American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Corrigendum on “H. Li and M. Ma, global dynamics of a virus infection model with repulsive effect, Discrete and Continuous Dynamical Systems, Series B, 24(9) 4783-4797, 2019”
  • DCDS-B Home
  • This Issue
  • Next Article
    A game-theoretic framework for autonomous vehicles velocity control: Bridging microscopic differential games and macroscopic mean field games
December  2020, 25(12): 4905-4923. doi: 10.3934/dcdsb.2020132

Numerical investigation of ensemble methods with block iterative solvers for evolution problems

Lili Ju 1, , Wei Leng 2, , Zhu Wang 1,, and  Shuai Yuan 1,

1. 

Department of Mathematics, University of South Carolina, Columbia, SC 29208
2. 

State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: Zhu Wang

Received  November 2019 Revised  January 2020 Published  December 2020 Early access  April 2020

Fund Project: The first author's research was partially supported by U.S. National Science Foundation under grant number DMS-1818438 and U.S. Department of Energy under grant numbers DE-SC0016540 and DE-SC0020270. The second author's research was partially supported the National Key Research and Development Program of China under grant number 2016YFB0201304, National Natural Science Foundation of China under grant numbers 91430215, 91530323, 11501553 and 11771440, State Key Laboratory of Scientific and Engineering Computing (LSEC), and National Center for Mathematics and Interdisciplinary Sciences of Chinese Academy of Sciences (NCMIS). The third author's research was partially supported by U.S. Department of Energy under grant numbers DE-SC0016540 and DE-SC0020270, U.S. National Science Foundation under grant number DMS-1913073 and Office of the Vice President for Research at the University of South Carolina through an ASPIRE grant

The ensemble method has been developed for accelerating a sequence of numerical simulations of evolution problems. Its main idea is, by manipulating the time stepping and grouping discrete problems, to make all members in the same group share a common coefficient matrix. Thus, at each time step, instead of solving a sequence of linear systems each of which contains only one right-hand-side vector, the ensemble method simultaneously solves a single linear system with multiple right-hand-side vectors for each group. Such a system could be solved efficiently by using direct linear solvers when the problems are of small scale, as the same LU factorization would work for the entire group members. However, for large-scale problems, iterative linear solvers often have to be used and then this appealing advantage becomes not obvious. In this paper we systematically investigate numerical performance of the ensemble method with block iterative solvers for two typical evolution problems: the heat equation and the incompressible Navier-Stokes equations. In particular, the block conjugate gradient (CG) solver is considered for the former and the block generalized minimal residual (GMRES) solver for the latter. Our numerical results demonstrate the effectiveness and efficiency of the ensemble method when working together with these block iterative solvers.

Keywords: Ensemble method, ensemble-based time stepping, block CG, block GMRES, iterative linear solver.
Mathematics Subject Classification: Primary: 65M60.
Citation: Lili Ju, Wei Leng, Zhu Wang, Shuai Yuan. Numerical investigation of ensemble methods with block iterative solvers for evolution problems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4905-4923. doi: 10.3934/dcdsb.2020132
References:
[1]

E. Agullo, L. Giraud and Y.-F. Jing, Block GMRES method with inexact breakdowns and deflated restarting, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 1625-1651.  doi: 10.1137/140961912.  Google Scholar

[2]

A. H. Baker, J. M. Dennis and E. R. Jessup, On improving linear solver performance: A block variant of GMRES, SIAM Journal on Scientific Computing, 27 (2006), 1608-1626.  doi: 10.1137/040608088.  Google Scholar

[3]

H. Calandra, S. Gratton, R. Lago, X. Vasseur and L. M. Carvalho, A modified block flexible GMRES method with deflation at each iteration for the solution of non-hermitian linear systems with multiple right-hand sides, SIAM Journal on Scientific Computing, 35 (2013), S345–S367. doi: 10.1137/120883037.  Google Scholar

[4]

H. Calandra, S. Gratton, J. Langou, X. Pinel and X. Vasseur, Flexible variants of block restarted GMRES methods with application to geophysics, SIAM Journal on Scientific Computing, 34 (2012), A714–A736. doi: 10.1137/10082364X.  Google Scholar

[5]

T. F. Chan and M. K. Ng, Galerkin projection methods for solving multiple linear systems, SIAM Journal on Scientific Computing, 21 (1999), 836-850.  doi: 10.1137/S1064827598310227.  Google Scholar

[6]

