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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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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 (July 28, 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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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 (July 30, 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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Qiu, Changkai, Changchun Yin, Yunhe Liu, Xiuyan Ren, Hui Chen, and Tingjie Yan. "Solution of large-scale 3D controlled-source electromagnetic modeling problem using efficient iterative solvers." GEOPHYSICS 86, no. 4 (June 30, 2021): E283—E296. http://dx.doi.org/10.1190/geo2020-0461.1.
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With geophysical surveys evolving from traditional 2D to 3D models, the large volume of data adds challenges to inversion, especially when aiming to resolve complex 3D structures. An iterative forward solver for the controlled-source electromagnetic (CSEM) method requires less memory than that for a direct solver; however, it is not easy to iteratively solve an ill-conditioned linear system of equations arising from finite-element discretization of Maxwell’s equations. To solve this problem, we have developed efficient and robust iterative solvers for frequency- and time-domain CSEM modeling problems. For the frequency-domain problem, we first transform the linear system into its equivalent real-number format, and then introduce an optimal block-diagonal preconditioner. Because the condition number of the preconditioned linear equation system has an upper bound of [Formula: see text], we can achieve fast solution convergence when applying a flexible generalized minimum residual solver. Applying the block preconditioner further results in solving two smaller linear systems with the same coefficient matrix. For the time-domain problem, we first discretize the partial differential equation for the electric field in time using an unconditionally stable backward Euler scheme. We then solve the resulting linear equation system iteratively at each time step. After the spatial discretization in the frequency domain, or space-time discretization in the time domain, we exploit the conjugate-gradient solver with auxiliary-space preconditioners derived from the Hiptmair-Xu decomposition to solve these real linear systems. Finally, we check the efficiency and effectiveness of our iterative methods by simulating complex CSEM models. The most significant advantage of our approach is that the iterative solvers we adopt have almost the same accuracy and robustness as direct solvers but require much less memory, rendering them more suitable for large-scale 3D CSEM forward modeling and inversion.
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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 (March 29, 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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Puglisi, Giuseppe, Davide Poletti, Giulio Fabbian, Carlo Baccigalupi, Luca Heltai, and Radek Stompor. "Iterative map-making with two-level preconditioning for polarized cosmic microwave background data sets." Astronomy & Astrophysics 618 (October 2018): A62. http://dx.doi.org/10.1051/0004-6361/201832710.
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Context. An estimation of the sky signal from streams of time ordered data (TOD) acquired by the cosmic microwave background (CMB) experiments is one of the most important steps in the context of CMB data analysis referred to as the map-making problem. The continuously growing CMB data sets render the CMB map-making problem progressively more challenging in terms of computational cost and memory in particular in the context of ground-based experiments with their operational limitations as well as the presence of contaminants. Aims. We study a recently proposed, novel class of the Preconditioned Conjugate Gradient (PCG) solvers which invoke two-level preconditioners in the context of the ground-based CMB experiments. We compare them against the PCG solvers commonly used in the map-making context considering their precision and time-to-solution. Methods. We compare these new methods on realistic, simulated data sets reflecting the characteristics of current and forthcoming CMB ground-based experiments. We develop a divide-and-conquer implementation of the approach where each processor performs a sequential map-making for a subset of the TOD. Results. We find that considering the map level residuals, the new class of solvers permits us to achieve a tolerance that is better than the standard approach by up to three orders of magnitude, where the residual level often saturates before convergence is reached. This often corresponds to an important improvement in the precision of the recovered power spectra in particular on the largest angular scales. The new method also typically requires fewer iterations to reach a required precision and therefore shorter run times are required for a single map-making solution. However, the construction of an appropriate two-level preconditioner can be as costly as a single standard map-making run. Nevertheless, if the same problem needs to be solved multiple times, for example, as in Monte Carlo simulations, this cost is incurred only once, and the method should be competitive, not only as far as its precision is concerned but also its performance.
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Ju, S. H., and H. H. Hsu. "An Out-of-Core Eigen-Solver with OpenMP Parallel Scheme for Large Spare Damped System." International Journal of Computational Methods 16, no. 07 (July 26, 2019): 1950038. http://dx.doi.org/10.1142/s0219876219500385.
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An out-of-core block Lanczos method with the OpenMP parallel scheme was developed to solve large spare damped eigenproblems. The symmetric generalized eigenproblem is first solved using the block Lanczos method with the preconditioned conjugate gradient (PCG) method, and the condensed damped eigenproblem is then solved to obtain the complex eigenvalues. Since the PCG solvers and out-of-core schemes are used, a large-scale eigenproblem can be solved using minimal computer memory. The out-of-core arrays only need to be read once in each Lanczos iteration, so the proposed method requires little extra CPU time. In addition, the second-level OpenMP parallel computation in the PCG solver is suggested to avoid using a large block size that often increases the number of iterations needed to achieve convergence.
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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 (February 13, 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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Betté, Srinivas, Julio C. Diaz, William R. Jines, and Trond Steihaug. "A block preconditioned conjugate gradient-type iterative solver for linear systems in thermal reservoir simulation." Journal of Computational Physics 67, no. 1 (November 1986): 37–58. http://dx.doi.org/10.1016/0021-9991(86)90114-2.
