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Journal articles on the topic 'Geometric preconditioner'

Author: Grafiati

Published: 23 November 2024

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1

Sun, Qingtao, Runren Zhang, Ke Chen, Naixing Feng, and Yunyun Hu. "Anisotropic modeling with geometric multigrid preconditioned finite-element method." GEOPHYSICS 87, no. 3 (February 24, 2022): A33—A36. http://dx.doi.org/10.1190/geo2021-0592.1.

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Formation anisotropy in complicated geophysical environments can have a significant impact on data interpretation of electromagnetic surveys. To facilitate full 3D modeling of arbitrary anisotropy, we have adopted an [Formula: see text]-version geometric multigrid preconditioned finite-element method (FEM) based on vector basis functions. By using a structured mesh, instead of an unstructured one, our method can conveniently construct the restriction and prolongation operators for multigrid implementation, and then recursively coarsen the grid with the F-cycle coarsening scheme. The geometric multigrid method is used as a preconditioner for the biconjugate-gradient stabilized method to efficiently solve the linear system resulting from the FEM. Our method avoids the need of interpolation for arbitrary anisotropy modeling as in Yee’s grid-based finite-difference method, and it is also more capable of large-scale modeling with respect to the [Formula: see text]-version geometric multigrid preconditioned finite-element method. A numerical example in geophysical well logging is included to demonstrate its numerical performance. Our [Formula: see text]-version geometric multigrid preconditioned FEM is expected to help formation anisotropy characterization with electromagnetic surveys in complicated geophysical environments.

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Cots, Olivier, Rémy Dutto, Sophie Jan, and Serge Laporte. "Geometric preconditioner for indirect shooting and application to hybrid vehicle." IFAC-PapersOnLine 58, no. 21 (2024): 43–48. http://dx.doi.org/10.1016/j.ifacol.2024.10.140.

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Pan, Guangdong, and Aria Abubakar. "Iterative solution of 3D acoustic wave equation with perfectly matched layer boundary condition and multigrid preconditioner." GEOPHYSICS 78, no. 5 (September 1, 2013): T133—T140. http://dx.doi.org/10.1190/geo2012-0287.1.

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We tested a biconjugate gradient stabilized (BiCGSTAB) solver using a multigrid-based preconditioner for solving the acoustic wave (Helmholtz) equation in the frequency domain. The perfectly matched layer (PML) method was used as the radiation boundary condition (RBC). The equation was discretized using either a second- or fourth-order finite-difference (FD) scheme. The convergence of an iterative solver depended strongly on the RBC used because the spectrum of the discretized equation also depends on it. We used a geometric multigrid approach to construct a preconditioner for our FD frequency-domain (FDFD) forward solver equipped with the PML boundary condition. For efficiency, this preconditioner was only constructed using a second-order FD scheme with negligible attenuation inside the PML domain. The preconditioner was used for accelerating the convergence rate of the FDFD forward solver for cases when the discretization grids were oversampled (i.e., when the number of discretization points per minimum wavelength was greater than 10). The number of multigrid levels was also chosen adaptively depending on the number of discretization grids. We found that the multigrid preconditioner can speed up the total computational time of the BiCGSTAB solver for oversampled cases or at low frequencies. We also observed that the BiCGSTAB solver using an accurate PML boundary condition converged for realistic SEG benchmark models at high frequencies.

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Cai, Mingchao, Andy Nonaka, John B. Bell, Boyce E. Griffith, and Aleksandar Donev. "Efficient Variable-Coefficient Finite-Volume Stokes Solvers." Communications in Computational Physics 16, no. 5 (November 2014): 1263–97. http://dx.doi.org/10.4208/cicp.070114.170614a.

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AbstractWe investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variable-coefficient Stokes equations on a uniform staggered grid. Building on the success of using the classical projection method as a preconditioner for the coupled velocity pressure system [B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 7565-7595], as well; established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems. We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems. Contrary to traditional wisdom, we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow, even for steady flow and in the presence of large density and viscosity contrasts. Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size, making them suitable for large-scale problems. Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.

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de Prenter, F., C. V. Verhoosel, E. H. van Brummelen, J. A. Evans, C. Messe, J. Benzaken, and K. Maute. "Multigrid solvers for immersed finite element methods and immersed isogeometric analysis." Computational Mechanics 65, no. 3 (November 26, 2019): 807–38. http://dx.doi.org/10.1007/s00466-019-01796-y.

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AbstractIll-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral properties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines.

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Yuan, Yu-Xin, A.-Man Li, Ting Hu, and Hong Liu. "An anisotropic multilevel preconditioner for solving the Helmholtz equation with unequal directional sampling intervals." GEOPHYSICS 85, no. 6 (October 13, 2020): T293—T300. http://dx.doi.org/10.1190/geo2019-0330.1.

