AbstractThe main focus in this paper is exact linesearch methods for minimizing a quadratic function whose Hessian is positive definite. We give a class of limited-memory quasi-Newton Hessian approximations which generate search directions parallel to those of the BFGS method, or equivalently, to those of the method of preconditioned conjugate gradients. In the setting of reduced Hessians, the class provides a dynamical framework for the construction of limited-memory quasi-Newton methods. These methods attain finite termination on quadratic optimization problems in exact arithmetic. We show performance of the methods within this framework in finite precision arithmetic by numerical simulations on sequences of related systems of linear equations, which originate from the CUTEst test collection. In addition, we give a compact representation of the Hessian approximations in the full Broyden class for the general unconstrained optimization problem. This representation consists of explicit matrices and gradients only as vector components.

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21

Hughes, Thomas J. R., Robert M. Ferencz, and John O. Hallquist. "Large-scale vectorized implicit calculations in solid mechanics on a Cray X-MP/48 utilizing EBE preconditioned conjugate gradients." Computer Methods in Applied Mechanics and Engineering 61, no. 2 (1987): 215–48. http://dx.doi.org/10.1016/0045-7825(87)90005-3.

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22

Bergamaschi, L., and A. Martínez. "Parallel RFSAI-BFGS Preconditioners for Large Symmetric Eigenproblems." Journal of Applied Mathematics 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/767042.

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We propose a parallel preconditioner for the Newton method in the computation of the leftmost eigenpairs of large and sparse symmetric positive definite matrices. A sequence of preconditioners starting from an enhanced approximate inverse RFSAI (Bergamaschi and Martínez, 2012) and enriched by a BFGS-like update formula is proposed to accelerate the preconditioned conjugate gradient solution of the linearized Newton system to solveAu=q(u)u,q(u)being the Rayleigh quotient. In a previous work (Bergamaschi and Martínez, 2013) the sequence of preconditioned Jacobians is proven to remain close to the identity matrix if the initial preconditioned Jacobian is so. Numerical results onto matrices arising from various realistic problems with size up to 1.5 million unknowns account for the efficiency and the scalability of the proposed low rank update of the RFSAI preconditioner. The overall RFSAI-BFGS preconditioned Newton algorithm has shown comparable efficiencies with a well-established eigenvalue solver on all the test problems.

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23

Bergamaschi, Luca, Angeles Martínez, and Giorgio Pini. "Parallel Rayleigh Quotient Optimization with FSAI-Based Preconditioning." Journal of Applied Mathematics 2012 (2012): 1–14. http://dx.doi.org/10.1155/2012/872901.

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The present paper describes a parallel preconditioned algorithm for the solution of partial eigenvalue problems for large sparse symmetric matrices, on parallel computers. Namely, we consider the Deflation-Accelerated Conjugate Gradient (DACG) algorithm accelerated by factorized-sparse-approximate-inverse- (FSAI-) type preconditioners. We present an enhanced parallel implementation of the FSAI preconditioner and make use of the recently developed Block FSAI-IC preconditioner, which combines the FSAI and the Block Jacobi-IC preconditioners. Results onto matrices of large size arising from finite element discretization of geomechanical models reveal that DACG accelerated by these type of preconditioners is competitive with respect to the available public parallelhyprepackage, especially in the computation of a few of the leftmost eigenpairs. The parallel DACG code accelerated by FSAI is written in MPI-Fortran 90 language and exhibits good scalability up to one thousand processors.

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24

XU, JINCHAO, and YUNRONG ZHU. "UNIFORM CONVERGENT MULTIGRID METHODS FOR ELLIPTIC PROBLEMS WITH STRONGLY DISCONTINUOUS COEFFICIENTS." Mathematical Models and Methods in Applied Sciences 18, no. 01 (2008): 77–105. http://dx.doi.org/10.1142/s0218202508002619.

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This paper gives a solution to an open problem concerning the performance of various multilevel preconditioners for the linear finite element approximation of second-order elliptic boundary value problems with strongly discontinuous coefficients. By analyzing the eigenvalue distribution of the BPX preconditioner and multigrid V-cycle preconditioner, we prove that only a small number of eigenvalues may deteriorate with respect to the discontinuous jump or meshsize, and we prove that all the other eigenvalues are bounded below and above nearly uniformly with respect to the jump and meshsize. As a result, we prove that the convergence rate of the preconditioned conjugate gradient methods is uniform with respect to the large jump and meshsize. We also present some numerical experiments to demonstrate the theoretical results.