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Bergamaschi, Luca. "A Survey of Low-Rank Updates of Preconditioners for Sequences of Symmetric Linear Systems." Algorithms 13, no. 4 (April 21, 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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Clemens, M., M. Wilke, and T. Weiland. "Efficient extrapolation methods for electro- and magnetoquasistatic field simulations." Advances in Radio Science 1 (May 5, 2003): 81–86. http://dx.doi.org/10.5194/ars-1-81-2003.
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Abstract. In magneto- and electroquasi-static time domain simulations with implicit time stepping schemes the iterative solvers applied to the large sparse (non-)linear systems of equations are observed to converge faster if more accurate start solutions are available. Different extrapolation techniques for such new time step solutions are compared in combination with the preconditioned conjugate gradient algorithm. Simple extrapolation schemes based on Taylor series expansion are used as well as schemes derived especially for multi-stage implicit Runge-Kutta time stepping methods. With several initial guesses available, a new subspace projection extrapolation technique is proven to produce an optimal initial value vector. Numerical tests show the resulting improvements in terms of computational efficiency for several test problems. In quasistatischen elektromagnetischen Zeitbereichsimulationen mit impliziten Zeitschrittverfahren zeigt sich, dass die iterativen Lösungsverfahren für die großen dünnbesetzten (nicht-)linearen Gleichungssysteme schneller konvergieren, wenn genauere Startlösungen vorgegeben werden. Verschiedene Extrapolationstechniken werden für jeweils neue Zeitschrittlösungen in Verbindung mit dem präkonditionierten Konjugierte Gradientenverfahren vorgestellt. Einfache Extrapolationsverfahren basierend auf Taylorreihenentwicklungen werden ebenso benutzt wie speziell für mehrstufige implizite Runge-Kutta-Verfahren entwickelte Verfahren. Sind verschiedene Startlösungen verfügbar, so erlaubt ein neues Unterraum-Projektion- Extrapolationsverfahren die Konstruktion eines optimalen neuen Startvektors. Numerische Tests zeigen die aus diesen Verfahren resultierenden Verbesserungen der numerischen Effizienz.
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Ren, Xiao Guang, Wen Hao Zhou, and Juan Chen. "Collective Communication Optimization for Solving Linear Algebraic Equations." Advanced Materials Research 989-994 (July 2014): 4934–39. http://dx.doi.org/10.4028/www.scientific.net/amr.989-994.4934.
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With the development of the electronic technology, the processors count in a supercomputer reaches million scales. However, the processes scale of a application is limited to several thousands, and the scalability face a bottle neck from several aspects, including I/O, communication, cache access .etc. In this paper, we focus on the communication bottleneck to the scalability of linear algebraic equation solve. We take preconditioned conjugate gradient (PCG) as an example, and analysis the feathers of the communication operations in the process of PCG solver. We find that reduce communication is the most critical issue for the scalability of the parallel iterative method for linear algebraic equation solve. We propose a local residual error optimization scheme to eliminate part of the reduce communication operations in the parallel iterative method, and improve the scalability of the parallel iterative method. Experimental results on the Tianhe-2 supercomputer demonstrate that our optimization scheme can achieve a much signally effect for the scalability of the linear algebraic equation solve.
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Iliev, O., R. Lazarov, and J. Willems. "Numerical Study of Two-grid Preconditioners for 1-d Elliptic Problems with Highly Oscillating Discontinuous Coefficients." Computational Methods in Applied Mathematics 7, no. 1 (2007): 48–67. http://dx.doi.org/10.2478/cmam-2007-0003.
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AbstractVarious advanced two-level iterative methods are studied numerically and compared with each other in conjunction with finite volume discretizations of symmetric 1-D elliptic problems with highly oscillatory discontinuous coefficients. Some of the methods considered rely on the homogenization approach for deriving the coarse grid operator. This approach is considered here as an alternative to the well-known Galerkin approach for deriving coarse grid operators. Different intergrid transfer operators are studied, primary consideration being given to the use of the so-called problemdependent prolongation. The two-grid methods considered are used as both solvers and preconditioners for the Conjugate Gradient method. The recent approaches, such as the hybrid domain decomposition method introduced by Vassilevski and the globallocal iterative procedure proposed by Durlofsky et al. are also discussed. A two-level method converging in one iteration in the case where the right-hand side is only a function of the coarse variable is introduced and discussed. Such a fast convergence for problems with discontinuous coefficients arbitrarily varying on the fine scale is achieved by a problem-dependent selection of the coarse grid combined with problem-dependent prolongation on a dual grid. The results of the numerical experiments are presented to illustrate the performance of the studied approaches.
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Turovets, Sergei, Vasily Volkov, Aleksej Zherdetsky, Alena Prakonina, and Allen D. Malony. "A 3D Finite-Difference BiCG Iterative Solver with the Fourier-Jacobi Preconditioner for the Anisotropic EIT/EEG Forward Problem." Computational and Mathematical Methods in Medicine 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/426902.
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The Electrical Impedance Tomography (EIT) and electroencephalography (EEG) forward problems in anisotropic inhomogeneous media like the human head belongs to the class of the three-dimensional boundary value problems for elliptic equations with mixed derivatives. We introduce and explore the performance of several new promising numerical techniques, which seem to be more suitable for solving these problems. The proposed numerical schemes combine the fictitious domain approach together with the finite-difference method and the optimally preconditioned Conjugate Gradient- (CG-) type iterative method for treatment of the discrete model. The numerical scheme includes the standard operations of summation and multiplication of sparse matrices and vector, as well as FFT, making it easy to implement and eligible for the effective parallel implementation. Some typical use cases for the EIT/EEG problems are considered demonstrating high efficiency of the proposed numerical technique.
