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Journal articles on the topic 'Adaptive Multigrid'

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Author: Grafiati

Published: 6 September 2023

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1

Brezina, M., R. Falgout, S. MacLachlan, T. Manteuffel, S. McCormick, and J. Ruge. "Adaptive Algebraic Multigrid." SIAM Journal on Scientific Computing 27, no. 4 (January 2006): 1261–86. http://dx.doi.org/10.1137/040614402.

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2

Rüde, Ulrich. "Fully Adaptive Multigrid Methods." SIAM Journal on Numerical Analysis 30, no. 1 (February 1993): 230–48. http://dx.doi.org/10.1137/0730011.

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3

Bittencourt, Marco L., Craig C. Douglas, and Raúl A. Feijóo. "Adaptive non‐nested multigrid methods." Engineering Computations 19, no. 2 (March 2002): 158–76. http://dx.doi.org/10.1108/02644400210419030.

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4

Bartsch, Guido, and Christian Wulf. "Adaptive Multigrid for Helmholtz Problems." Journal of Computational Acoustics 11, no. 03 (September 2003): 341–50. http://dx.doi.org/10.1142/s0218396x03001997.

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Solving Helmholtz problems for low frequency sound fields by a truncated modal basis approach is very efficient. The most time-consuming process is the calculation of the undamped modes. Using traditional FE solvers, the user has to provide a mesh which has at least six nodes per wavelength in each spatial direction to achieve acceptable results. Because the mesh size increases with the 3rd power of the highest frequency of interest, this uniform dense mesh approach is a very expensive way of creating a modal space. However, the number of modes and the accuracy of the modal basis directly influences the solution quality. It is well known that the representation of sound fields by modal basis functions φi is optimal with respect to the L2 error norm. This means that having a modal basis Φ := {φi, i = 1⋯n}, the distance between true and approximated sound field takes its minimum in the mean square. So, it is necessary to have a FE basis which also minimizes the discretization error when computing the modal basis. One can reach this goal by applying adaptive mesh refinements. Additionally, this yields the opportunity of using fast multigrid methods to solve discrete eigenvalue problems. In context of this presentation we will discuss the results of our adaptive multigrid algorithms.

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5

Belouafi, M. E., M. Beggas, and M. Haiour. "MAXIMUM NORM CONVERGENCE OF NEWTON-MULTIGRID METHODS FOR ELLIPTIC QUASI-VARIATIONAL INEQUALITIES WITH NONLINEAR SOURCE TERMS." Advances in Mathematics: Scientific Journal 11, no. 10 (October 29, 2022): 969–83. http://dx.doi.org/10.37418/amsj.11.10.12.

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In this paper, Newton-multigrid scheme on adaptive finite element discretisation is employed for solving elliptic quasi-variational inequalities with nonlinear source terms. We use Newton’s method as the outer iteration for the standard linearization, and using standard multigrid as the inner iteration for the solution of the Jacobian system at each step. The uniform convergence of Newton-multigrid methods is shown in the sense that the multigrid methods have a contraction number with respect to the maximum norm.

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6

Wada, Yoshitaka, Takuji Hayashi, Masanori Kikuchi, and Fei Xu. "Improvement of Unstructured Quadrilateral Mesh Quality for Multigrid Analysis." Advanced Materials Research 33-37 (March 2008): 833–38. http://dx.doi.org/10.4028/www.scientific.net/amr.33-37.833.

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Due to more complex and severe design restrictions, more effective and faster finite element analyses are demanded. There are several ways to compute FE analysis efficiently: parallel computing, fast iterative or direct solvers, adaptive analysis and so on. One of the most effective analysis ways is the combination of adaptive analysis and multigrid iterative solver, because an adaptive analysis requires several meshes with difference resolutions and multigrid solver utilizes such meshes to accelerate its computation. However, convergence of multigrid solver is largely affected by initial shape of each element. An effective mesh improvement method is proposed here. It is the combination of mesh coarsening and refinement. A good mesh can be obtained by the method to be applied to an initial mesh, and better convergence is achieved by the improved initial mesh.

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7

Brezina, M., R. Falgout, S. MacLachlan, T. Manteuffel, S. McCormick, and J. Ruge. "Adaptive Smoothed Aggregation ($\alpha$SA) Multigrid." SIAM Review 47, no. 2 (January 2005): 317–46. http://dx.doi.org/10.1137/050626272.