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An efficient finite-difference method for solving the isotropic Helmholtz equation relies on a discretization scheme and an appropriate solver. Accordingly, we have adopted an average-derivative optimal scheme that has two advantages: (1) it can be applied to unequal directional sampling intervals and (2) it requires less than four grid points of sampling per wavelength. Direct methods are not of interest for industry-sized problems due to the high memory requirements; Krylov subspace methods such as the biconjugate gradient stabilized method and the flexible generalized minimal residual method that combine a multigrid-based preconditioner are better alternatives. However, standard geometric multigrid algorithms fail to converge when there exist unequal directional sampling intervals; this is called anisotropic grids in terms of the multigrid. We first review our previous research on 2D anisotropic grids: the semicoarsening strategy, line-wise relaxation operator, and matrix-dependent interpolation were used to modify the standard V-cycle multigrid algorithms, resulting in convergence. Although directly extending to the 3D case by substituting line relaxation for plane relaxation deteriorates the convergence rate considerably, we then find that a multilevel generalized minimal residual preconditioner-combined semicoarsening strategy is more suitable for anisotropic grids and the convergence rate is faster in the 2D and 3D cases. The results of the numerical experiments indicate that the standard geometric multigrid does not work for anisotropic grids, whereas our method demonstrates a faster convergence rate than the previous method.

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FENG, QUANDONG, JINGFANG HUANG, NINGMING NIE, ZAIJIU SHANG, and YIFA TANG. "IMPLEMENTING ARBITRARILY HIGH-ORDER SYMPLECTIC METHODS VIA KRYLOV DEFERRED CORRECTION TECHNIQUE." International Journal of Modeling, Simulation, and Scientific Computing 01, no. 02 (June 2010): 277–301. http://dx.doi.org/10.1142/s1793962310000171.

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In this paper, an efficient numerical procedure is presented to implement the Gaussian Runge–Kutta (GRK) methods (also called Gauss methods). The GRK technique first discretizes each marching step of the initial value problem using collocation formulations based on Gaussian quadrature. As is well known, it preserves the geometric structures of Hamiltonian systems. Existing analysis shows that the GRK discretization with s nodes is of order 2s, A-stable, B-stable, symplectic and symmetric, and hence "optimal" for solving initial value problems of general ordinary differential equations (ODEs). However, as the unknowns at different collocation points are coupled in the discretized system, direct solution of the resulting algebraic equations is in general inefficient. Instead, we use the Krylov deferred correction (KDC) method in which the spectral deferred correction (SDC) scheme is applied as a preconditioner to decouple the original system, and the resulting preconditioned nonlinear system is solved efficiently using Newton–Krylov schemes such as Newton–GMRES method. The KDC accelerated GRK methods have been applied to several Hamiltonian systems and preliminary numerical results are presented to show the accuracy, stability, and efficiency features of these methods for different accuracy requirements in short- and long-time simulations.

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8

Martynenko, S. I. "Potentialities of the Robust Multigrid Technique." Computational Methods in Applied Mathematics 10, no. 1 (2010): 87–94. http://dx.doi.org/10.2478/cmam-2010-0004.

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AbstractThe present paper discusses the parallelization of the robust multigrid technique (RMT) and the possible way of applying this to unstructured grids. As opposed to the classical multigrid methods, the RMT is a trivial method of parallelization on coarse grids independent of the smoothing iterations. Estimates of the minimum speed-up and parallelism efficiency are given. An almost perfect load balance is demonstrated in a 3D illustrative test. To overcome the geometric nature of the technique, the RMT is used as a preconditioner in solving PDEs on unstructured grids. The procedure of auxiliary structured grids generation is considered in details.

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Nammour, Rami, and William W. Symes. "Multiparameter Inversion: Cramer's Rule for Pseudodifferential Operators." International Journal of Geophysics 2011 (2011): 1–12. http://dx.doi.org/10.1155/2011/780291.

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Linearized multiparameter inversion is a model-driven variant of amplitude-versus-offset analysis, which seeks to separately account for the influences of several model parameters on the seismic response. Previous approaches to this class of problems have included geometric optics-based (Kirchhoff, GRT) inversion and iterative methods suitable for large linear systems. In this paper, we suggest an approach based on the mathematical nature of the normal operator of linearized inversion—it is a scaling operator in phase space—and on a very old idea from linear algebra, namely, Cramer's rule for computing the inverse of a matrix. The approximate solution of the linearized multiparameter problem so produced involves no ray theory computations. It may be sufficiently accurate for some purposes; for others, it can serve as a preconditioner to enhance the convergence of standard iterative methods.