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Yue, Yubo, Paul Sava, Zhongping Qian, Jidong Yang, and Zhen Zou. "Least-squares Gaussian beam migration in elastic media." GEOPHYSICS 84, no. 4 (July 1, 2019): S329—S340. http://dx.doi.org/10.1190/geo2018-0391.1.
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Gaussian beam migration (GBM) is an effective imaging method that has the ability to image multiple arrivals while preserving the advantages of ray-based methods. We have extended this method to linearized least-squares imaging for elastic waves in isotropic media. We have dynamically transformed the multicomponent data to the principal components of different wave modes using the polarization information available in the beam migration process, and then we use Gaussian beams as wavefield propagator to construct the forward modeling and adjoint migration operators. Based on the constructed operators, we formulate a least-squares migration scheme that is iteratively solved using a preconditioned conjugate gradient method. With this method, we can obtain crosstalk-attenuated multiwave images with better subsurface illumination and higher resolution than those of the conventional elastic Gaussian beam migration. This method also allows us to achieve a good balance between computational cost and imaging accuracy, which are both important requirements for iterative least-squares migrations. Numerical tests on two synthetic data sets demonstrate the validity and effectiveness of our proposed method.
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CHONG, FREDERIC T., and ANANT AGARWAL. "SHARED MEMORY VERSUS MESSAGE PASSING FOR ITERATIVE SOLUTION OF SPARSE, IRREGULAR PROBLEMS." Parallel Processing Letters 09, no. 01 (March 1999): 159–70. http://dx.doi.org/10.1142/s0129626499000177.
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The benefits of hardware support for shared memory versus those for message passing are difficult to evaluate without an in-depth study of real applications on a common platform. We evaluate the communication mechanisms of the MIT Alewife machine, a multiprocessor which provides integrated cache-coherent shared memory, massage passing, and DMA. We perform this evaluation with "best-effort" implementations which solve several sparse, irregular benchmark problems with a preconditioned conjugate gradient sparse matrix solver (ICCG). We find that machines with fast global memory operations do not need message passing or bulk transfer to suport our irregular problems. This is primarily due to three reasons. First, a 5-to-1 ratio between global and local cache misses makes memory copies in bulk communication expensive relati to communication via shared memory. Second, although message passing has synchronization semantics superior to shared memory for data-driven computation, efficient shared memory can overcome this handicap by using global read-modify-writes to change from the traditional owner-computers model to a producer-computes model. Third, bulk transfers can result in high processor idle times in irregular applications.
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Pail, R., and G. Plank. "Comparison of numerical solution strategies for gravity field recovery from GOCE SGG observations implemented on a parallel platform." Advances in Geosciences 1 (June 17, 2003): 39–45. http://dx.doi.org/10.5194/adgeo-1-39-2003.
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Abstract. The recovery of a full set of gravity field parameters from satellite gravity gradiometry (SGG) is a huge numerical and computational task. In practice, parallel computing has to be applied to estimate the more than 90 000 harmonic coefficients parameterizing the Earth’s gravity field up to a maximum spherical harmonic degree of 300. Three independent solution strategies, i.e. two iterative methods (preconditioned conjugate gradient method, semi-analytic approach) and a strict solver (Distributed Non-approximative Adjustment), which are operational on a parallel platform (‘Graz Beowulf Cluster’), are assessed and compared both theoretically and on the basis of a realistic-as-possible numerical simulation, regarding the accuracy of the results, as well as the computational effort. Special concern is given to the correct treatment of the coloured noise characteristics of the gradiometer. The numerical simulations show that there are no significant discrepancies among the solutions of the three methods. The newly proposed Distributed Nonapproximative Adjustment approach, which is the only one of the three methods that solves the inverse problem in a strict sense, also turns out to be a feasible method for practical applications.Key words. Spherical harmonics – satellite gravity gradiometry – GOCE – parallel computing – Beowulf cluster
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Kalateh, Farhoud, and Ali Koosheh. "Finite Element Analysis of Flexible Structure and Cavitating Nonlinear Acoustic Fluid Interaction under Shock Wave Loading." International Journal of Nonlinear Sciences and Numerical Simulation 19, no. 5 (July 26, 2018): 459–73. http://dx.doi.org/10.1515/ijnsns-2016-0135.
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AbstractThis paper describes a numerical model and its finite element implementation that used to compute the cavitation effects on nonlinear acoustic fluid and adjacent flexible structure interaction. The system is composed of two sub-systems, namely, the fluid and the flexible flat plate. A fully coupled approach using iterative implicit partitioned scheme was implemented in the present work which can account for the effects associated whit a mutual interaction. This approach included a compressible nonlinear acoustic fluid Eulerian solver and a Lagrangian solver for the flexible structure both in finite element formulation. A novel implementation of acoustic cavitation was made possible with the introduction of a simplified one-fluid cavitation model. The element-by-element PCG (Preconditioned Conjugate Gradient) solver together with diagonal preconditioning is used to solve the large equation system resulting from the finite element discretization of the governing equation of fluid domain. The capability of three different cavitation model, as the cut-off model, Modified Schmidt model and developed model are compared with each other in the evaluation of plate vibration response. Simulation results are presented on a large size shock tube, in which planar shock waves were impacting in “face on” configuration flat plates mounted at tube's end. Results are presented to demonstrate the capability of proposed solver in simulating cavitating nonlinear acoustic fluid. Obtained results show that impact forces caused impinging shock wave and reloading by cavitating region collapse have a considerable effect on the dynamic response of flexible plate.