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25

Arany, I. "The preconditioned conjugate gradient method with incomplete factorization preconditioners." Computers & Mathematics with Applications 31, no. 4-5 (1996): 1–5. http://dx.doi.org/10.1016/0898-1221(95)00210-3.

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26

Papež, J., L. Grigori, and R. Stompor. "Solving linear equations with messenger-field and conjugate gradient techniques: An application to CMB data analysis." Astronomy & Astrophysics 620 (November 29, 2018): A59. http://dx.doi.org/10.1051/0004-6361/201832987.

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We discuss linear system solvers invoking a messenger-field and compare them with (preconditioned) conjugate gradient approaches. We show that the messenger-field techniques correspond to fixed point iterations of an appropriately preconditioned initial system of linear equations. We then argue that a conjugate gradient solver applied to the same preconditioned system, or equivalently a preconditioned conjugate gradient solver using the same preconditioner and applied to the original system, will in general ensure at least a comparable and typically better performance in terms of the number of iterations to convergence and time-to-solution. We illustrate our conclusions with two common examples drawn from the cosmic microwave background (CMB) data analysis: Wiener filtering and map-making. In addition, and contrary to the standard lore in the CMB field, we show that the performance of the preconditioned conjugate gradient solver can depend significantly on the starting vector. This observation seems of particular importance in the cases of map-making of high signal-to-noise ratio sky maps and therefore should be of relevance for the next generation of CMB experiments.

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27

Kazemi, Nasser. "Shot-record extended model domain preconditioners for least-squares migration." GEOPHYSICS 84, no. 4 (2019): S285—S299. http://dx.doi.org/10.1190/geo2018-0475.1.

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We have studied the preconditioned conjugate gradient (CG) algorithm in the context of shot-record extended model domain least-squares migration. The CG algorithm is a powerful iterative technique that can solve the least-squares migration problem efficiently; however, to see the merits of least-squares migration, one needs to apply the algorithm for several iterations. Generally speaking, the convergence rate of the CG algorithm depends on the condition number of the operator. Preconditioners are a family of operators that are easy to build and invert. Proper preconditioners can cluster the eigenvalues of the original operator; hence, they reduce the condition number of the operator that one wishes to invert. Accordingly, preconditioning the operator can, in theory, improve the convergence rate of the algorithm. In least-squares migration, the diagonal scaling of the Hessian and the approximated inverse of the Hessian are proven to work well as a preconditioner. We develop and apply two types of preconditioners for the shot-record extended model domain least-squares migration problem. The first preconditioner belongs to the diagonal scaling category, and a second preconditioner is a filter-based approach, which approximates the partial Hessian operators by local convolutional filters. The goal is to increase the convergence rate of the shot-record extended model domain least-squares migration using the reformulated cost function with a preconditioned operator. Experiments with a synthetic Sigsbee model and a real data example from the Gulf of Mexico, Mississippi Canyon data set, indicate that preconditioning the linear system of the equations improves the convergence rate of the algorithm.

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Liu, Zhiyong, and Yinnian He. "Restricted Additive Schwarz Preconditioner for Elliptic Equations with Jump Coefficients." Advances in Applied Mathematics and Mechanics 8, no. 6 (2016): 1072–83. http://dx.doi.org/10.4208/aamm.2014.m669.

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AbstractThis paper provides a proof of robustness of the restricted additive Schwarz preconditioner with harmonic overlap (RASHO) for the second order elliptic problems with jump coefficients. By analyzing the eigenvalue distribution of the RASHO preconditioner, we prove that the convergence rate of preconditioned conjugate gradient method with RASHO preconditioner is uniform with respect to the large jump and meshsize.

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Bertolazzi, Enrico, and Marco Frego. "Preconditioning Complex Symmetric Linear Systems." Mathematical Problems in Engineering 2015 (2015): 1–20. http://dx.doi.org/10.1155/2015/548609.

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A new preconditioner for symmetric complex linear systems based on Hermitian and skew-Hermitian splitting (HSS) for complex symmetric linear systems is herein presented. It applies to conjugate orthogonal conjugate gradient (COCG) or conjugate orthogonal conjugate residual (COCR) iterative solvers and does not require any estimation of the spectrum of the coefficient matrix. An upper bound of the condition number of the preconditioned linear system is provided. To reduce the computational cost the preconditioner is approximated with an inexact variant based on incomplete Cholesky decomposition or on orthogonal polynomials. Numerical results show that the present preconditioner and its inexact variant are efficient and robust solvers for this class of linear systems. A stability analysis of the inexact polynomial version completes the description of the preconditioner.