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Chen, Shu-Wen, Feng Lu, and Yao Ma. "Fitting Green’s Function FFT Acceleration Applied to Anisotropic Dielectric Scattering Problems." International Journal of Antennas and Propagation 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/123739.

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A volume integral equation based fast algorithm using the Fast Fourier Transform of fitting Green’s function (FG-FFT) is proposed in this paper for analysis of electromagnetic scattering from 3D anisotropic dielectric objects. For the anisotropic VIE model, geometric discretization is still implemented by tetrahedron cells and the Schaubert-Wilton-Glisson (SWG) basis functions are also used to represent the electric flux density vectors. Compared with other Fast Fourier Transform based fast methods, using fitting Green’s function technique has higher accuracy and can be applied to a relatively coarse grid, so the Fast Fourier Transform of fitting Green’s function is selected to accelerate anisotropic dielectric model of volume integral equation for solving electromagnetic scattering problems. Besides, the near-field matrix elements in this method are used to construct preconditioner, which has been proved to be effective. At last, several representative numerical experiments proved the validity and efficiency of the proposed method.

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A. Gravvanis, George, and Christos K. Filelis-Papadopoulos. "On the multigrid cycle strategy with approximate inverse smoothing." Engineering Computations 31, no. 1 (February 25, 2014): 110–22. http://dx.doi.org/10.1108/ec-03-2012-0055.

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Purpose – The purpose of this paper is to propose multigrid methods in conjunction with explicit approximate inverses with various cycles strategies and comparison with the other smoothers. Design/methodology/approach – The main motive for the derivation of the various multigrid schemes lies in the efficiency of the multigrid methods as well as the explicit approximate inverses. The combination of the various multigrid cycles with the explicit approximate inverses as smoothers in conjunction with the dynamic over/under relaxation (DOUR) algorithm results in efficient schemes for solving large sparse linear systems derived from the discretization of partial differential equations (PDE). Findings – Application of the proposed multigrid methods on two-dimensional boundary value problems is discussed and numerical results are given concerning the convergence behavior and the convergence factors. The results are comparatively better than the V-cycle multigrid schemes presented in a recent report (Filelis-Papadopoulos and Gravvanis). Research limitations/implications – The limitations of the proposed scheme lie in the fact that the explicit finite difference approximate inverse matrix used as smoother in the multigrid method is a preconditioner for specific sparsity pattern. Further research is carried out in order to derive a generic explicit approximate inverse for any type of sparsity pattern. Originality/value – A novel smoother for the geometric multigrid method is proposed, based on optimized banded approximate inverse matrix preconditioner, the Richardson method in conjunction with the DOUR scheme, for solving large sparse linear systems derived from finite difference discretization of PDEs. Moreover, the applicability and convergence behavior of the proposed scheme is examined based on various cycles and comparative results are given against the damped Jacobi smoother.

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Jakub, Fabian, and Bernhard Mayer. "3-D radiative transfer in large-eddy simulations – experiences coupling the TenStream solver to the UCLA-LES." Geoscientific Model Development 9, no. 4 (April 15, 2016): 1413–22. http://dx.doi.org/10.5194/gmd-9-1413-2016.

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Abstract. The recently developed 3-D TenStream radiative transfer solver was integrated into the University of California, Los Angeles large-eddy simulation (UCLA-LES) cloud-resolving model. This work documents the overall performance of the TenStream solver as well as the technical challenges of migrating from 1-D schemes to 3-D schemes. In particular the employed Monte Carlo spectral integration needed to be reexamined in conjunction with 3-D radiative transfer. Despite the fact that the spectral sampling has to be performed uniformly over the whole domain, we find that the Monte Carlo spectral integration remains valid. To understand the performance characteristics of the coupled TenStream solver, we conducted weak as well as strong-scaling experiments. In this context, we investigate two matrix preconditioner: geometric algebraic multigrid preconditioning (GAMG) and block Jacobi incomplete LU (ILU) factorization and find that algebraic multigrid preconditioning performs well for complex scenes and highly parallelized simulations. The TenStream solver is tested for up to 4096 cores and shows a parallel scaling efficiency of 80–90 % on various supercomputers. Compared to the widely employed 1-D delta-Eddington two-stream solver, the computational costs for the radiative transfer solver alone increases by a factor of 5–10.

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Turek, Vojtěch. "Improving Performance of Simplified Computational Fluid Dynamics Models via Symmetric Successive Overrelaxation." Energies 12, no. 12 (June 25, 2019): 2438. http://dx.doi.org/10.3390/en12122438.