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Mora, Peter. "Nonlinear two‐dimensional elastic inversion of multioffset seismic data." GEOPHYSICS 52, no. 9 (September 1987): 1211–28. http://dx.doi.org/10.1190/1.1442384.
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The treatment of multioffset seismic data as an acoustic wave field is becoming increasingly disturbing to many geophysicists who see a multitude of wave phenomena, such as amplitude‐offset variations and shearwave events, which can only be explained by using the more correct elastic wave equation. Not only are such phenomena ignored by acoustic theory, but they are also treated as undesirable noise when they should be used to provide extra information, such as S‐wave velocity, about the subsurface. The problems of using the conventional acoustic wave equation approach can be eliminated via an elastic approach. In this paper, equations have been derived to perform an inversion for P‐wave velocity, S‐wave velocity, and density as well as the P‐wave impedance, S‐wave impedance, and density. These are better resolved than the Lamé parameters. The inversion is based on nonlinear least squares and proceeds by iteratively updating the earth parameters until a good fit is achieved between the observed data and the modeled data corresponding to these earth parameters. The iterations are based on the preconditioned conjugate gradient algorithm. The fundamental requirement of such a least‐squares algorithm is the gradient direction which tells how to update the model parameters. The gradient direction can be derived directly from the wave equation and it may be computed by several wave propagations. Although in principle any scheme could be chosen to perform the wave propagations, the elastic finite‐ difference method is used because it directly simulates the elastic wave equation and can handle complex, and thus realistic, distributions of elastic parameters. This method of inversion is costly since it is similar to an iterative prestack shot‐profile migration. However, it has greater power than any migration since it solves for the P‐wave velocity, S‐wave velocity, and density and can handle very general situations including transmission problems. Three main weaknesses of this technique are that it requires fairly accurate a priori knowledge of the low‐ wavenumber velocity model, it assumes Gaussian model statistics, and it is very computer‐intensive. All these problems seem surmountable. The low‐wavenumber information can be obtained either by a prior tomographic step, by the conventional normal‐moveout method, by a priori knowledge and empirical relationships, or by adding an additional inversion step for low wavenumbers to each iteration. The Gaussian statistics can be altered by preconditioning the gradient direction, perhaps to make the solution blocky in appearance like well logs, or by using large model variances in the inversion to reduce the effect of the Gaussian model constraints. Moreover, with some improvements to the algorithm and more parallel computers, it is hoped the technique will soon become routinely feasible.
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Ma, Huan, Handong Tan, and Yue Guo. "Three-Dimensional Induced Polarization Parallel Inversion Using Nonlinear Conjugate Gradients Method." Mathematical Problems in Engineering 2015 (2015): 1–12. http://dx.doi.org/10.1155/2015/464793.
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Four kinds of array of induced polarization (IP) methods (surface, borehole-surface, surface-borehole, and borehole-borehole) are widely used in resource exploration. However, due to the presence of large amounts of the sources, it will take much time to complete the inversion. In the paper, a new parallel algorithm is described which uses message passing interface (MPI) and graphics processing unit (GPU) to accelerate 3D inversion of these four methods. The forward finite differential equation is solved by ILU0 preconditioner and the conjugate gradient (CG) solver. The inverse problem is solved by nonlinear conjugate gradients (NLCG) iteration which is used to calculate one forward and two “pseudo-forward” modelings and update the direction, space, and model in turn. Because each source is independent in forward and “pseudo-forward” modelings, multiprocess modes are opened by calling MPI library. The iterative matrix solver within CULA is called in each process. Some tables and synthetic data examples illustrate that this parallel inversion algorithm is effective. Furthermore, we demonstrate that the joint inversion of surface and borehole data produces resistivity and chargeability results are superior to those obtained from inversions of individual surface data.
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Jiang, Zhong Hua, Ning Xu, and Chun Xiang Wu. "Thermal Floorplan Base on Conjugate Gradient Solver in HotSpot." Applied Mechanics and Materials 608-609 (October 2014): 908–12. http://dx.doi.org/10.4028/www.scientific.net/amm.608-609.908.
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In this paper, we introduce an effective iterative method to solve the thermal linear system in HotSpot thermal floorplan, the iterative Conjugate Gradient Method is suitable to solve the traditional sparse matrix linear equations. We define a class of dummy sparse linear systems in iterative thermal floorplan algorithm, the iterative methods for linear system can be extended to apply to other iterative framework algorithm. We apply the conjugate gradient method to solve the thermal model in floorplan of VLSI physical design. The experiments' result shows that thermal floorplan using Conjugate gradient method is effective. The running time of our incremental conjugate gradient thermal solver with Jocabi Precondition is approximate 0.59 comparing with LU decomposition method.
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Aubry, R., F. Mut, S. Dey, and R. Löhner. "Deflated preconditioned conjugate gradient solvers for linear elasticity." International Journal for Numerical Methods in Engineering 88, no. 11 (April 28, 2011): 1112–27. http://dx.doi.org/10.1002/nme.3209.