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Prakonina, A. U. "Spectral properties of discrete models of multi-dimensional elliptic problems with mixed derivatives." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 55, no. 2 (2019): 207–15. http://dx.doi.org/10.29235/1561-2430-2019-55-2-207-215.

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The influence of the spectrum of original and preconditioned matrices on a convergence rate of iterative methods for solving systems of finite-difference equations applicable to two-dimensional elliptic equations with mixed derivatives is investigated. It is shown that the efficiency of the bi-conjugate gradient iterative methods for systems with asymmetric matrices significantly depends not only on the matrix spectrum boundaries, but also on the heterogeneity of the distribution of the spectrum components, as well as on the magnitude of the imaginary part of complex eigenvalues. For test matrices with a fixed condition number, three variants of the spectral distribution were studied and the dependences of the number of iterations on the dimension of matrices were estimated. It is shown that the non-uniformity in the eigenvalue distribution within the fixed spectrum boundaries leads to a significant increase in the number of iterations with increasing dimension of the matrices. The increasing imaginary part of the eigenvalues has a similar effect on the convergence rate. Using as an example the model potential distribution problem in a square domain, including anisotropic ring inhomogeneity, a comparative analysis of the matrix structure and the convergence rate of the bi-conjugate gradient method with Fourier – Jacobi and incomplete LU factorization preconditioners is performed. It is shown that the advantages of the Fourier – Jacobi preconditioner are associated with a more uniform distribution of the spectrum of the preconditioned matrix along the real axis and a better suppression of the imaginary part of the spectrum compared to the preconditioner based on the incomplete LU factorization.

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31

Zítko, Jan. "Combining the preconditioned conjugate gradient method and a matrix iterative method." Applications of Mathematics 41, no. 1 (1996): 19–39. http://dx.doi.org/10.21136/am.1996.134311.

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Pang, Hong-Kui, Ying-Ying Zhang, and Xiao-Qing Jin. "Tri-Diagonal Preconditioner for Toeplitz Systems from Finance." East Asian Journal on Applied Mathematics 1, no. 1 (2011): 82–88. http://dx.doi.org/10.4208/eajam.260609.190510a.

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AbstractWe consider a nonsymmetric Toeplitz system which arises in the discretization of a partial integro-differential equation in option pricing problems. The preconditioned conjugate gradient method with a tri-diagonal preconditioner is used to solve this system. Theoretical analysis shows that under certain conditions the tri-diagonal preconditioner leads to a superlinear convergence rate. Numerical results exemplify our theoretical analysis.

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Beuchler, Sven, and Martin Purrucker. "Schwarz Type Solvers for -FEM Discretizations of Mixed Problems." Computational Methods in Applied Mathematics 12, no. 4 (2012): 369–90. http://dx.doi.org/10.2478/cmam-2012-0030.

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AbstractThis paper investigates the discretization of mixed variational formulation as, e.g., the Stokes problem by means of the hp-version of the finite element method. The system of linear algebraic equations is solved by the preconditioned Bramble-Pasciak conjugate gradient method. The development of an efficient preconditioner requires three ingredients, a preconditioner related to the components of the velocity modes, a preconditioner for the Schur complement related to the components of the pressure modes and a discrezation by a stable finite element pair which satisfies the discrete inf-sup-condition. The last condition is also important in order to obtain a stable discretization scheme. The preconditioner for the velocity modes is adapted from fast $hp$-FEM preconditioners for the potential equation. Moreover, we will prove that the preconditioner for the Schur complement can be chosen as a diagonal matrix if the pressure is discretized by discontinuous finite elements. We will prove that the system of linear algebraic equations can be solved in almost optimal complexity. This yields quasioptimal hp-FEM solvers for the Stokes problems and the linear elasticity problems. The latter are robust with respect to the contraction ratio ν. The efficiency of the presented solver is shown in several numerical examples.

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Новиков, А. К., C. П. Копысов, and Н. С. Недожогин. "Parallel forming of preconditioners based on the approximation of the Sherman-Morrison inversion formula." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 1 (April 2, 2015): 86–93. http://dx.doi.org/10.26089/nummet.v16r109.