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8

Nägel, Arne, Robert D. Falgout, and Gabriel Wittum. "Filtering algebraic multigrid and adaptive strategies." Computing and Visualization in Science 11, no. 3 (March 6, 2007): 159–67. http://dx.doi.org/10.1007/s00791-007-0066-9.

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9

Bastian, Peter. "Load Balancing for Adaptive Multigrid Methods." SIAM Journal on Scientific Computing 19, no. 4 (July 1998): 1303–21. http://dx.doi.org/10.1137/s1064827596297562.

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10

Zheng, Y., and L. He. "Multigrid upwind Euler/Navier-Stokes computation on adaptive unstructured meshes." Aeronautical Journal 105, no. 1046 (April 2001): 173–84. http://dx.doi.org/10.1017/s0001924000025410.

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Abstract An unstructured flow solver with adaptive mesh refinement and multigrid acceleration is developed to efficiently compute two-dimensional inviscid and viscous steady flows about complex configurations. High resolution is achieved by using the upwind scheme coupled with adaptive mesh refinement. An aspect-ratio adaptive multigrid method is developed and applied to effectively accelerate the solution convergence of the explicit time-marching in the near wall regions with high aspect mesh ratios. Numerical examples are presented for configurations and conditions ranging from transonic to low speed flows to demonstrate accuracy, speed, and robustness of the method.

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11

Liu, Tao. "An adaptive multigrid conjugate gradient method for the inversion of a nonlinear convection-diffusion equation." Journal of Inverse and Ill-posed Problems 26, no. 5 (October 1, 2018): 623–31. http://dx.doi.org/10.1515/jiip-2016-0062.

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AbstractThis paper considers the problem of estimating the permeability in a nonlinear convection-diffusion equation. To overcome the large calculation burden of conventional methods, we apply an adaptive multigrid conjugate gradient method to solve this inverse problem. This new method combines the multigrid multiscale idea with the conjugate gradient method, and adopts the necessary condition that the optimum solution should be the fixed point of the multigrid inversion method. Some numerical results verify that the proposed method both dramatically reduces the required computations and improves the inversion quality.

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12

Sun, J., and P. Monk. "An Adaptive Algebraic Multigrid Algorithm for Micromagnetism." IEEE Transactions on Magnetics 42, no. 6 (June 2006): 1643–47. http://dx.doi.org/10.1109/tmag.2006.872004.

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13

Peherstorfer, Benjamin, Stefan Zimmer, Christoph Zenger, and Hans-Joachim Bungartz. "A Multigrid Method for Adaptive Sparse Grids." SIAM Journal on Scientific Computing 37, no. 5 (January 2015): S51—S70. http://dx.doi.org/10.1137/140974985.

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14

Feodoritova, O. B., and V. T. Zhukov. "An adaptive multigrid on block-structured grids." Journal of Physics: Conference Series 1640 (October 2020): 012020. http://dx.doi.org/10.1088/1742-6596/1640/1/012020.

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15

Becker, R., C. Johnson, and R. Rannacher. "Adaptive error control for multigrid finite element." Computing 55, no. 4 (December 1995): 271–88. http://dx.doi.org/10.1007/bf02238483.

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16

van der Maarel, H. T. M. "Adaptive multigrid for the steady euler equations." Communications in Applied Numerical Methods 8, no. 10 (October 1992): 749–60. http://dx.doi.org/10.1002/cnm.1630081005.

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17

Ji, Hua, Fue-Sang Lien, and Eugene Yee. "Parallel Adaptive Mesh Refinement Combined with Additive Multigrid for the Efficient Solution of the Poisson Equation." ISRN Applied Mathematics 2012 (March 12, 2012): 1–24. http://dx.doi.org/10.5402/2012/246491.

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Three different speed-up methods (viz., additive multigrid method, adaptive mesh refinement (AMR), and parallelization) have been combined in order to provide a highly efficient parallel solver for the Poisson equation. Rather than using an ordinary tree data structure to organize the information on the adaptive Cartesian mesh, a modified form of the fully threaded tree (FTT) data structure is used. The Hilbert space-filling curve (SFC) approach has been adopted for dynamic grid partitioning (resulting in a partitioning that is near optimal with respect to load balancing on a parallel computational platform). Finally, an additive multigrid method (BPX preconditioner), which itself is parallelizable to a certain extent, has been used to solve the linear equation system arising from the discretization. Our numerical experiments show that the proposed parallel AMR algorithm based on the FTT data structure, Hilbert SFC for grid partitioning, and additive multigrid method is highly efficient.