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The ability to model fluid flow and heat transfer in process equipment (e.g., shell-and-tube heat exchangers) is often critical. What is more, many different geometric variants may need to be evaluated during the design process. Although this can be done using detailed computational fluid dynamics (CFD) models, the time needed to evaluate a single variant can easily reach tens of hours on powerful computing hardware. Simplified CFD models providing solutions in much shorter time frames may, therefore, be employed instead. Still, even these models can prove to be too slow or not robust enough when used in optimization algorithms. Effort is thus devoted to further improving their performance by applying the symmetric successive overrelaxation (SSOR) preconditioning technique in which, in contrast to, e.g., incomplete lower–upper factorization (ILU), the respective preconditioning matrix can always be constructed. Because the efficacy of SSOR is influenced by the selection of forward and backward relaxation factors, whose direct calculation is prohibitively expensive, their combinations are experimentally investigated using several representative meshes. Performance is then compared in terms of the single-core computational time needed to reach a converged steady-state solution, and recommendations are made regarding relaxation factor combinations generally suitable for the discussed purpose. It is shown that SSOR can be used as a suitable fallback preconditioner for the fast-performing, but numerically sensitive, incomplete lower–upper factorization.

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Roberts, Nathan V. "Camellia: A Rapid Development Framework for Finite Element Solvers." Computational Methods in Applied Mathematics 19, no. 3 (July 1, 2019): 581–602. http://dx.doi.org/10.1515/cmam-2018-0218.

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AbstractThe discontinuous Petrov–Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan guarantees the optimality of the finite element solution in a user-controllable energy norm, and provides several features supporting adaptive schemes. The approach provides stability automatically; there is no need for carefully derived numerical fluxes (as in DG schemes) or for mesh-dependent stabilization terms (as in stabilized methods). In this paper, we focus on features of Camellia that facilitate implementation of new DPG formulations; chief among these is a rich set of features in support of symbolic manipulation, which allow, e.g., bilinear formulations in the code to appear much as they would on paper. Many of these features are general in the sense that they can also be used in the implementation of other finite element formulations. In fact, because DPG’s requirements are essentially a superset of those of other finite element methods, Camellia provides built-in support for most common methods. We believe, however, that the combination of an essentially “hands-free” finite element methodology as found in DPG with the rapid development features of Camellia are particularly winsome, so we focus on use cases in this class. In addition to the symbolic manipulation features mentioned above, Camellia offers support for one-irregular adaptive meshes in 1D, 2D, 3D, and space-time. It provides a geometric multigrid preconditioner particularly suited for DPG problems, and supports distributed parallel execution using MPI. For its load balancing and distributed data structures, Camellia relies on packages from the Trilinos project, which simplifies interfacing with other computational science packages. Camellia also allows loading of standard mesh formats through an interface with the MOAB package. Camellia includes support for static condensation to eliminate element-interior degrees of freedom locally, usually resulting in substantial reduction of the cost of the global problem. We include a discussion of the variational formulations built into Camellia, with references to those formulations in the literature, as well as an MPI performance study.

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Neymeyr, Klaus. "A Geometric Convergence Theory for the Preconditioned Steepest Descent Iteration." SIAM Journal on Numerical Analysis 50, no. 6 (January 2012): 3188–207. http://dx.doi.org/10.1137/11084488x.

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Knyazev, Andrew V., and Klaus Neymeyr. "Gradient Flow Approach to Geometric Convergence Analysis of Preconditioned Eigensolvers." SIAM Journal on Matrix Analysis and Applications 31, no. 2 (January 2009): 621–28. http://dx.doi.org/10.1137/080727567.

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17

Neymeyr, Klaus. "A geometric theory for preconditioned inverse iteration applied to a subspace." Mathematics of Computation 71, no. 237 (September 17, 2001): 197–217. http://dx.doi.org/10.1090/s0025-5718-01-01357-6.

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18

Drmač, Zlatko, and Krešimir Veselić. "Approximate eigenvectors as preconditioner." Linear Algebra and its Applications 309, no. 1-3 (April 2000): 191–215. http://dx.doi.org/10.1016/s0024-3795(00)00046-x.

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Chen, Jian-Long, and Xiao-Qing Jin. "The generalized superoptimal preconditioner." Linear Algebra and its Applications 432, no. 1 (January 2010): 203–17. http://dx.doi.org/10.1016/j.laa.2009.06.045.

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Neymeyr, Klaus. "A geometric theory for preconditioned inverse iteration I: Extrema of the Rayleigh quotient." Linear Algebra and its Applications 322, no. 1-3 (January 2001): 61–85. http://dx.doi.org/10.1016/s0024-3795(00)00239-1.

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21

Neymeyr, Klaus. "A geometric theory for preconditioned inverse iteration IV: On the fastest convergence cases." Linear Algebra and its Applications 415, no. 1 (May 2006): 114–39. http://dx.doi.org/10.1016/j.laa.2005.06.022.

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22

Bender, M., and A. Raabe. "Preconditions to ground based GPS water vapour tomography." Annales Geophysicae 25, no. 8 (August 29, 2007): 1727–34. http://dx.doi.org/10.5194/angeo-25-1727-2007.