A. T. Chronopoulos and A. B. Kucherov, Block-$s$-step Krylov iterative methods, Numerical Linear Algebra with Applications, 17 (2010), 3-15.  doi: 10.1002/nla.643.  Google Scholar

[7]

D. Darnell, R. B. Morgan and W. Wilcox, Deflated GMRES for systems with multiple shifts and multiple right-hand sides, Linear Algebra and its Applications, 429 (2008), 2415-2434.  doi: 10.1016/j.laa.2008.04.019.  Google Scholar

[8]

I. S. Duff, A. M. Erisman and J. K. Reid, Direct Methods for Sparse Matrices, Oxford University Press, 2017. doi: 10.1093/acprof:oso/9780198508380.001.0001.  Google Scholar

[9]

H. C. Elman, A. Ramage and D. J. Silvester, IFISS: A computational laboratory for investigating incompressible flow problems, SIAM Review, 56 (2014), 261-273.  doi: 10.1137/120891393.  Google Scholar

[10]

H. C. Elman, D. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Oxford University Press, 2005.  Google Scholar

[11]

J. A. Fiordilino, A second order ensemble timestepping algorithm for natural convection, SIAM Journal on Numerical Analysis, 56 (2018), 816-837.  doi: 10.1137/17M1135104.  Google Scholar

[12]

W. D. Gropp, D. K. Kaushik, D. E. Keyes and B. F. Smith, Toward realistic performance bounds for implicit CFD codes, Parallel Computational Fluid Dynamics, (2000), 241–248. doi: 10.1016/B978-044482851-4.50030-X.  Google Scholar

[13]

G.-D. Gu and Z.-H. Cao, A block GMRES method augmented with eigenvectors, Applied Mathematics and Computation, 121 (2001), 271-289.  doi: 10.1016/S0096-3003(99)00294-5.  Google Scholar

[14]

M. Gunzburger, N. Jiang and M. Schneier, An ensemble-proper orthogonal decomposition method for the nonstationary Navier-Stokes equations, SIAM Journal on Numerical Analysis, 55 (2017), 286-304.  doi: 10.1137/16M1056444.  Google Scholar

[15]

M. Gunzburger, N. Jiang and Z. Wang, A second-order time-stepping scheme for simulating ensembles of parameterized flow problems, Computational Methods in Applied Mathematics, 19 (2019), 681-701.  doi: 10.1515/cmam-2017-0051.  Google Scholar

[16]

M. Gunzburger, N. Jiang and Z. Wang, An efficient algorithm for simulating ensembles of parameterized flow problems, IMA Journal of Numerical Analysis, 39 (2019), 1180-1205.  doi: 10.1093/imanum/dry029.  Google Scholar

[17] M. D. Gunzburger, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989.   Google Scholar
[18]

M. H. Gutknecht, Block Krylov space methods for linear systems with multiple right-hand sides: An introduction, Modern Mathematical Models, Methods and Algorithms for Real World Systems, 420–447. Google Scholar

[19]

H. Ji and Y. H. Li, A breakdown-free block conjugate gradient method, BIT Numerical Mathematics, 57 (2017), 379-403.  doi: 10.1007/s10543-016-0631-z.  Google Scholar

[20]

N. Jiang, S. Kaya and W. Layton, Analysis of model variance for ensemble based turbulence modeling, Computational Methods in Applied Mathematics, 15 (2015), 173-188.  doi: 10.1515/cmam-2014-0029.  Google Scholar

[21]

N. Jiang and W. Layton, An algorithm for fast calculation of flow ensembles, International Journal for Uncertainty Quantification, 4 (2014), 273-301.  doi: 10.1615/Int.J.UncertaintyQuantification.2014007691.  Google Scholar

[22]

N. Jiang and W. Layton, Numerical analysis of two ensemble eddy viscosity numerical regularizations of fluid motion, Numerical Methods for Partial Differential Equations, 31 (2015), 630-651.  doi: 10.1002/num.21908.  Google Scholar

[23]

Y. Luo and Z. Wang, An ensemble algorithm for numerical solutions to deterministic and random parabolic PDEs, SIAM Journal on Numerical Analysis, 56 (2018), 859-876.  doi: 10.1137/17M1131489.  Google Scholar

[24]

Y. Luo and Z. Wang, A multilevel Monte Carlo ensemble scheme for solving random parabolic PDEs, SIAM Journal on Scientific Computing, 41 (2019), A622–A642. doi: 10.1137/18M1174635.  Google Scholar

[25]

J. McCarthy, Block-conjugate-gradient method, Physical Review D, 40 (1989), 2149. doi: 10.1103/PhysRevD.40.2149.  Google Scholar