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Исследуются возможности ускорения предобусловленных методов бисопряженных градиентов (BiCGStab, Bi-Conjugate Gradient Stabilized) с предобусловливателем на основе аппроксимации обращения матрицы по формуле Шермана-Моррисона. Рассмотрена новая форма параллельного алгоритма, использующая матрично-векторные произведения при формирования матриц предобусловливателя. Показана эффективность распараллеливания наиболее ресурсоемких операций этого предобусловливателя на графических процессорах. Acceleration of preconditioned bi-conjugate gradient stabilized (BiCGStab) methods with preconditioners based on the matrix approximation by the Sherman-Morrison inversion formula is studied. A new form of the parallel algorithm using matrix-vector products to generate preconditioning matrices is proposed. A parallelization efficiency of the most resource-intensive operations of such preconditioners on multi-core central and graphics processing units (CPUs and GPUs) is shown.

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Jin, Xiao-Qing, Fu-Rong Lin, and Zhi Zhao. "Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations." Communications in Computational Physics 18, no. 2 (2015): 469–88. http://dx.doi.org/10.4208/cicp.120314.230115a.

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AbstractIn this paper, preconditioned iterative methods for solving two-dimensional space-fractional diffusion equations are considered. The fractional diffusion equation is discretized by a second-order finite difference scheme, namely, the Crank-Nicolson weighted and shifted Grünwald difference (CN-WSGD) scheme proposed in [W. Tian, H. Zhou and W. Deng, A class of second order difference approximation for solving space fractional diffusion equations, Math. Comp., 84 (2015) 1703-1727]. For the discretized linear systems, we first propose preconditioned iterative methods to solve them. Then we apply the D’Yakonov ADI scheme to split the linear systems and solve the obtained splitting systems by iterative methods. Two preconditioned iterative methods, the preconditioned generalized minimal residual (preconditioned GMRES) method and the preconditioned conjugate gradient normal residual (preconditioned CGNR) method, are proposed to solve relevant linear systems. By fully exploiting the structure of the coefficient matrix, we design two special kinds of preconditioners, which are easily constructed and are able to accelerate convergence of iterative solvers. Numerical results show the efficiency of our preconditioners.

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Hu, Zixiang, Shi Zhang, Yun Zhang, Huamin Zhou, and Dequn Li. "An efficient preconditioned Krylov subspace method for large-scale finite element equations with MPC using Lagrange multiplier method." Engineering Computations 31, no. 7 (2014): 1169–97. http://dx.doi.org/10.1108/ec-03-2013-0077.

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Purpose – The purpose of this paper is to propose an efficient iterative method for large-scale finite element equations of bad numerical stability arising from deformation analysis with multi-point constraint using Lagrange multiplier method. Design/methodology/approach – In this paper, taking warpage analysis of polymer injection molding based on surface model as an example, the performance of several popular Krylov subspace methods, including conjugate gradient, BiCGSTAB and generalized minimal residual (GMRES), with diffident Incomplete LU (ILU)-type preconditions is investigated and compared. For controlling memory usage, GMRES(m) is also considered. And the ordering technique, commonly used in the direct method, is introduced into the presented iterative method to improve the preconditioner. Findings – It is found that the proposed preconditioned GMRES method is robust and effective for solving problems considered in this paper, and approximate minimum degree (AMD) ordering is most beneficial for the reduction of fill-ins in the ILU preconditioner and acceleration of the convergence, especially for relatively accurate ILU-type preconditioning. And because of concerns about memory usage, GMRES(m) is a good choice if necessary. Originality/value – In this paper, for overcoming difficulties of bad numerical stability resulting from Lagrange multiplier method, together with increasing scale of problems in engineering applications and limited hardware conditions of computer, a stable and efficient preconditioned iterative method is proposed for practical purpose. Before the preconditioning, AMD reordering, commonly used in the direct method, is introduced to improve the preconditioner. The numerical experiments show the good performance of the proposed iterative method for practical cases, which is implemented in in-house and commercial codes on PC.

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Camargos, Ana Flávia P., Viviane C. Silva, Jean-M. Guichon, and Gérard Meunier. "GPU-accelerated iterative solution of complex-entry systems issued from 3D edge-FEA of electromagnetics in the frequency domain." International Journal of High Performance Computing Applications 31, no. 2 (2016): 119–33. http://dx.doi.org/10.1177/1094342015584476.