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Riyanti, C. D., Y. A. Erlangga, R. E. Plessix, W. A. Mulder, C. Vuik, and C. Oosterlee. "A new iterative solver for the time-harmonic wave equation." GEOPHYSICS 71, no. 5 (September 2006): E57—E63. http://dx.doi.org/10.1190/1.2231109.
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The time-harmonic wave equation, also known as the Helmholtz equation, is obtained if the constant-density acoustic wave equation is transformed from the time domain to the frequency domain. Its discretization results in a large, sparse, linear system of equations. In two dimensions, this system can be solved efficiently by a direct method. In three dimensions, direct methods cannot be used for problems of practical sizes because the computational time and the amount of memory required become too large. Iterative methods are an alternative. These methods are often based on a conjugate gradient iterative scheme with a preconditioner that accelerates its convergence. The iterative solution of the time-harmonic wave equation has long been a notoriously difficult problem in numerical analysis. Recently, a new preconditioner based on a strongly damped wave equation has heralded a breakthrough. The solution of the linear system associated with the preconditioner is approximated by another iterative method, the multigrid method. The multigrid method fails for the original wave equation but performs well on the damped version. The performance of the new iterative solver is investigated on a number of 2D test problems. The results suggest that the number of required iterations increases linearly with frequency, even for a strongly heterogeneous model where earlier iterative schemes fail to converge. Complexity analysis shows that the new iterative solver is still slower than a time-domain solver to generate a full time series. We compare the time-domain numeric results obtained using the new iterative solver with those using the direct solver and conclude that they agree very well quantitatively. The new iterative solver can be applied straightforwardly to 3D problems.
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Pan, V., A. L. Zheng, O. Dias, and X. H. Huang. "A fast, preconditioned conjugate gradient Toeplitz and Toeplitz-like solvers." Computers & Mathematics with Applications 30, no. 8 (October 1995): 57–63. http://dx.doi.org/10.1016/0898-1221(95)00137-n.
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Aubry, Romain, Fernando Mut, Rainald Löhner, and Juan R. Cebral. "Deflated preconditioned conjugate gradient solvers for the Pressure–Poisson equation." Journal of Computational Physics 227, no. 24 (December 2008): 10196–208. http://dx.doi.org/10.1016/j.jcp.2008.08.025.
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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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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 (October 25, 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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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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Löhner, Rainald, Fernando Mut, Juan Raul Cebral, Romain Aubry, and Guillaume Houzeaux. "Deflated preconditioned conjugate gradient solvers for the pressure-Poisson equation: Extensions and improvements." International Journal for Numerical Methods in Engineering 87, no. 1-5 (June 22, 2010): 2–14. http://dx.doi.org/10.1002/nme.2932.
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Zhang, Lei, Guoxin Zhang, Lixiang Wang, Zhaosong Ma, and Shihai Li. "A Comparative Study on Different Parallel Solvers for Nonlinear Analysis of Complex Structures." Mathematical Problems in Engineering 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/764237.
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The parallelization of 2D/3D software SAPTIS is discussed for nonlinear analysis of complex structures. A comparative study is made on different parallel solvers. The numerical models are presented, including hydration models, water cooling models, modulus models, creep model, and autogenous deformation models. A finite element simulation is made for the whole process of excavation and pouring of dams using these models. The numerical results show a good agreement with the measured ones. To achieve a better computing efficiency, four parallel solvers utilizing parallelization techniques are employed: (1) a parallel preconditioned conjugate gradient (PCG) solver based on OpenMP, (2) a parallel preconditioned Krylov subspace solver based on MPI, (3) a parallel sparse equation solver based on OpenMP, and (4) a parallel GPU equation solver. The parallel solvers run either in a shared memory environment OpenMP or in a distributed memory environment MPI. A comparative study on these parallel solvers is made, and the results show that the parallelization makes SAPTIS more efficient, powerful, and adaptable.
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Hou, Junxia, Quanyi Lv, and Manyu Xiao. "A Parallel Preconditioned Modified Conjugate Gradient Method for Large Sylvester Matrix Equation." Mathematical Problems in Engineering 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/598716.
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Computational effort of solving large-scale Sylvester equationsAX+XB+F=Ois frequently hindered in dealing with many complex control problems. In this work, a parallel preconditioned algorithm for solving it is proposed based on combination of a parameter iterative preconditioned method and modified form of conjugate gradient (MCG) method. Furthermore, Schur’s inequality and modified conjugate gradient method are employed to overcome the involved difficulties such as determination of parameter and calculation of inverse matrix. Several numerical results finally show that high performance of proposed parallel algorithm is obtained both in convergent rate and in parallel efficiency.
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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 (March 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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Gazzola, Silvia, and Paolo Novati. "Some transpose-free CG-like solvers for nonsymmetric ill-posed problems." Journal of Numerical Mathematics 28, no. 1 (March 26, 2020): 15–32. http://dx.doi.org/10.1515/jnma-2018-0107.
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AbstractThis paper introduces and analyzes an original class of Krylov subspace methods that provide an efficient alternative to many well-known conjugate-gradient-like (CG-like) Krylov solvers for square nonsymmetric linear systems arising from discretizations of inverse ill-posed problems. The main idea underlying the new methods is to consider some rank-deficient approximations of the transpose of the system matrix, obtained by running the (transpose-free) Arnoldi algorithm, and then apply some Krylov solvers to a formally right-preconditioned system of equations. Theoretical insight is given, and many numerical tests show that the new solvers outperform classical Arnoldi-based or CG-like methods in a variety of situations.