[26]

M. Mohebujjaman and L. G. Rebholz, An efficient algorithm for computation of MHD flow ensembles, Computational Methods in Applied Mathematics, 17 (2017), 121-137.  doi: 10.1515/cmam-2016-0033.  Google Scholar

[27]

R. B. Morgan, Restarted block-GMRES with deflation of eigenvalues, Applied Numerical Mathematics, 54 (2005), 222-236.  doi: 10.1016/j.apnum.2004.09.028.  Google Scholar

[28]

D. P. O'Leary, The block conjugate gradient algorithm and related methods, Linear Algebra and its Applications, 29 (1980), 293-322.  doi: 10.1016/0024-3795(80)90247-5.  Google Scholar

[29]

D. P. O'Leary, Parallel implementation of the block conjugate gradient algorithm, Parallel Computing, 5 (1987), 127-139.  doi: 10.1016/0167-8191(87)90013-5.  Google Scholar

[30]

M. L. Parks, E. De Sturler, G. Mackey, D. D. Johnson and S. Maiti, Recycling Krylov subspaces for sequences of linear systems, SIAM Journal on Scientific Computing, 28 (2006), 1651-1674.  doi: 10.1137/040607277.  Google Scholar

[31]

M. L. Parks, K. M. Soodhalter and D. B. Szyld, A block recycled GMRES method with investigations into aspects of solver performance, preprint, arXiv: 1604.01713. Google Scholar

[32]

V. Puzyrev and J. M. Cela, A review of block Krylov subspace methods for multisource electromagnetic modelling, Geophysical Journal International, 202 (2015), 1241-1252.   Google Scholar

[33]

Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing, 7 (1986), 856-869.  doi: 10.1137/0907058.  Google Scholar

[34]

Y. Saad, Iterative Methods for Sparse Linear Systems, Second edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718003.  Google Scholar

[35]

V. Simoncini and E. Gallopoulos, Convergence properties of block GMRES and matrix polynomials, Linear Algebra and its Applications, 247 (1996), 97-119.  doi: 10.1016/0024-3795(95)00093-3.  Google Scholar

[36]

V. Simoncini and E. Gallopoulos, A hybrid block GMRES method for nonsymmetric systems with multiple right-hand sides, Journal of Computational and Applied Mathematics, 66 (1996), 457-469.  doi: 10.1016/0377-0427(95)00198-0.  Google Scholar

[37]

L. N. Trefethen and D. Bau III, Numerical Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. doi: 10.1137/1.9780898719574.  Google Scholar

show all references

References:
[1]

E. Agullo, L. Giraud and Y.-F. Jing, Block GMRES method with inexact breakdowns and deflated restarting, SIAM Journal on Matrix Analysis and Applications, 35 (2014), 1625-1651.  doi: 10.1137/140961912.  Google Scholar

[2]

A. H. Baker, J. M. Dennis and E. R. Jessup, On improving linear solver performance: A block variant of GMRES, SIAM Journal on Scientific Computing, 27 (2006), 1608-1626.  doi: 10.1137/040608088.  Google Scholar

[3]

H. Calandra, S. Gratton, R. Lago, X. Vasseur and L. M. Carvalho, A modified block flexible GMRES method with deflation at each iteration for the solution of non-hermitian linear systems with multiple right-hand sides, SIAM Journal on Scientific Computing, 35 (2013), S345–S367. doi: 10.1137/120883037.  Google Scholar

[4]

H. Calandra, S. Gratton, J. Langou, X. Pinel and X. Vasseur, Flexible variants of block restarted GMRES methods with application to geophysics, SIAM Journal on Scientific Computing, 34 (2012), A714–A736. doi: 10.1137/10082364X.  Google Scholar

[5]

T. F. Chan and M. K. Ng, Galerkin projection methods for solving multiple linear systems, SIAM Journal on Scientific Computing, 21 (1999), 836-850.  doi: 10.1137/S1064827598310227.  Google Scholar

[6]

A. T. Chronopoulos and A. B. Kucherov, Block-$s$-step Krylov iterative methods, Numerical Linear Algebra with Applications, 17 (2010), 3-15.  doi: 10.1002/nla.643.  Google Scholar

[7]

D. Darnell, R. B. Morgan and W. Wilcox, Deflated GMRES for systems with multiple shifts and multiple right-hand sides, Linear Algebra and its Applications, 429 (2008), 2415-2434.  doi: 10.1016/j.laa.2008.04.019.  Google Scholar