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18

Shao, Han, Libo Huang, and Dominik L. Michels. "A fast unsmoothed aggregation algebraic multigrid framework for the large-scale simulation of incompressible flow." ACM Transactions on Graphics 41, no. 4 (July 2022): 1–18. http://dx.doi.org/10.1145/3528223.3530109.

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Multigrid methods are quite efficient for solving the pressure Poisson equation in simulations of incompressible flow. However, for viscous liquids, geometric multigrid turned out to be less efficient for solving the variational viscosity equation. In this contribution, we present an Unsmoothed Aggregation Algebraic MultiGrid (UAAMG) method with a multi-color Gauss-Seidel smoother, which consistently solves the variational viscosity equation in a few iterations for various material parameters. Moreover, we augment the OpenVDB data structure with Intel SIMD intrinsic functions to perform sparse matrix-vector multiplications efficiently on all multigrid levels. Our framework is 2.0 to 14.6 times faster compared to the state-of-the-art adaptive octree solver in commercial software for the large-scale simulation of both non-viscous and viscous flow. The code is available at http://computationalsciences.org/publications/shao-2022-multigrid.html.

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19

Bacchio, Simone, Constantia Alexandrou, and Jacob Finkerath. "Multigrid accelerated simulations for Twisted Mass fermions." EPJ Web of Conferences 175 (2018): 02002. http://dx.doi.org/10.1051/epjconf/201817502002.

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Simulations at physical quark masses are affected by the critical slowing down of the solvers. Multigrid preconditioning has proved to deal effectively with this problem. Multigrid accelerated simulations at the physical value of the pion mass are being performed to generate Nf = 2 and Nf = 2 + 1 + 1 gauge ensembles using twisted mass fermions. The adaptive aggregation-based domain decomposition multigrid solver, referred to as DD-αAMG method, is employed for these simulations. Our simulation strategy consists of an hybrid approach of different solvers, involving the Conjugate Gradient (CG), multi-mass-shift CG and DD-αAMG solvers. We present an analysis of the multigrid performance during the simulations discussing the stability of the method. This significant speeds up the Hybrid Monte Carlo simulation by more than a factor 4 at physical pion mass compared to the usage of the CG solver.

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20

Miraçi, Ani, Jan Papež, and Martin Vohralík. "Contractive Local Adaptive Smoothing Based on Dörfler’s Marking in A-Posteriori-Steered p-Robust Multigrid Solvers." Computational Methods in Applied Mathematics 21, no. 2 (February 5, 2021): 445–68. http://dx.doi.org/10.1515/cmam-2020-0024.

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Abstract In this work, we study a local adaptive smoothing algorithm for a-posteriori-steered p-robust multigrid methods. The solver tackles a linear system which is generated by the discretization of a second-order elliptic diffusion problem using conforming finite elements of polynomial order p ≥ 1 {p\geq 1} . After one V-cycle (“full-smoothing” substep) of the solver of [A. Miraçi, J. Papež, and M. Vohralík, A-posteriori-steered p-robust multigrid with optimal step-sizes and adaptive number of smoothing steps, SIAM J. Sci. Comput. 2021, 10.1137/20M1349503], we dispose of a reliable, efficient, and localized estimation of the algebraic error. We use this existing result to develop our new adaptive algorithm: thanks to the information of the estimator and based on a bulk-chasing criterion, cf. [W. Dörfler, A convergent adaptive algorithm for Poisson’s equation, SIAM J. Numer. Anal. 33 1996, 3, 1106–1124], we mark patches of elements with increased estimated error on all levels. Then, we proceed by a modified and cheaper V-cycle (“adaptive-smoothing” substep), which only applies smoothing in the marked regions. The proposed adaptive multigrid solver picks autonomously and adaptively the optimal step-size per level as in our previous work but also the type of smoothing per level (weighted restricted additive or additive Schwarz) and concentrates smoothing to marked regions with high error. We prove that, under a numerical condition that we verify in the algorithm, each substep (full and adaptive) contracts the error p-robustly, which is confirmed by numerical experiments. Moreover, the proposed algorithm behaves numerically robustly with respect to the number of levels as well as to the diffusion coefficient jump for a uniformly-refined hierarchy of meshes.

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21

Fulton, Scott R. "An Adaptive Multigrid Barotropic Tropical Cyclone Track Model." Monthly Weather Review 129, no. 1 (January 2001): 138–51. http://dx.doi.org/10.1175/1520-0493(2001)129<0138:aambtc>2.0.co;2.