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We present a performance analysis of a parallel implementation for both preconditioned conjugate gradient and preconditioned bi-conjugate gradient solvers running on graphic processing units (GPUs) with CUDA programming model. The solvers were mainly optimized for the solution of sparse systems of algebraic equations at complex entries, arising from the three-dimensional edge-finite element analysis of the electromagnetic phenomena involved in the open-bound earth diffusion of currents under time-harmonic excitation. We used a shifted incomplete Cholesky (IC) factorization as preconditioner. Results show a significant speedup by using either a single-GPU or a multi-GPU device, compared to a serial central processing unit (CPU) implementation, thereby allowing the simulations of large-scale problems in low-cost personal computers. Additional experiments of the optimized solvers show that its use can be extended successfully to other complex systems of equations arising in electrical engineering, such as those obtained in power–system analysis.

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Greenbaum, Anne, Congming Li, and Han Zheng Chao. "Parallelizing preconditioned conjugate gradient algorithms." Computer Physics Communications 53, no. 1-3 (1989): 295–309. http://dx.doi.org/10.1016/0010-4655(89)90167-7.

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39

Cai, Mingchao, and Luca F. Pavarino. "Hybrid and Multiplicative Overlapping Schwarz Algorithms with Standard Coarse Spaces for Mixed Linear Elasticity and Stokes Problems." Communications in Computational Physics 20, no. 4 (2016): 989–1015. http://dx.doi.org/10.4208/cicp.020815.080316a.

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AbstractThe goal of this work is to construct and study hybrid and multiplicative two-level overlapping Schwarz algorithms with standard coarse spaces for the almost incompressible linear elasticity and Stokes systems, discretized by mixed finite and spectral element methods with discontinuous pressures. Two different approaches are considered to solve the resulting saddle point systems: a) a preconditioned conjugate gradient (PCG) method applied to the symmetric positive definite reformulation of the almost incompressible linear elasticity system obtained by eliminating the pressure unknowns; b) a GMRES method with indefinite overlapping Schwarz preconditioner applied directly to the saddle point formulation of both the elasticity and Stokes systems. Condition number estimates and convergence properties of the proposed hybrid and multiplicative overlapping Schwarz algorithms are proven for the positive definite reformulation of almost incompressible elasticity. These results are based on our previous study [8] where only additive Schwarz preconditioners were considered for almost incompressible elasticity. Extensive numerical experiments with both finite and spectral elements show that the proposed overlapping Schwarz preconditioners are scalable, quasi-optimal in the number of unknowns across individual subdomains and robust with respect to discontinuities of the material parameters across subdomains interfaces. The results indicate that the proposed preconditioners retain a good performance also when the quasi-monotonicity assumption, required by the available theory, does not hold.

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Moutafis, Byron E., George A. Gravvanis, and Christos K. Filelis-Papadopoulos. "Hybrid multi-projection method using sparse approximate inverses on GPU clusters." International Journal of High Performance Computing Applications 34, no. 3 (2020): 282–305. http://dx.doi.org/10.1177/1094342020905637.

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The state-of-the-art supercomputing infrastructures are equipped with accelerators, such as graphics processing units (GPUs), that operate as coprocessors for each workstation of the distributed memory system. The multi-projection type methods are a class of algebraic domain decomposition methods based on semi-aggregation techniques. The multi-projection type methods have improved convergence behavior, as the number of subdomains increases, due to the corresponding augmentation of the semi-aggregated local linear systems with more coarse components, while the number of fine components is reduced. Moreover, limited amount of communications among the workstations is required by the proposed method. The utilization of the available GPUs allows an increase in the number of subdomains along with finer-grained parallelism, leading to improved performance. A load-balancing algorithm that ensures the concurrency of the computations on multicore processors and GPUs is proposed. Flexible parallel preconditioned Krylov subspace iterative methods enhanced with multi-projection type methods have been designed appropriately in order to have improved performance, compared to CPU-only or GPU-only executions, by exploiting the available CPUs and GPUs of the distributed memory system concurrently. The unsymmetric local linear systems are solved by the preconditioned Bi-Conjugate Gradient STABilized (BiCGSTAB) method enhanced with the modified generic factored approximate sparse inverse preconditioner, whereas the preconditioned conjugate gradient (CG) method along with the symmetric factored approximate sparse inverse preconditioner is used for the symmetric positive definite local coefficient matrices. Numerical results regarding the convergence behavior, the performance, and the scalability of the proposed method for several problems are given.