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Abstract. The GPS water vapour tomography is a new technique which provides spatially resolved water vapour distributions in the atmosphere under all weather conditions. This work investigates the information contained in a given set of GPS signals as a precondition to an optimal tomographic reconstruction. The spatial distribution of the geometric intersection points between different ray paths is used to estimate the information density. Different distributions of intersection points obtained from hypothetical GPS networks with varying densities of GPS stations are compared with respect to the horizontal and vertical resolution of a subsequent tomographic reconstruction. As a result some minimum requirements for continuously operating extensive GPS networks for meteorological applications are given.

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Cai, Ming-Chao, and Xiao-Qing Jin. "A note on T. Chan’s preconditioner." Linear Algebra and its Applications 376 (January 2004): 283–90. http://dx.doi.org/10.1016/j.laa.2003.07.005.

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Cai, Ming-Chao, Xiao-Qing Jin, and Yi-Min Wei. "A generalization of T. Chan’s preconditioner." Linear Algebra and its Applications 407 (September 2005): 11–18. http://dx.doi.org/10.1016/j.laa.2005.04.014.

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25

Kamatani, K. "Ergodicity of Markov chain Monte Carlo with reversible proposal." Journal of Applied Probability 54, no. 2 (June 2017): 638–54. http://dx.doi.org/10.1017/jpr.2017.22.

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Abstract We describe the ergodic properties of some Metropolis–Hastings algorithms for heavy-tailed target distributions. The results of these algorithms are usually analyzed under a subgeometric ergodic framework, but we prove that the mixed preconditioned Crank–Nicolson (MpCN) algorithm has geometric ergodicity even for heavy-tailed target distributions. This useful property comes from the fact that, under a suitable transformation, the MpCN algorithm becomes a random-walk Metropolis algorithm.

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Laird, Avery, Bangtian Liu, Nikolaj Bjørner, and Maryam Mehri Dehnavi. "SpEQ: Translation of Sparse Codes using Equivalences." Proceedings of the ACM on Programming Languages 8, PLDI (June 20, 2024): 1680–703. http://dx.doi.org/10.1145/3656445.

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We present SpEQ, a quick and correct strategy for detecting semantics in sparse codes and enabling automatic translation to high-performance library calls or domain-specific languages (DSLs). When sparse linear algebra codes contain implicit preconditions about how data is stored that hamper direct translation, SpEQ identifies the high-level computation along with storage details and related preconditions. A run-time check guards the translation and ensures that required preconditions are met. We implement SpEQ using the LLVM framework, the Z3 solver, and egglog library and correctly translate sparse linear algebra codes into two high-performance libraries, NVIDIA cuSPARSE and Intel MKL, and OpenMP (OMP). We evaluate SpEQ on ten diverse benchmarks against two state-of-the-art translation tools. SpEQ achieves geometric mean speedups of 3.25×, 5.09×, and 8.04× on OpenMP, MKL, and cuSPARSE backends, respectively. SpEQ is the only tool that can guarantee the correct translation of sparse computations.

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Boman, E., and I. Koltracht. "Computing preconditioners via subspace projection." Linear Algebra and its Applications 302-303 (December 1999): 347–53. http://dx.doi.org/10.1016/s0024-3795(99)00143-3.

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Pang, Hong-Kui, Ying-Ying Zhang, Seak-Weng Vong, and Xiao-Qing Jin. "Circulant preconditioners for pricing options." Linear Algebra and its Applications 434, no. 11 (June 2011): 2325–42. http://dx.doi.org/10.1016/j.laa.2010.03.034.

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Cheng, Che-Man, and Xiao-Qing Jin. "Some stability properties of T. Chan’s preconditioner." Linear Algebra and its Applications 395 (January 2005): 361–65. http://dx.doi.org/10.1016/j.laa.2004.09.010.

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Chen, C., and O. Bíró. "Three-dimensional time-harmonic eddy current problems solved by the geometric multigrid preconditioned conjugate gradient method." IET Science, Measurement & Technology 6, no. 5 (2012): 319. http://dx.doi.org/10.1049/iet-smt.2011.0117.

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Cho, Durkbin. "Isogeometric Schwarz Preconditioners with Generalized B-Splines for the Biharmonic Problem." Axioms 12, no. 5 (May 4, 2023): 452. http://dx.doi.org/10.3390/axioms12050452.

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We construct an overlapping additive Schwarz preconditioner for the biharmonic Dirichlet problems discretized by isogeometric analysis based on generalized B-splines (GB-splines) and analyze its optimal convergence rate bound that is cubic in the ratio between subdomains and overlap sizes. Our analysis is validated through a set of numerical experiments that illustrate good behavior of the proposed preconditioner with respect to the model parameters.