[8]

I. S. Duff, A. M. Erisman and J. K. Reid, Direct Methods for Sparse Matrices, Oxford University Press, 2017. doi: 10.1093/acprof:oso/9780198508380.001.0001.  Google Scholar

[9]

H. C. Elman, A. Ramage and D. J. Silvester, IFISS: A computational laboratory for investigating incompressible flow problems, SIAM Review, 56 (2014), 261-273.  doi: 10.1137/120891393.  Google Scholar

[10]

H. C. Elman, D. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics, Oxford University Press, 2005.  Google Scholar

[11]

J. A. Fiordilino, A second order ensemble timestepping algorithm for natural convection, SIAM Journal on Numerical Analysis, 56 (2018), 816-837.  doi: 10.1137/17M1135104.  Google Scholar

[12]

W. D. Gropp, D. K. Kaushik, D. E. Keyes and B. F. Smith, Toward realistic performance bounds for implicit CFD codes, Parallel Computational Fluid Dynamics, (2000), 241–248. doi: 10.1016/B978-044482851-4.50030-X.  Google Scholar

[13]

G.-D. Gu and Z.-H. Cao, A block GMRES method augmented with eigenvectors, Applied Mathematics and Computation, 121 (2001), 271-289.  doi: 10.1016/S0096-3003(99)00294-5.  Google Scholar

[14]

M. Gunzburger, N. Jiang and M. Schneier, An ensemble-proper orthogonal decomposition method for the nonstationary Navier-Stokes equations, SIAM Journal on Numerical Analysis, 55 (2017), 286-304.  doi: 10.1137/16M1056444.  Google Scholar

[15]

M. Gunzburger, N. Jiang and Z. Wang, A second-order time-stepping scheme for simulating ensembles of parameterized flow problems, Computational Methods in Applied Mathematics, 19 (2019), 681-701.  doi: 10.1515/cmam-2017-0051.  Google Scholar

[16]

M. Gunzburger, N. Jiang and Z. Wang, An efficient algorithm for simulating ensembles of parameterized flow problems, IMA Journal of Numerical Analysis, 39 (2019), 1180-1205.  doi: 10.1093/imanum/dry029.  Google Scholar

[17] M. D. Gunzburger, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1989.   Google Scholar
[18]

M. H. Gutknecht, Block Krylov space methods for linear systems with multiple right-hand sides: An introduction, Modern Mathematical Models, Methods and Algorithms for Real World Systems, 420–447. Google Scholar

[19]

H. Ji and Y. H. Li, A breakdown-free block conjugate gradient method, BIT Numerical Mathematics, 57 (2017), 379-403.  doi: 10.1007/s10543-016-0631-z.  Google Scholar

[20]

N. Jiang, S. Kaya and W. Layton, Analysis of model variance for ensemble based turbulence modeling, Computational Methods in Applied Mathematics, 15 (2015), 173-188.  doi: 10.1515/cmam-2014-0029.  Google Scholar

[21]

N. Jiang and W. Layton, An algorithm for fast calculation of flow ensembles, International Journal for Uncertainty Quantification, 4 (2014), 273-301.  doi: 10.1615/Int.J.UncertaintyQuantification.2014007691.  Google Scholar

[22]

N. Jiang and W. Layton, Numerical analysis of two ensemble eddy viscosity numerical regularizations of fluid motion, Numerical Methods for Partial Differential Equations, 31 (2015), 630-651.  doi: 10.1002/num.21908.  Google Scholar

[23]

Y. Luo and Z. Wang, An ensemble algorithm for numerical solutions to deterministic and random parabolic PDEs, SIAM Journal on Numerical Analysis, 56 (2018), 859-876.  doi: 10.1137/17M1131489.  Google Scholar

[24]

Y. Luo and Z. Wang, A multilevel Monte Carlo ensemble scheme for solving random parabolic PDEs, SIAM Journal on Scientific Computing, 41 (2019), A622–A642. doi: 10.1137/18M1174635.  Google Scholar

[25]

J. McCarthy, Block-conjugate-gradient method, Physical Review D, 40 (1989), 2149. doi: 10.1103/PhysRevD.40.2149.  Google Scholar

[26]

M. Mohebujjaman and L. G. Rebholz, An efficient algorithm for computation of MHD flow ensembles, Computational Methods in Applied Mathematics, 17 (2017), 121-137.  doi: 10.1515/cmam-2016-0033.  Google Scholar

[27]