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Bergamaschi, Luca. "A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems." Algorithms 13, no. 4 (2020): 100. http://dx.doi.org/10.3390/a13040100.

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The aim of this survey is to review some recent developments in devising efficient preconditioners for sequences of symmetric positive definite (SPD) linear systems A k x k = b k , k = 1 , … arising in many scientific applications, such as discretization of transient Partial Differential Equations (PDEs), solution of eigenvalue problems, (Inexact) Newton methods applied to nonlinear systems, rational Krylov methods for computing a function of a matrix. In this paper, we will analyze a number of techniques of updating a given initial preconditioner by a low-rank matrix with the aim of improving the clustering of eigenvalues around 1, in order to speed-up the convergence of the Preconditioned Conjugate Gradient (PCG) method. We will also review some techniques to efficiently approximate the linearly independent vectors which constitute the low-rank corrections and whose choice is crucial for the effectiveness of the approach. Numerical results on real-life applications show that the performance of a given iterative solver can be very much enhanced by the use of low-rank updates.

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Pan, Wenyong, Kristopher A. Innanen, and Wenyuan Liao. "Accelerating Hessian-free Gauss-Newton full-waveform inversion via l-BFGS preconditioned conjugate-gradient algorithm." GEOPHYSICS 82, no. 2 (2017): R49—R64. http://dx.doi.org/10.1190/geo2015-0595.1.

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Full-waveform inversion (FWI) has emerged as a powerful strategy for estimating subsurface model parameters by iteratively minimizing the difference between synthetic data and observed data. The Hessian-free (HF) optimization method represents an attractive alternative to Newton-type and gradient-based optimization methods. At each iteration, the HF approach obtains the search direction by approximately solving the Newton linear system using a matrix-free conjugate-gradient (CG) algorithm. The main drawback with HF optimization is that the CG algorithm requires many iterations. In our research, we develop and compare different preconditioning schemes for the CG algorithm to accelerate the HF Gauss-Newton (GN) method. Traditionally, preconditioners are designed as diagonal Hessian approximations. We additionally use a new pseudo diagonal GN Hessian as a preconditioner, making use of the reciprocal property of Green’s function. Furthermore, we have developed an [Formula: see text]-BFGS inverse Hessian preconditioning strategy with the diagonal Hessian approximations as an initial guess. Several numerical examples are carried out. We determine that the quasi-Newton [Formula: see text]-BFGS preconditioning scheme with the pseudo diagonal GN Hessian as the initial guess is most effective in speeding up the HF GN FWI. We examine the sensitivity of this preconditioning strategy to random noise with numerical examples. Finally, in the case of multiparameter acoustic FWI, we find that the [Formula: see text]-BFGS preconditioned HF GN method can reconstruct velocity and density models better and more efficiently compared with the nonpreconditioned method.

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43

Adams, Loyce. "m-Step Preconditioned Conjugate Gradient Methods." SIAM Journal on Scientific and Statistical Computing 6, no. 2 (1985): 452–63. http://dx.doi.org/10.1137/0906032.

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Cloutier, J. R., and R. F. Wilson. "Periodically preconditioned conjugate gradient-restoration algorithm." Journal of Optimization Theory and Applications 70, no. 1 (1991): 79–95. http://dx.doi.org/10.1007/bf00940505.

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Sues, Robert H., Heh-Chyun Chen, and Francis M. Lavelle. "The Stochastic Preconditioned Conjugate Gradient method." Probabilistic Engineering Mechanics 7, no. 3 (1992): 175–82. http://dx.doi.org/10.1016/0266-8920(92)90021-9.

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CHANG, MING-YUE, and CHING-HAN HSU. "AN INVESTIGATION OF GRADIENT-BASED RECONSTRUCTION ALGORITHMS FOR STATISTICAL PET TRANSMISSION IMAGING." Biomedical Engineering: Applications, Basis and Communications 15, no. 05 (2003): 179–85. http://dx.doi.org/10.4015/s1016237203000274.