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ZOU, QI, XIANG-LIN HUANG, and SI-WEI LUO. "MULTIRESOLUTION IMAGE PERCEPTUAL GROUPING USING TOPOLOGICAL STRUCTURE EMBEDDED IN MANIFOLD." International Journal of Wavelets, Multiresolution and Information Processing 05, no. 01 (January 2007): 39–49. http://dx.doi.org/10.1142/s0219691307001641.

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Contour grouping is the precondition for high-level visual tasks such as region-of-interest detection and object recognition. Those methods adopting local geometric features and simple photometric attributes always lead to unreliable result or bad robustness. In this paper, we propose a grouping method using multiresolution analysis and one-dimensional manifold. By using multiresolution analysis, grouping seeds and initial clusters are detected as precondition for describing topological structure. Then one-dimensional manifold is applied to discovering intrinsic order of topological structure, which corresponds to coordinates of edges in a closed contour. These two improvements enhance our method's robustness relative to those using local features or linear combinations of them. Experiments on different classes of real images show competence of our method.

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Chan, Raymond H., and Kwok-Po Ng. "Toeplitz preconditioners for Hermitian Toeplitz systems." Linear Algebra and its Applications 190 (September 1993): 181–208. http://dx.doi.org/10.1016/0024-3795(93)90226-e.

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Taher, Mardeen Sh, and Salah G. Shareef. "A Combined Conjugate Gradient Quasi-Newton Method with Modification BFGS Formula." International Journal of Analysis and Applications 21 (April 3, 2023): 31. http://dx.doi.org/10.28924/2291-8639-21-2023-31.

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The conjugate gradient and Quasi-Newton methods have advantages and drawbacks, as although quasi-Newton algorithm has more rapid convergence than conjugate gradient, they require more storage compared to conjugate gradient algorithms. In 1976, Buckley designed a method that combines the CG method with QN updates, which is better than that observed for conjugate gradient algorithms but not as good as the quasi-Newton approach. This type of method is called the preconditioned conjugate gradient (PCG) method. In this paper, we introduce two new preconditioned conjugate gradient (PCG) methods that combine conjugate gradient with a new update of quasi-Newton methods. The new quasi-Newton method satisfied the positive define, and the direction of the new preconditioned conjugate gradient is descent direction. In numerical results, it is showing the new preconditioned conjugate gradient method is more effective on several high-dimension test problems than standard preconditioning.

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Hyka (Xhako), Dafina, and Rudina Osmanaj (Zeqirllari). "Fast algorithms for chiral fermions in 2 dimensions." EPJ Web of Conferences 175 (2018): 14005. http://dx.doi.org/10.1051/epjconf/201817514005.

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In lattice QCD simulations the formulation of the theory in lattice should be chiral in order that symmetry breaking happens dynamically from interactions. In order to guarantee this symmetry on the lattice one uses overlap and domain wall fermions. On the other hand high computational cost of lattice QCD simulations with overlap or domain wall fermions remains a major obstacle of research in the field of elementary particles. We have developed the preconditioned GMRESR algorithm as fast inverting algorithm for chiral fermions in U(1) lattice gauge theory. In this algorithm we used the geometric multigrid idea along the extra dimension.The main result of this work is that the preconditioned GMRESR is capable to accelerate the convergence 2 to 12 times faster than the other optimal algorithms (SHUMR) for different coupling constant and lattice 32x32. Also, in this paper we tested it for larger lattice size 64x64. From the results of simulations we can see that our algorithm is faster than SHUMR. This is a very promising result that this algorithm can be adapted also in 4 dimension.

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Lin, Fu-Rong, and De-Cai Zhang. "BTTB preconditioners for BTTB least squares problems." Linear Algebra and its Applications 434, no. 11 (June 2011): 2285–95. http://dx.doi.org/10.1016/j.laa.2009.10.035.

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Neymeyr, Klaus. "On preconditioned eigensolvers and Invert–Lanczos processes." Linear Algebra and its Applications 430, no. 4 (February 2009): 1039–56. http://dx.doi.org/10.1016/j.laa.2008.10.016.

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Jin, Xiao-Qing, and Yi-Min Wei. "A survey and some extensions of T. Chan’s preconditioner." Linear Algebra and its Applications 428, no. 2-3 (January 2008): 403–12. http://dx.doi.org/10.1016/j.laa.2007.01.021.

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Knyazev, Andrew V., and Klaus Neymeyr. "A geometric theory for preconditioned inverse iteration III: A short and sharp convergence estimate for generalized eigenvalue problems." Linear Algebra and its Applications 358, no. 1-3 (January 2003): 95–114. http://dx.doi.org/10.1016/s0024-3795(01)00461-x.