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22

Zaslavsky, Leonid Yu. "An Adaptive Algebraic Multigrid for Reactor Criticality Calculations." SIAM Journal on Scientific Computing 16, no. 4 (July 1995): 840–47. http://dx.doi.org/10.1137/0916049.

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23

Ekevid, Torbjörn, Per Kettil, and Nils-Erik Wiberg. "Adaptive multigrid for finite element computations in plasticity." Computers & Structures 82, no. 28 (November 2004): 2413–24. http://dx.doi.org/10.1016/j.compstruc.2004.04.013.

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24

Wu, Jinbiao, and Hui Zheng. "Uniform convergence of multigrid methods for adaptive meshes." Applied Numerical Mathematics 113 (March 2017): 109–23. http://dx.doi.org/10.1016/j.apnum.2016.11.005.

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25

Brown, J. David, and Lisa L. Lowe. "Multigrid elliptic equation solver with adaptive mesh refinement." Journal of Computational Physics 209, no. 2 (November 2005): 582–98. http://dx.doi.org/10.1016/j.jcp.2005.03.026.

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26

Watanabe, Kota, Seiji Fujino, and Hajime Igarashi. "Multigrid Method With Adaptive IDR-Based Jacobi Smoother." IEEE Transactions on Magnetics 47, no. 5 (May 2011): 1210–13. http://dx.doi.org/10.1109/tmag.2010.2092754.

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27

Hu, Xiaozhe, Junyuan Lin, and Ludmil T. Zikatanov. "An Adaptive Multigrid Method Based on Path Cover." SIAM Journal on Scientific Computing 41, no. 5 (January 2019): S220—S241. http://dx.doi.org/10.1137/18m1194493.

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28

Mehl, M., T. Weinzierl, and Chr Zenger. "A cache-oblivious self-adaptive full multigrid method." Numerical Linear Algebra with Applications 13, no. 2-3 (2006): 275–91. http://dx.doi.org/10.1002/nla.481.

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29

Molenaar, J. "Adaptive multigrid applied to a bipolar transistor problem." Applied Numerical Mathematics 17, no. 1 (March 1995): 61–83. http://dx.doi.org/10.1016/0168-9274(94)00061-k.

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30

ELIAS, S. R., G. D. STUBLEY, and G. D. RAITHBY. "AN ADAPTIVE AGGLOMERATION METHOD FOR ADDITIVE CORRECTION MULTIGRID." International Journal for Numerical Methods in Engineering 40, no. 5 (March 15, 1997): 887–903. http://dx.doi.org/10.1002/(sici)1097-0207(19970315)40:5<887::aid-nme93>3.0.co;2-i.

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31

LOPEZ, S., and R. CASCIARO. "ALGORITHMIC ASPECTS OF ADAPTIVE MULTIGRID FINITE ELEMENT ANALYSIS." International Journal for Numerical Methods in Engineering 40, no. 5 (March 15, 1997): 919–36. http://dx.doi.org/10.1002/(sici)1097-0207(19970315)40:5<919::aid-nme95>3.0.co;2-u.

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32

Kočvara, Michal. "An adaptive multigrid technique for three-dimensional elasticity." International Journal for Numerical Methods in Engineering 36, no. 10 (May 30, 1993): 1703–16. http://dx.doi.org/10.1002/nme.1620361006.

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33

Dom�nguez, M. del Carmen, and Luis Ferragut. "Adaptive multigrid method using duality in plane elasticity." International Journal for Numerical Methods in Engineering 50, no. 1 (2000): 95–118. http://dx.doi.org/10.1002/1097-0207(20010110)50:1<95::aid-nme23>3.0.co;2-d.

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34

Tucker, P. G. "Novel multigrid orientated solution adaptive time-step approaches." International Journal for Numerical Methods in Fluids 40, no. 3-4 (2002): 507–19. http://dx.doi.org/10.1002/fld.308.

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35

Xu, Liangkun, and Hai Bi. "A multigrid discretization scheme of discontinuous Galerkin method for the Steklov-Lamé eigenproblem." AIMS Mathematics 8, no. 6 (2023): 14207–31. http://dx.doi.org/10.3934/math.2023727.

