• 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

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  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.

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. AgulloL. 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. BakerJ. 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. DarnellR. 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. ElmanA. 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. GunzburgerN. 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. GunzburgerN. 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. GunzburgerN. 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. JiangS. 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. ParksE. De SturlerG. MackeyD. 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. AgulloL. 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. BakerJ. 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. DarnellR. 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. ElmanA. 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. GunzburgerN. 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. GunzburgerN. 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. GunzburgerN. 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. JiangS. 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. ParksE. De SturlerG. MackeyD. 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 2.  Time evolution of the rank of RHS vectors using the BGMRES-D solver in Problem 2
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 4.  Time evolutions of velocity magnitude for three problems in the ensemble simulation (left column) and individual simulations (right column) in Problem 3
Figure 5.  Time evolution of the rank of RHS vectors using BGMRES-D solver in Problem 3
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
($ 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
($ 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} $
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
($ 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
($ 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} $
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} $
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]

Håkon Hoel, Gaukhar Shaimerdenova, Raúl Tempone. Multilevel Ensemble Kalman Filtering based on a sample average of independent EnKF estimators. Foundations of Data Science, 2020  doi: 10.3934/fods.2020017

[2]

Zhiyan Ding, Qin Li, Jianfeng Lu. Ensemble Kalman Inversion for nonlinear problems: Weights, consistency, and variance bounds. Foundations of Data Science, 2020  doi: 10.3934/fods.2020018

[3]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[4]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[5]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[6]

Yongge Tian, Pengyang Xie. Simultaneous optimal predictions under two seemingly unrelated linear random-effects models. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020168

[7]

Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013

[8]

Shuyang Dai, Fengru Wang, Jerry Zhijian Yang, Cheng Yuan. A comparative study of atomistic-based stress evaluation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020322

[9]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074

[10]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[11]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[12]

Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078

[13]

Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462

[14]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[15]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120

[16]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[17]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[18]

Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336

[19]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[20]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

2019 Impact Factor: 1.27

Article outline

Figures and Tables

[Back to Top]