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Feng, Xiao Mei, Jun Qing Zhan, and Li Shun Li. "Study of the Loading Arms of the Side-Crane Used for Container." Advanced Materials Research 503-504 (April 2012): 896–99. http://dx.doi.org/10.4028/www.scientific.net/amr.503-504.896.

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As the important device, self-loading device plays a key role in the side-crane used for container. And as the key component of the self-loading device, loading arm’s strength analysis is very important. FEA is introduced into the arm’s strength and rigidity analysis, and under the precondition of satisfying allowable stress, minimum weight of the arm is taken as the design target. Ulteriorly, by obtaining optimal geometric parameters, the loading arm’s optimization design can be realized and its manufacture cost can be reduced.

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MANDAL, J. C., and J. BALLMANN. "UNSTEADY FLOW COMPUTATIONS OVER MOVING BODY USING DYNAMIC MESHES." International Journal of Computational Methods 01, no. 03 (December 2004): 507–18. http://dx.doi.org/10.1142/s0219876204000253.

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An efficient implicit unstructured grid algorithm for solving unsteady inviscid compressible flows over moving body employing an Arbitrary Lagrangian Eulerian formulation is presented. In the present formulation, the time discretization is performed using a second-order accurate 3-point time integration scheme and the upwind-biased space discretization using second-order accurate finite volume formulation with Venkatakrishnan limiter. The face-velocities of the control volumes are computed using Geometric Conservation Laws. The nonlinear system arising from the implicit formulation is solved using an ILU preconditioned Newton–Krylov iteration at every time step. The computed results for two test cases involving harmonically oscillating NACA0012 airfoil are presented in order to demonstrate the efficacy of the present solver.

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Guan, Jianbo, Yu Li, and Guohua Liu. "Preconditioned Conjugate Gradient Algorithm-Based 2D Waveform Inversion for Shallow-Surface Site Characterization." Shock and Vibration 2021 (December 21, 2021): 1–16. http://dx.doi.org/10.1155/2021/3164358.

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The full-waveform inversion (FWI) of a Love wave has become a powerful tool for shallow-surface site characterization. In classic conjugate gradient algorithm- (CG) based FWI, the energy distribution of the gradient calculated with the adjoint state method does not scale with increasing depth, resulting in diminished illumination capability and insufficient model updating. The inverse Hessian matrix (HM) can be used as a preprocessing operator to balance, filter, and regularize the gradient to strengthen the model illumination capabilities at depth and improve the inversion accuracy. However, the explicit calculation of the HM is unacceptable due to its large dimension in FWI. In this paper, we present a new method for obtaining the inverse HM of the Love wave FWI by referring to HM determination in inverse scattering theory to achieve a preconditioned gradient, and the preconditioned CG (PCG) is developed. This method uses the Love wave wavefield stress components to construct a pseudo-HM to avoid the huge calculation cost. It can effectively alleviate the influence of nonuniform coverage from source to receiver, including double scattering, transmission, and geometric diffusion, thus improving the inversion result. The superiority of the proposed algorithm is verified with two synthetic tests. The inversion results indicate that the PCG significantly improves the imaging accuracy of deep media, accelerates the convergence rate, and has strong antinoise ability, which can be attributed to the use of the pseudo-HM.

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Ng, Michael K., Raymond H. Chan, Tony F. Chan, and Andy M. Yip. "Cosine transform preconditioners for high resolution image reconstruction." Linear Algebra and its Applications 316, no. 1-3 (September 2000): 89–104. http://dx.doi.org/10.1016/s0024-3795(99)00274-8.

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Ovtchinnikov, E. "Cluster robustness of preconditioned gradient subspace iteration eigensolvers." Linear Algebra and its Applications 415, no. 1 (May 2006): 140–66. http://dx.doi.org/10.1016/j.laa.2005.06.039.

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Rafiei, A. "Left-looking version of AINV preconditioner with complete pivoting strategy." Linear Algebra and its Applications 445 (March 2014): 103–26. http://dx.doi.org/10.1016/j.laa.2013.11.046.

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Гурьева, Я. Л., В. П. Ильин, and Д. В. Перевозкин. "Algebraic-geometric and information structures of domain decomposition methods." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 2 (May 26, 2016): 132–46. http://dx.doi.org/10.26089/nummet.v17r213.