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BASERMANN, ACHIM. "CONJUGATE GRADIENTS PARALLELIZED ON THE HYPERCUBE." International Journal of Modern Physics C 04, no. 06 (December 1993): 1295–306. http://dx.doi.org/10.1142/s0129183193001014.
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For the solution of discretized ordinary or partial differential equations it is necessary to solve systems of equations with coefficient matrices of different sparsity pattern, depending on the discretization method; using the finite element method (FE) results in largely unstructured systems of equations. A frequently used iterative solver for systems of equations is the method of conjugate gradients (CG) with different preconditioners. On a multiprocessor system with distributed memory, in particular the data distribution and the communication scheme depending on the used data struture are of greatest importance for the efficient execution of this method. Here, a data distribution and a communication scheme are presented which are based on the analysis of the column indices of the non-zero matrix elements. The performance of the developed parallel CG-method was measured on the distributed-memory-system INTEL iPSC/860 of the Research Centre Jülich with systems of equations from FE-models. The parallel CG-algorithm has been shown to be well suited for both regular and irregular discretization meshes, i.e. for coefficient matrices of very different sparsity pattern.
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Karatson, Janos, and Tamas Kurics. "A PRECONDITIONED ITERATIVE SOLUTION SCHEME FOR NONLINEAR PARABOLIC SYSTEMS ARISING IN AIR POLLUTION MODELING." Mathematical Modelling and Analysis 18, no. 5 (December 1, 2013): 641–53. http://dx.doi.org/10.3846/13926292.2013.868841.
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A preconditioned iterative solution method is presented for nonlinear parabolic transport systems. The ingredients are implicit Euler discretization in time and finite element discretization in space, then an outer-inner (outer damped inexact Newton method with inner preconditioned conjugate gradient) iteration, further, as a main part, preconditioning via an l-tuple of independent elliptic operators. Numerical results show that the suggested method works properly for a test problem in air pollution modeling.
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Bartels, Simon, Jon Cockayne, Ilse C. F. Ipsen, and Philipp Hennig. "Probabilistic linear solvers: a unifying view." Statistics and Computing 29, no. 6 (September 10, 2019): 1249–63. http://dx.doi.org/10.1007/s11222-019-09897-7.
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Abstract Several recent works have developed a new, probabilistic interpretation for numerical algorithms solving linear systems in which the solution is inferred in a Bayesian framework, either directly or by inferring the unknown action of the matrix inverse. These approaches have typically focused on replicating the behaviour of the conjugate gradient method as a prototypical iterative method. In this work, surprisingly general conditions for equivalence of these disparate methods are presented. We also describe connections between probabilistic linear solvers and projection methods for linear systems, providing a probabilistic interpretation of a far more general class of iterative methods. In particular, this provides such an interpretation of the generalised minimum residual method. A probabilistic view of preconditioning is also introduced. These developments unify the literature on probabilistic linear solvers and provide foundational connections to the literature on iterative solvers for linear systems.
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Meyer, A. "A parallel preconditioned conjugate gradient method using domain decomposition and inexact solvers on each subdomain." Computing 45, no. 3 (September 1990): 217–34. http://dx.doi.org/10.1007/bf02250634.
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Ocłoń, Paweł, Stanisław Łopata, and Marzena Nowak. "Comparative study of conjugate gradient algorithms performance on the example of steady-state axisymmetric heat transfer problem." Archives of Thermodynamics 34, no. 3 (September 1, 2013): 15–44. http://dx.doi.org/10.2478/aoter-2013-0013.
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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.
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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 (March 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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Huang, Yin, Rami Nammour, and William Symes. "Flexibly preconditioned extended least-squares migration in shot-record domain." GEOPHYSICS 81, no. 5 (September 2016): S299—S315. http://dx.doi.org/10.1190/geo2016-0023.1.
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We have developed a method for accelerating the convergence of iterative least-squares migration. The algorithm uses a pseudodifferential scaling (dip and spatially varying filter) preconditioner together with a variant of conjugate gradient (CG) iteration with iterate-dependent (flexible) preconditioning. The migration is formulated without the image stack, thus producing a shot-dependent image volume that retains offset information useful for velocity updating and amplitude variation with offset analysis. Numerical experiments indicate that flexible preconditioning with pseudodifferential scaling not only attains considerably smaller data misfit and gradient error for a given computational effort, but also produces higher resolution image volumes with more balanced amplitude and fewer artifacts than is achieved with a nonpreconditioned CG method.
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Li, Chuang, Zhaoqi Gao, Jinghuai Gao, Feipeng Li, and Tao Yang. "Angle-domain common-image gathers from plane-wave least-squares reverse time migration." GEOPHYSICS 86, no. 5 (August 3, 2021): S311—S324. http://dx.doi.org/10.1190/geo2020-0511.1.