R. B. Morgan, Restarted block-GMRES with deflation of eigenvalues, Applied Numerical Mathematics, 54 (2005), 222-236.  doi: 10.1016/j.apnum.2004.09.028.  Google Scholar

[28]

D. P. O'Leary, The block conjugate gradient algorithm and related methods, Linear Algebra and its Applications, 29 (1980), 293-322.  doi: 10.1016/0024-3795(80)90247-5.  Google Scholar

[29]

D. P. O'Leary, Parallel implementation of the block conjugate gradient algorithm, Parallel Computing, 5 (1987), 127-139.  doi: 10.1016/0167-8191(87)90013-5.  Google Scholar

[30]

M. L. Parks, E. De Sturler, G. Mackey, D. D. Johnson and S. Maiti, Recycling Krylov subspaces for sequences of linear systems, SIAM Journal on Scientific Computing, 28 (2006), 1651-1674.  doi: 10.1137/040607277.  Google Scholar

[31]

M. L. Parks, K. M. Soodhalter and D. B. Szyld, A block recycled GMRES method with investigations into aspects of solver performance, preprint, arXiv: 1604.01713. Google Scholar

[32]

V. Puzyrev and J. M. Cela, A review of block Krylov subspace methods for multisource electromagnetic modelling, Geophysical Journal International, 202 (2015), 1241-1252.   Google Scholar

[33]

Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on Scientific and Statistical Computing, 7 (1986), 856-869.  doi: 10.1137/0907058.  Google Scholar

[34]

Y. Saad, Iterative Methods for Sparse Linear Systems, Second edition, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2003. doi: 10.1137/1.9780898718003.  Google Scholar

[35]

V. Simoncini and E. Gallopoulos, Convergence properties of block GMRES and matrix polynomials, Linear Algebra and its Applications, 247 (1996), 97-119.  doi: 10.1016/0024-3795(95)00093-3.  Google Scholar

[36]

V. Simoncini and E. Gallopoulos, A hybrid block GMRES method for nonsymmetric systems with multiple right-hand sides, Journal of Computational and Applied Mathematics, 66 (1996), 457-469.  doi: 10.1016/0377-0427(95)00198-0.  Google Scholar

[37]

L. N. Trefethen and D. Bau III, Numerical Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997. doi: 10.1137/1.9780898719574.  Google Scholar

Figure 1.  Time evolution of the dimension of the search space at different iterations in BFBCG
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Time evolution of the rank of RHS vectors using the BGMRES-D solver in Problem 2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Speed fields at $ t = 60 $ for three problems in the ensemble simulation (left column) and individual simulations (right column) in Problem 3
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Time evolutions of velocity magnitude for three problems in the ensemble simulation (left column) and individual simulations (right column) in Problem 3
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Time evolution of the rank of RHS vectors using BGMRES-D solver in Problem 3
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  The $ L_2 $ errors at final time: first-order ensemble, $ Q_1 $ elements in Problem 1
($ N_x $, $ N_y $, $ K $) $ \mathcal{E}_1^N $ Rate $ \mathcal{E}_{50}^N $ Rate $ \mathcal{E}_{100}^N $ Rate
(16, 32, 50) 5.8005$ \times 10^{-2} $ – 4.3544$ \times 10^{-2} $ – 4.8908$ \times 10^{-2} $ –
(32, 64,100) 2.9140$ \times 10^{-2} $ 0.99 2.1972$ \times 10^{-2} $ 0.99 2.4615$ \times 10^{-2} $ 0.99
(64,128,200) 1.4629$ \times 10^{-2} $ 0.99 1.1061$ \times 10^{-2} $ 0.99 1.2371$ \times 10^{-2} $ 0.99
(128,256,400) 7.3326$ \times 10^{-3} $ 1.00 5.5529$ \times 10^{-3} $ 1.00 6.2053$ \times 10^{-3} $ 1.00
Table Options