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For PET transmission imaging, the conventional iterative algorithms based on expectation maximization type algorithms, could not effectively converge to optimal image solution. In this study, we suggest a statistical model PET transmission data, and then investigate a class of gradient-based optimization algorithms for transmission image reconstruction including steepest ascent, conjugate gradient, and preconditioned conjugate gradient. From phantom studies, the preconditioned conjugate algorithms can converge to good image results within limited number of iteration. Combined with the suggested statistical model of transmission data, the preconditioned conjugate algorithms can also produce attenuation maps with accurate linear attenuation coefficients for clinical data.

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Bello, Musa, Jianxin Liu, and Rongwen Guo. "Three-Dimensional Wide-Band Electromagnetic Forward Modelling Using Potential Technique." Applied Sciences 9, no. 7 (2019): 1328. http://dx.doi.org/10.3390/app9071328.

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The efficacy of Krylov subspace solvers is strongly dependent on the preconditioner applied to solve the large sparse linear systems of equation for electromagnetic problems. In this study, we present a three-dimensional (3-D) plane wave electromagnetic forward simulation over a broadband frequency range. The Maxwell’s equation is solved in a secondary formulation of the Lorentz gauge coupled-potential technique. A finite-volume scheme is employed for discretizing the system of equations on a structured rectilinear mesh. We employed a block incomplete lower-upper factorization (ILU) preconditioner that is suitable for our potential formulation to enhance the computing time and convergence of the systems of equation by comparing with other preconditioners. Furthermore, we observe their effect on the iterative solvers such as the quasi-minimum residual and bi-conjugate gradient stabilizer. Several applications were used to validate and test the effectiveness of our method. Our scheme shows good agreement with the analytical solution. Notably, from the marine hydrocarbon and the crustal model, the utilisation of the bi-conjugate gradient stabilizer with block ILU preconditioner is the most appropriate. Thus, our approach can be incorporated to optimize the inversion process.

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Milyukova, Olga Yurievna. "MPI+OpenMP parallel implementation of conjugate gradient method with preconditioner of block partial inverse triangular decomposition of IC2S and IC1." Keldysh Institute Preprints, no. 48 (2021): 1–32. http://dx.doi.org/10.20948/prepr-2021-48.

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The paper proposes a new preconditioner for solving systems of linear algebraic equations with a symmetric positively defined matrix by the method of conjugate gradients – Block Incomplete Inverse Cholesky BIIC preconditioner in combination with a triangular first-order decomposition "by value" - BIIC-IC1. The algorithm based on MPI+OpenMP techniques is proposed for the construction and application of the BIIC preconditioner combined with stabilized triangular decomposition of the second order "by value" (BIIC-IS2S). In this case, the BIIC-IC2S preconditioner uses the number of blocks multiple of the number of processors used and the number of threads used. Two algorithms based on MPI+OpenMP techniques are proposed for the construction and application of the BIIC-IC1 preconditioner. Comparative timing results for the MPI+OpenMP and MPI implementations of the proposed preconditioning used with the conjugate gradient method for a model problem and the sparse matrix collections SuiteSparse are presented.

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Zhang, Jianfei, and Lei Zhang. "Efficient CUDA Polynomial Preconditioned Conjugate Gradient Solver for Finite Element Computation of Elasticity Problems." Mathematical Problems in Engineering 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/398438.

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Graphics processing unit (GPU) has obtained great success in scientific computations for its tremendous computational horsepower and very high memory bandwidth. This paper discusses the efficient way to implement polynomial preconditioned conjugate gradient solver for the finite element computation of elasticity on NVIDIA GPUs using compute unified device architecture (CUDA). Sliced block ELLPACK (SBELL) format is introduced to store sparse matrix arising from finite element discretization of elasticity with fewer padding zeros than traditional ELLPACK-based formats. Polynomial preconditioning methods have been investigated both in convergence and running time. From the overall performance, the least-squares (L-S) polynomial method is chosen as a preconditioner in PCG solver to finite element equations derived from elasticity for its best results on different example meshes. In the PCG solver, mixed precision algorithm is used not only to reduce the overall computational, storage requirements and bandwidth but to make full use of the capacity of the GPU devices. With SBELL format and mixed precision algorithm, the GPU-based L-S preconditioned CG can get a speedup of about 7–9 to CPU-implementation.

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Čiegis, Raimondas. "Analysis of Parallel Preconditioned Conjugate Gradient Algorithms." Informatica 16, no. 3 (2005): 317–32. http://dx.doi.org/10.15388/informatica.2005.101.

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