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Angle-domain common-image gathers (ADCIGs) that can be used for migration velocity analysis and amplitude-versus-angle analysis are important for seismic exploration. However, because of the limited acquisition geometry and seismic frequency band, ADCIGs extracted by reverse time migration (RTM) suffer from illumination gaps, migration artifacts, and low resolution. We have developed a reflection angle-domain pseudoextended plane-wave least-squares RTM method for obtaining high-quality ADCIGs. We build the mapping relations between the ADCIGs and the plane-wave sections using an angle-domain pseudoextended Born modeling operator and an adjoint operator, based on which we formulate the extraction of ADCIGs as an inverse problem. The inverse problem is iteratively solved by a preconditioned stochastic conjugate-gradient method, allowing for reduction in computational cost by migrating only a subset instead of the whole data set and improving the image quality thanks to preconditioners. Numerical tests on synthetic and field data verify that our method can compensate for illumination gaps, suppress migration artifacts, and improve resolution of the ADCIGs and the stacked images. Therefore, compared to RTM, our method provides a more reliable input for migration velocity analysis and amplitude-versus-angle analysis. Moreover, it also provides much better stacked images for seismic interpretation.
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Ke, Yi-Fen, and Chang-Feng Ma. "A preconditioned nested splitting conjugate gradient iterative method for the large sparse generalized Sylvester equation." Computers & Mathematics with Applications 68, no. 10 (November 2014): 1409–20. http://dx.doi.org/10.1016/j.camwa.2014.09.009.
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Lin, Zhipeng, Wenjing Yang, Houcun Zhou, Xinhai Xu, Liaoyuan Sun, Yongjun Zhang, and Yuhua Tang. "Communication Optimization for Multiphase Flow Solver in the Library of OpenFOAM." Water 10, no. 10 (October 16, 2018): 1461. http://dx.doi.org/10.3390/w10101461.
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Multiphase flow solvers are widely-used applications in OpenFOAM, whose scalability suffers from the costly communication overhead. Therefore, we establish communication-optimized multiphase flow solvers in OpenFOAM. In this paper, we first deliver a scalability bottleneck test on the typical multiphase flow case damBreak and reveal that the Message Passing Interface (MPI) communication in a Multidimensional Universal Limiter for Explicit Solution (MULES) and a Preconditioned Conjugate Gradient (PCG) algorithm is the short slab of multiphase flow solvers. Furthermore, an analysis of the communication behavior is carried out. We find that the redundant communication in MULES and the global synchronization in PCG are the performance limiting factors. Based on the analysis, we propose our communication optimization algorithm. For MULES, we remove the redundant communication and obtain optMULES. For PCG, we import several intermediate variables and rearrange PCG to reduce the global communication. We also overlap the computation of matrix-vector multiply and vector update with the non-blocking computation. The resulting algorithms are respectively referred to as OFPiPePCG and OFRePiPePCG. Extensive experiments show that our proposed method could dramatically increase the parallel scalability and solving speed of multiphase flow solvers in OpenFOAM approximately without the loss of accuracy.
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Cayrols, Sébastien, Iain S. Duff, and Florent Lopez. "Parallelization of the solve phase in a task-based Cholesky solver using a sequential task flow model." International Journal of High Performance Computing Applications 34, no. 3 (November 29, 2019): 340–56. http://dx.doi.org/10.1177/1094342019888567.
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We describe the parallelization of the solve phase in the sparse Cholesky solver SpLLT when using a sequential task flow model. In the context of direct methods, the solution of a sparse linear system is achieved through three main phases: the analyse, the factorization and the solve phases. In the last two phases, which involve numerical computation, the factorization corresponds to the most computationally costly phase, and it is therefore crucial to parallelize this phase in order to reduce the time-to-solution on modern architectures. As a consequence, the solve phase is often not as optimized as the factorization in state-of-the-art solvers, and opportunities for parallelism are often not exploited in this phase. However, in some applications, the time spent in the solve phase is comparable to or even greater than the time for the factorization, and the user could dramatically benefit from a faster solve routine. This is the case, for example, for a conjugate gradient (CG) solver using a block Jacobi preconditioner. The diagonal blocks are factorized once only, but their factors are used to solve subsystems at each CG iteration. In this study, we design and implement a parallel version of a task-based solve routine for an OpenMP version of the SpLLT solver. We show that we can obtain good scalability on a multicore architecture enabling a dramatic reduction of the overall time-to-solution in some applications.
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OKAMOTO, NORIKO, REIJI TOMIKU, TORU OTSURU, and YOSUKE YASUDA. "NUMERICAL ANALYSIS OF LARGE-SCALE SOUND FIELDS USING ITERATIVE METHODS PART II: APPLICATION OF KRYLOV SUBSPACE METHODS TO FINITE ELEMENT ANALYSIS." Journal of Computational Acoustics 15, no. 04 (December 2007): 473–93. http://dx.doi.org/10.1142/s0218396x07003512.
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Krylov subspace iterative solvers are applied to large-scale finite element sound-field analyses of architectural rooms. First, convergence behaviors are compared among four iterative solvers. Results show that the Conjugate Orthogonal Conjugate Gradient (COCG) method offers the best characteristics for finite-element (FE) analysis from the viewpoint of robustness of convergence and computation time. Two investigations to reduce the computation time of the COCG method were carried out. Results show the following. (1) The mean residual of sound pressure levels between COCG method and direct method is less than 0.1 dB if the convergence criterion is set to 10-4 and the maximum residual of those between COCG method and direct method is less than 0.2 dB if the convergence criterion is set to 10-6. (2) The computation time of the COCG method with diagonal preconditioning is about 30% shorter than that of COCG method without preconditioning. Finally, sound pressure level distributions obtained using the authors' FEM are compared to those obtained using fast multipole BEM (FMBEM) and measurements.