Download as excel

($ N_x $, $ N_y $, $ K $) $ \mathcal{E}_1^N $ Rate $ \mathcal{E}_{50}^N $ Rate $ \mathcal{E}_{100}^N $ Rate
(16, 32, 50) 5.8005$ \times 10^{-2} $ – 4.3544$ \times 10^{-2} $ – 4.8908$ \times 10^{-2} $ –
(32, 64,100) 2.9140$ \times 10^{-2} $ 0.99 2.1972$ \times 10^{-2} $ 0.99 2.4615$ \times 10^{-2} $ 0.99
(64,128,200) 1.4629$ \times 10^{-2} $ 0.99 1.1061$ \times 10^{-2} $ 0.99 1.2371$ \times 10^{-2} $ 0.99
(128,256,400) 7.3326$ \times 10^{-3} $ 1.00 5.5529$ \times 10^{-3} $ 1.00 6.2053$ \times 10^{-3} $ 1.00
Table 2.  The $ L_2 $ errors at final time: second-order ensemble, $ Q_2 $ elements in Problem 1
($ N_x $, $ N_y $, $ K $) $ \mathcal{E}_1^N $ Rate $ \mathcal{E}_{50}^N $ Rate $ \mathcal{E}_{100}^N $ Rate
(8, 16, 50) 3.1827$ \times 10^{-3} $ – 2.4799$ \times 10^{-3} $ – 2.7259$ \times 10^{-3} $ –
(16, 32,100) 7.6003$ \times 10^{-4} $ 2.07 5.8014$ \times 10^{-4} $ 2.10 6.4617$ \times 10^{-4} $ 2.08
(32, 64,200) 1.9288$ \times 10^{-4} $ 1.98 1.4695$ \times 10^{-4} $ 1.98 1.6366$ \times 10^{-4} $ 1.98
(64,128,400) 4.9629$ \times 10^{-5} $ 1.96 3.7682$ \times 10^{-5} $ 1.96 4.2046$ \times 10^{-5} $ 1.96
Table Options

Download as excel

($ N_x $, $ N_y $, $ K $) $ \mathcal{E}_1^N $ Rate $ \mathcal{E}_{50}^N $ Rate $ \mathcal{E}_{100}^N $ Rate
(8, 16, 50) 3.1827$ \times 10^{-3} $ – 2.4799$ \times 10^{-3} $ – 2.7259$ \times 10^{-3} $ –
(16, 32,100) 7.6003$ \times 10^{-4} $ 2.07 5.8014$ \times 10^{-4} $ 2.10 6.4617$ \times 10^{-4} $ 2.08
(32, 64,200) 1.9288$ \times 10^{-4} $ 1.98 1.4695$ \times 10^{-4} $ 1.98 1.6366$ \times 10^{-4} $ 1.98
(64,128,400) 4.9629$ \times 10^{-5} $ 1.96 3.7682$ \times 10^{-5} $ 1.96 4.2046$ \times 10^{-5} $ 1.96
Table 3.  CPU time comparison in Problem 1
Iteration & CPU time BFBCG 100 CG
Average iteration number per time step 4 4$ \times $ 100
Average execution time per step (seconds) 5.6362$ \times 10^{-1} $ (1.1482$ \times 10^{-2} $) $ \times $ 100
Total CPU time for integration (seconds) 5.658$ \times 10^{2} $ 1.464$ \times 10^{3} $
Table Options

Download as excel

Iteration & CPU time BFBCG 100 CG
Average iteration number per time step 4 4$ \times $ 100
Average execution time per step (seconds) 5.6362$ \times 10^{-1} $ (1.1482$ \times 10^{-2} $) $ \times $ 100
Total CPU time for integration (seconds) 5.658$ \times 10^{2} $ 1.464$ \times 10^{3} $
Table 4.  The $ L_2 $ errors at final time: first-order ensemble, $ Q_2/Q_1 $ elements in Problem 2
($ N_x $, $ N_y $, $ K $) $ \mathcal{E}_1^N $ Rate $ \mathcal{E}_{20}^N $ Rate $ \mathcal{E}_{40}^N $ Rate
(128,128, 5) 3.7899$ \times 10^{-3} $ – 3.4660$ \times 10^{-3} $ – 3.6324$ \times 10^{-3} $ –
(128,128, 10) 1.9331$ \times 10^{-3} $ 0.97 1.7626$ \times 10^{-3} $ 0.98 1.8478$ \times 10^{-3} $ 0.98
(128,128, 20) 9.7905$ \times 10^{-4} $ 0.98 8.9096$ \times 10^{-4} $ 0.98 9.3418$ \times 10^{-4} $ 0.98
(128,128, 40) 4.8947$ \times 10^{-4} $ 1.00 4.4499$ \times 10^{-4} $ 1.00 4.6666$ \times 10^{-4} $ 1.00
Table Options