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<abstract><p>In this paper, for the Steklov-Lamé eigenvalue problem, we propose a multigrid discretization scheme of discontinuous Galerkin method based on the shifted-inverse iteration. Based on the existing a priori error estimates, we give the error estimates for the proposed scheme and prove that the resulting approximations can achieve the optimal convergence order when the mesh sizes fit into some relationships. Finally, we combine the multigrid scheme and adaptive procedure to present some numerical examples which indicate that our scheme are locking-free and efficient for computing Steklov-Lamé eigenvalues.</p></abstract>

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36

Suisalu, I., and E. Saar. "An adaptive multigrid solver for high-resolution cosmological simulations." Monthly Notices of the Royal Astronomical Society 274, no. 1 (May 1995): 287–99. http://dx.doi.org/10.1093/mnras/274.1.287.

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37

Dargaville, Steven, Andrew Buchan, Richard Smedley-Stevenson, Paul Smith, and Chris Pain. "Adaptive angle and parallel multigrid for deterministic shielding problems." EPJ Web of Conferences 153 (2017): 06026. http://dx.doi.org/10.1051/epjconf/201715306026.

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38

Ferm, Lars, and Per Lotstedt. "Blockwise Adaptive Grids with Multigrid Acceleration for Compressible Flow." AIAA Journal 37, no. 1 (January 1999): 121–23. http://dx.doi.org/10.2514/2.674.

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39

Parthasarathy, Vijayan, and Y. Kallinderis. "New multigrid approach for three-dimensional unstructured, adaptive grids." AIAA Journal 32, no. 5 (May 1994): 956–63. http://dx.doi.org/10.2514/3.12080.

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40

Clevenger, Thomas C., Timo Heister, Guido Kanschat, and Martin Kronbichler. "A Flexible, Parallel, Adaptive Geometric Multigrid Method for FEM." ACM Transactions on Mathematical Software 47, no. 1 (January 6, 2021): 1–27. http://dx.doi.org/10.1145/3425193.

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41

Bernaschi, Massimo, Pasqua D’Ambra, and Dario Pasquini. "BootCMatchG: An adaptive Algebraic MultiGrid linear solver for GPUs." Software Impacts 6 (November 2020): 100041. http://dx.doi.org/10.1016/j.simpa.2020.100041.

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42

Hoppe, Hans-Christian, and Heinz Mühlenbein. "Parallel adaptive full-multigrid methods on message-based multiprocessors." Parallel Computing 3, no. 4 (October 1986): 269–87. http://dx.doi.org/10.1016/0167-8191(86)90011-6.

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43

Ramage, Alison, and Lina von Sydow. "A multigrid preconditioner for an adaptive Black-Scholes solver." BIT Numerical Mathematics 51, no. 1 (February 1, 2011): 217–33. http://dx.doi.org/10.1007/s10543-011-0316-6.

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44

Metzner, M., and G. Wittum. "Computing low Mach number flows by parallel adaptive multigrid." Computing and Visualization in Science 9, no. 4 (October 20, 2006): 259–69. http://dx.doi.org/10.1007/s00791-006-0025-x.

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45

Kornhuber, Ralf, and Rolf Krause. "Adaptive multigrid methods for Signorini's problem in linear elasticity." Computing and Visualization in Science 4, no. 1 (November 1, 2001): 9–20. http://dx.doi.org/10.1007/s007910100052.

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46

Ferm, Lars, and Per Lotstedt. "Blockwise adaptive grids with multigrid acceleration for compressible flow." AIAA Journal 37 (January 1999): 121–23. http://dx.doi.org/10.2514/3.14132.

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47

Li, Ming-Hsu, Hwai-Ping Cheng, and Gour-Tsyh Yeh. "Solving 3D Subsurface Flow and Transport with Adaptive Multigrid." Journal of Hydrologic Engineering 5, no. 1 (January 2000): 74–81. http://dx.doi.org/10.1061/(asce)1084-0699(2000)5:1(74).

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48

Dahmen, Wolfgang, Siegfried Müller, and Thomas Schlinkmann. "On an Adaptive Multigrid Solver for Convection-Dominated Problems." SIAM Journal on Scientific Computing 23, no. 3 (January 2001): 781–804. http://dx.doi.org/10.1137/s106482759935544x.

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49

Bastian, Peter, Stefan Lang, and Knut Eckstein. "Parallel adaptive multigrid methods in plane linear elasticity problems." Numerical Linear Algebra with Applications 4, no. 3 (May 1997): 153–76. http://dx.doi.org/10.1002/(sici)1099-1506(199705/06)4:3<153::aid-nla108>3.0.co;2-j.

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50

Mohamed, S. A. "An adaptive multigrid iterative approach for frictional contact problems." Communications in Numerical Methods in Engineering 22, no. 7 (December 13, 2005): 677–97. http://dx.doi.org/10.1002/cnm.839.

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