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Axelsson, Owe. "Milestones in the Development of Iterative Solution Methods." Journal of Electrical and Computer Engineering 2010 (2010): 1–33. http://dx.doi.org/10.1155/2010/972794.
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Iterative solution methods to solve linear systems of equations were originally formulated as basic iteration methods of defect-correction type, commonly referred to as Richardson's iteration method. These methods developed further into various versions of splitting methods, including the successive overrelaxation (SOR) method. Later, immensely important developments included convergence acceleration methods, such as the Chebyshev and conjugate gradient iteration methods and preconditioning methods of various forms. A major strive has been to find methods with a total computational complexity of optimal order, that is, proportional to the degrees of freedom involved in the equation. Methods that have turned out to have been particularly important for the further developments of linear equation solvers are surveyed. Some of them are presented in greater detail.
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Ju, S. H. "OpenMp solvers for parallel finite element and meshless analyses." Engineering Computations 31, no. 1 (February 25, 2014): 2–17. http://dx.doi.org/10.1108/ec-02-2012-0032.
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Purpose – This paper develops C++ and Fortran-90 solvers to establish parallel solution procedures in a finite element or meshless analysis program using shared memory computers. The paper aims to discuss these issues. Design/methodology/approach – The stiffness matrix can be symmetrical or unsymmetrical, and the solution schemes include sky-line Cholesky and parallel preconditioned conjugate gradient-like methods. Findings – By using the features of C++ or Fortran-90, the stiffness matrix and its auxiliary arrays can be encapsulated into a class or module as private arrays. This class or module will handle how to allocate, renumber, assemble, parallelize and solve these complicated arrays automatically. Practical implications – The source codes can be obtained online at http//myweb.ncku.edu.tw/∼juju. The major advantage of the scheme is that it is simple and systematic, so an efficient parallel finite element or meshless program can be established easily. Originality/value – With the minimum requirement of computer memory, an object-oriented C++ class and a Fortran-90 module were established to allocate, renumber, assemble, parallel, and solve the global stiffness matrix, so that the programmer does not need to handle them directly.
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WU, S. C., H. O. ZHANG, C. ZHENG, and J. H. ZHANG. "A HIGH PERFORMANCE LARGE SPARSE SYMMETRIC SOLVER FOR THE MESHFREE GALERKIN METHOD." International Journal of Computational Methods 05, no. 04 (December 2008): 533–50. http://dx.doi.org/10.1142/s0219876208001613.
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One main disadvantage of meshfree methods is that their memory requirement and computational cost are much higher than those of the usual finite element method (FEM). This paper presents an efficient and reliable solver for the large sparse symmetric positive definite (SPD) system resulting from the element-free Galerkin (EFG) approach. A compact mathematical model of heat transfer problems is first established using the EFG procedure. Based on the widely used Successive Over-Relaxation–Preconditioned Conjugate Gradient (SSOR–PCG) scheme, a novel solver named FastPCG is then proposed for solving the SPD linear system. To decrease the computational time in each iteration step, a new algorithm for realizing multiplication of the global stiffness matrix by a vector is presented for this solver. The global matrix and load vector are changed in accordance with a special rule and, in this way, a large account of calculation is avoided on the premise of not decreasing the solution's accuracy. In addition, a double data structure is designed to tackle frequent and unexpected operations of adding or removing nodes in problems of dynamic adaptive or moving high-gradient field analysis. An information matrix is also built to avoid drastic transformation of the coefficient matrix caused by the initial-boundary values. Numerical results show that the memory requirement of the FastPCG solver is only one-third of that of the well-developed AGGJE solver, and the computational cost is comparable with the traditional method with the increas of solution scale and order.
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Greengard, Leslie, Denis Gueyffier, Per-Gunnar Martinsson, and Vladimir Rokhlin. "Fast direct solvers for integral equations in complex three-dimensional domains." Acta Numerica 18 (May 2009): 243–75. http://dx.doi.org/10.1017/s0962492906410011.
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Methods for the solution of boundary integral equations have changed significantly during the last two decades. This is due, in part, to improvements in computer hardware, but more importantly, to the development of fast algorithms which scale linearly or nearly linearly with the number of degrees of freedom required. These methods are typically iterative, based on coupling fast matrix-vector multiplication routines with conjugate-gradient-type schemes. Here, we discuss methods that are currently under development for the fast, direct solution of boundary integral equations in three dimensions. After reviewing the mathematical foundations of such schemes, we illustrate their performance with some numerical examples, and discuss the potential impact of the overall approach in a variety of settings.
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Kanayama, H. "An Industrial Application of Thermal Convection Analysis." International Journal of Computational Methods 13, no. 02 (March 2016): 1640005. http://dx.doi.org/10.1142/s0219876216400053.
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A coupling analysis of thermal convection problems is performed in this work. By approximating the material derivative along the trajectory of fluid particle, the characteristic curve (CC) method can be considered. The most attractive advantage of this method is the symmetry of the linear system, which enables some classic symmetric linear iterative solvers, like the conjugate gradient (CG) method or the minimal residual (MINRES) method, to be used to solve the interface problem of the domain decomposition system. An application to industrial problems is demonstrated to show the effectiveness of our approach.