Download as excel

($ N_x $, $ N_y $, $ K $) $ \mathcal{E}_1^N $ Rate $ \mathcal{E}_{20}^N $ Rate $ \mathcal{E}_{40}^N $ Rate
(128,128, 5) 3.7899$ \times 10^{-3} $ – 3.4660$ \times 10^{-3} $ – 3.6324$ \times 10^{-3} $ –
(128,128, 10) 1.9331$ \times 10^{-3} $ 0.97 1.7626$ \times 10^{-3} $ 0.98 1.8478$ \times 10^{-3} $ 0.98
(128,128, 20) 9.7905$ \times 10^{-4} $ 0.98 8.9096$ \times 10^{-4} $ 0.98 9.3418$ \times 10^{-4} $ 0.98
(128,128, 40) 4.8947$ \times 10^{-4} $ 1.00 4.4499$ \times 10^{-4} $ 1.00 4.6666$ \times 10^{-4} $ 1.00
Table 5.  The $ L_2 $ errors at final time: second-order ensemble, $ Q_2/Q_1 $ elements in Problem 2
($ N_x $, $ N_y $, $ K $) $ \mathcal{E}_1^N $ Rate $ \mathcal{E}_{20}^N $ Rate $ \mathcal{E}_{40}^N $ Rate
(256,256, 5) 1.0597$ \times 10^{-3} $ – 9.5139$ \times 10^{-4} $ – 9.9694$ \times 10^{-4} $ –
(256,256, 10) 2.5992$ \times 10^{-4} $ 2.03 2.3215$ \times 10^{-4} $ 2.03 2.4328$ \times 10^{-4} $ 2.03
(256,256, 20) 6.3997$ \times 10^{-5} $ 2.02 5.7010$ \times 10^{-5} $ 2.03 5.9748$ \times 10^{-5} $ 2.02
(256,256, 40) 1.5905$ \times 10^{-5} $ 2.00 1.4112$ \times 10^{-5} $ 2.01 1.4796$ \times 10^{-5} $ 2.01
Table Options

Download as excel

($ N_x $, $ N_y $, $ K $) $ \mathcal{E}_1^N $ Rate $ \mathcal{E}_{20}^N $ Rate $ \mathcal{E}_{40}^N $ Rate
(256,256, 5) 1.0597$ \times 10^{-3} $ – 9.5139$ \times 10^{-4} $ – 9.9694$ \times 10^{-4} $ –
(256,256, 10) 2.5992$ \times 10^{-4} $ 2.03 2.3215$ \times 10^{-4} $ 2.03 2.4328$ \times 10^{-4} $ 2.03
(256,256, 20) 6.3997$ \times 10^{-5} $ 2.02 5.7010$ \times 10^{-5} $ 2.03 5.9748$ \times 10^{-5} $ 2.02
(256,256, 40) 1.5905$ \times 10^{-5} $ 2.00 1.4112$ \times 10^{-5} $ 2.01 1.4796$ \times 10^{-5} $ 2.01
Table 6.  CPU time comparison in Problem 2
Iteration & CPU time BGMRES-D 40 GMRES
average iteration number per time step 5 5$ \times $ 40
average execution time per step (seconds) 19.658 12.032 $ \times $ 40
total CPU time for integration (seconds) 1.965$ \times 10^{3} $ 2.103$ \times 10^{4} $
Table Options

Download as excel

Iteration & CPU time BGMRES-D 40 GMRES
average iteration number per time step 5 5$ \times $ 40
average execution time per step (seconds) 19.658 12.032 $ \times $ 40
total CPU time for integration (seconds) 1.965$ \times 10^{3} $ 2.103$ \times 10^{4} $
Table 7.  CPU time comparison in Problem 3
Iteration & CPU time BGMRES-D 40 GMRES
Average iteration number per time step 7 8$ \times $ 40
Average execution time per step (seconds) 4.7894 1.7313 $ \times $ 40
Total CPU time for integration (seconds) 7.106$ \times 10^{4} $ 4.836$ \times 10^{5} $
Table Options

Download as excel

Iteration & CPU time BGMRES-D 40 GMRES
Average iteration number per time step 7 8$ \times $ 40
Average execution time per step (seconds) 4.7894 1.7313 $ \times $ 40
Total CPU time for integration (seconds) 7.106$ \times 10^{4} $ 4.836$ \times 10^{5} $
[1]

Fang Chen, Ning Gao, Yao- Lin Jiang. On product-type generalized block AOR method for augmented linear systems. Numerical Algebra, Control & Optimization, 2012, 2 (4) : 797-809. doi: 10.3934/naco.2012.2.797
[2]

Gillala Rekha, V Krishna Reddy, Amit Kumar Tyagi. A novel approach for solving skewed classification problem using cluster based ensemble method. Mathematical Foundations of Computing, 2020, 3 (1) : 1-9. doi: 10.3934/mfc.2020001
[3]