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Рассматриваются алгебраические, геометрические и информационные аспекты параллельных методов декомпозиции для решения больших систем линейных уравнений с разреженными матрицами, возникающими при аппроксимации многомерных краевых задач на неструктурированных сетках. Алгоритмы базируются на разбиении сеточной расчетной области на подобласти с параметризованной величиной пересечений и различными интерфейсными условиями на смежных границах. Рассматриваются вопросы, возникающие при алгебраической декомпозиции исходной матрицы. Применяются различные двухуровневые итерационные процессы, включающие в себя предобусловленные крыловские методы с использованием грубосеточной коррекции, а также синхронное решение вспомогательных систем в подобластях с помощью прямых или итерационных алгоритмов. Распараллеливание алгоритмов реализуется средствами гибридного программирования с формированием MPI-процессов для каждой подобласти и использованием в них многопотоковых вычислений над общей памятью. Информационные коммуникации между соседними подобластями осуществляются на каждой внешней итерации путем предварительной организации буферов обмена и применения неблокирующих операций с возможностями проведения арифметических действий на фоне передачи данных. Algebraic, geometric, and informational aspects of parallel decomposition methods are considered to solve large systems of linear equations with sparse matrices arising after approximation of multidimensional boundary value problems on unstructured grids. Algorithms are based on partitioning a grid computational domain into its subdomains with a parameterized value of overlapping and various interface conditions on the adjacent boundaries. Some questions arising in algebraic decomposition of the original matrix are discussed. Various two-level iterative processes are used. They include both preconditioned Krylov methods with a coarse grid correction and the parallel solution of auxiliary subsystems in subdomains by direct or iterative algorithms. Parallelization of algorithms is implemented by means of hybrid programming with separate MPI processes for each subdomain and by multithreaded computations over shared memory in each of the subdomains. Communications between adjacent subdomains are performed on each external iteration via the preliminary creation of some exchange buffers and using non-blocking operations, which makes it possible to combine both the arithmetic operations and data transfer.

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Jiao, Niangang, Feng Wang, Hongjian You, Xiaolan Qiu, and Mudan Yang. "Geo-Positioning Accuracy Improvement of Multi-Mode GF-3 Satellite SAR Imagery Based on Error Sources Analysis." Sensors 18, no. 7 (July 18, 2018): 2333. http://dx.doi.org/10.3390/s18072333.

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The GaoFen-3 (GF-3) satellite is the only synthetic aperture radar (SAR) satellite in the High-Resolution Earth Observation System Project, which is the first C-band full-polarization SAR satellite in China. In this paper, we proposed some error sources-based weight strategies to improve the geometric performance of multi-mode GF-3 satellite SAR images without using ground control points (GCPs). To get enough tie points, a robust SAR image registration method and the SAR-features from accelerated segment test (SAR-FAST) method is used to achieve the image registration and tie point extraction. Then, the original position of these tie points in object-space is calculated with the help of the space intersection method. With the dataset clustered by the density-based spatial clustering of applications with noise (DBSCAN) algorithm, we undertake the block adjustment with a bias-compensated rational function model (RFM) aided to improve the geometric performance of these multi-mode GF-3 satellite SAR images. Different weight strategies are proposed to develop the normal equation matrix according to the error sources analysis of GF-3 satellite SAR images, and the preconditioned conjugate gradient (PCG) method is utilized to solve the normal equation. The experimental results indicate that our proposed method can improve the geometric positioning accuracy of GF-3 satellite SAR images within 2 pixels.

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Wu, Yanze, and Joseph E. Subotnik. "A quantum-classical Liouville formalism in a preconditioned basis and its connection with phase-space surface hopping." Journal of Chemical Physics 158, no. 2 (January 14, 2023): 024115. http://dx.doi.org/10.1063/5.0124835.

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We revisit a recent proposal to model nonadiabatic problems with a complex-valued Hamiltonian through a phase-space surface hopping (PSSH) algorithm employing a pseudo-diabatic basis. Here, we show that such a pseudo-diabatic PSSH (PD-PSSH) ansatz is consistent with a quantum-classical Liouville equation (QCLE) that can be derived following a preconditioning process, and we demonstrate that a proper PD-PSSH algorithm is able to capture some geometric magnetic effects (whereas the standard fewest switches surface hopping approach cannot capture such effects). We also find that a preconditioned QCLE can outperform the standard QCLE in certain cases, highlighting the fact that there is no unique QCLE. Finally, we also point out that one can construct a mean-field Ehrenfest algorithm using a phase-space representation similar to what is done for PSSH. These findings would appear extremely helpful as far as understanding and simulating nonadiabatic dynamics with complex-valued Hamiltonians and/or spin degeneracy.

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Jbilou, K. "ADI preconditioned Krylov methods for large Lyapunov matrix equations." Linear Algebra and its Applications 432, no. 10 (May 2010): 2473–85. http://dx.doi.org/10.1016/j.laa.2009.12.025.

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Vecharynski, Eugene, and Andrew Knyazev. "Preconditioned steepest descent-like methods for symmetric indefinite systems." Linear Algebra and its Applications 511 (December 2016): 274–95. http://dx.doi.org/10.1016/j.laa.2016.09.011.

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