Neil K. Chada, Yuming Chen, Daniel Sanz-Alonso. Iterative ensemble Kalman methods: A unified perspective with some new variants. Foundations of Data Science, 2021, 3 (3) : 331-369. doi: 10.3934/fods.2021011
[4]

Zhongming Wu, Xingju Cai, Deren Han. Linearized block-wise alternating direction method of multipliers for multiple-block convex programming. Journal of Industrial & Management Optimization, 2018, 14 (3) : 833-855. doi: 10.3934/jimo.2017078
[5]

Wei Wan, Haiyang Huang, Jun Liu. Local block operators and TV regularization based image inpainting. Inverse Problems & Imaging, 2018, 12 (6) : 1389-1410. doi: 10.3934/ipi.2018058
[6]

Jianguo Huang, Sen Lin. A $ C^0P_2 $ time-stepping virtual element method for linear wave equations on polygonal meshes. Electronic Research Archive, 2020, 28 (2) : 911-933. doi: 10.3934/era.2020048
[7]

Foxiang Liu, Lingling Xu, Yuehong Sun, Deren Han. A proximal alternating direction method for multi-block coupled convex optimization. Journal of Industrial & Management Optimization, 2019, 15 (2) : 723-737. doi: 10.3934/jimo.2018067
[8]

Susanne Pumplün, Thomas Unger. Space-time block codes from nonassociative division algebras. Advances in Mathematics of Communications, 2011, 5 (3) : 449-471. doi: 10.3934/amc.2011.5.449
[9]

Junyoung Jang, Kihoon Jang, Hee-Dae Kwon, Jeehyun Lee. Feedback control of an HBV model based on ensemble kalman filter and differential evolution. Mathematical Biosciences & Engineering, 2018, 15 (3) : 667-691. doi: 10.3934/mbe.2018030
[10]

Håkon Hoel, Gaukhar Shaimerdenova, Raúl Tempone. Multilevel Ensemble Kalman Filtering based on a sample average of independent EnKF estimators. Foundations of Data Science, 2020, 2 (4) : 351-390. doi: 10.3934/fods.2020017
[11]

Nguyen Buong. Steepest-descent block-iterative methods for a finite family of quasi-nonexpansive mappings. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021133
[12]

Theresa Lange, Wilhelm Stannat. Mean field limit of Ensemble Square Root filters - discrete and continuous time. Foundations of Data Science, 2021, 3 (3) : 563-588. doi: 10.3934/fods.2021003
[13]

Chuandong Li, Fali Ma, Tingwen Huang. 2-D analysis based iterative learning control for linear discrete-time systems with time delay. Journal of Industrial & Management Optimization, 2011, 7 (1) : 175-181. doi: 10.3934/jimo.2011.7.175
[14]

Sho Nanao, Hiroyuki Masuyama, Shoji Kasahara, Yutaka Takahashi. Queueing analysis of data block synchronization mechanism in peer-to-peer based video streaming system. Journal of Industrial & Management Optimization, 2011, 7 (3) : 699-716. doi: 10.3934/jimo.2011.7.699
[15]

Xiaoxiao Yuan, Jing Liu, Xingxing Hao. A moving block sequence-based evolutionary algorithm for resource investment project scheduling problems. Big Data & Information Analytics, 2017, 2 (1) : 39-58. doi: 10.3934/bdia.2017007
[16]

Tian Ma, Shouhong Wang. Block structure and block stability of two-dimensional incompressible flows. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 169-184. doi: 10.3934/dcdsb.2006.6.169
[17]

Su-Hong Jiang, Min Li. A modified strictly contractive peaceman-rachford splitting method for multi-block separable convex programming. Journal of Industrial & Management Optimization, 2018, 14 (1) : 397-412. doi: 10.3934/jimo.2017052
[18]

Hassan Khodaiemehr, Dariush Kiani. High-rate space-time block codes from twisted Laurent series rings. Advances in Mathematics of Communications, 2015, 9 (3) : 255-275. doi: 10.3934/amc.2015.9.255
[19]

Susanne Pumplün, Andrew Steele. The nonassociative algebras used to build fast-decodable space-time block codes. Advances in Mathematics of Communications, 2015, 9 (4) : 449-469. doi: 10.3934/amc.2015.9.449
[20]

Susanne Pumplün. How to obtain division algebras used for fast-decodable space-time block codes. Advances in Mathematics of Communications, 2014, 8 (3) : 323-342. doi: 10.3934/amc.2014.8.323

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (280)
  • HTML views (310)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy